BIFURCATIONANDCOMPETITIVEEVOLUTIONOFNETWORK MORPHOLOGIESINTHESTRONGFUNCTIONALIZEDCAHN-HILLIARD EQUATION By NoaKraitzman ADISSERTATION Submittedto MichiganStateUniversity inpartialentoftherequirements forthedegreeof AppliedMathematics{DoctorofPhilosophy 2015 ABSTRACT BIFURCATIONANDCOMPETITIVEEVOLUTIONOFNETWORK MORPHOLOGIESINTHESTRONGFUNCTIONALIZED CAHN-HILLIARDEQUATION By NoaKraitzman TheFunctionalizedCahn-Hilliard(FCH)energyisahigher-orderfreeenergythathasbeenproposedto describephaseseparationinblendsofamphiphilicpolymersandsolvent.Itbalancesinterfacialsolvation energyofionicgroupsandvolumetriccounter-ionandpolymerchainself-interactionenergyagainstelastic energyoftheunderlyingpolymerbackbone.Itishopedthatitsgradientwsdescribetheformationof solventaccessiblenetworkstructures,suchasfoundinpolymerelectrolytemembranes,lipidmembranes,and amphiphlicdiblockcopolymers.TheFCHgradientwspossesslong-livednetworkmorphologiesofdistinct co-dimensionandwecharacterizetheirgeometricevolution,bifurcationandcompetitionthroughaformal asymptoticreduction.Thisreductionencompassesabroadclassofcoexistingnetworkmorphologieswith tinnerstructure,suchasbilayersandpores.Thestabilityofthetnetworkmorphologiesis characterizedbythemeanderingandpearlingmodesassociatedtothelinearizedsystem.Forthe H 1 gra- dientwoftheFCHenergy,usingfunctionalanalysisandasymptoticmethods,wederiveasharp-interface geometricmotionwhichcouplesthewofco-dimension1andco-dimension2networkmorphologies,in R 3 , throughthespatiallyconstantchemicalpotential.Inparticular,werigorouslycharacterizethe pearlingeigenvaluesforaclassofadmissibleco-dimension1andco-dimension2networks. ! . hnwknhKrdbytw'KyrdmdymtŸ , yb'l ! . yytwmwlxt'MyŸghlxkht'ylNtwnŸ , yl`bl ! . ylŸrŸw'h , ynbl Tomyfather,whoalwaysguidesmealongtheright path. Tomyhusband,whogivesmethestrengthtofollow mydreams. Tomyson,myjoy. iii ACKNOWLEDGMENTS Iwouldliketoexpressmygratitudeandappreciationtomyadvisor,Prof.KeithPromislow,forhisfunda- mentalroleinmydoctoralwork.Keithprovidedmewithguidance,assistance,andexpertisethatIneeded duringmyeyearsatMichiganState.Thisfeatwaspossibleonlybecauseoftheunconditionalsupport providedbyKeith.Apersonwithawonderfulsenseofhumorandpositivedisposition,Keithhasalways madehimselfavailabledespitehisbusyscheduleandIconsidermyselfluckytosuchanincredibleadvisor. Iwouldliketothankmycommitteemembers:Prof.PeterBates,Prof.DiLiu,Prof.Schenker,Prof. RussellSchwab,andProf.BaishengYan,fortheirtime,patienceandencouragementthroughouttheyears. IextendmygratitudetoallmembersintheDepartmentofMathematicsatMichiganStateUniversityfor givingmethiswonderfulopportunity. Lastnotleast,Iwouldliketothankmyhusband,Alon,forhisendlessloveandsupport. iv TABLEOFCONTENTS LISTOFTABLES ................................................ vii LISTOFFIGURES ............................................... viii KEYTOABBREVIATIONS ......................................... xi Chapter1Introduction ............................................ 1 1.1AmphiphilicMaterials ...........................................1 1.2TheFunctionalizedCahn-HilliardFreeEnergy ............................4 1.3OverviewofMainResults .........................................8 1.4Quasi-StationarysolutionsofthestrongFunctionalizedCahn-HilliardFreeEnergy ......10 1.4.1Constructionofco-dimension1Quasi-stationarysolutionsofFCH ............11 1.4.2Constructionofco-dimension2andco-dimension3Quasi-stationarysolutionsofthe FCH .................................................15 1.4.3MinimizationofthestrongFCHfreeenergyoverco-dimension1quasi-stationary 16 1.5NetworkBifurcationintheFCH .....................................19 1.6CompetitiveGeometricEvolutionofBilayersandPores ......................23 Chapter2CoordinateSystem,DetionsandNotation ...................... 27 2.1Co-dimensionOneMorphologyin R d , d C 2 ..............................27 2.2Co-dimensionTwoMorphologyin R 3 .................................37 Chapter3GeometricEvolutionofBilayersin R d ........................... 46 3.1InnerandOuterExpansions .......................................47 3.2MatchingConditions ............................................48 3.3Expansionofthechemicalpotential ..................................49 3.3.1OuterExpansionoftheChemicalPotential ..........................49 3.3.2InnerExpansionoftheChemicalPotential ..........................50 3.4Timescale ˝ t :aGradientFlow ...................................51 3.4.1Outerexpansion ..........................................51 3.4.2Innerexpansion ..........................................51 3.4.3JumpConditionsontheOuterSolution:GradientFlow .................52 3.5Timescale ˝ "t :MeanCurvatureDrivenFlow ...........................55 3.5.1OuterExpansion ..........................................55 3.5.2InnerExpansion ..........................................56 3.5.2.1SolvingEquation( 3.69 ) ................................57 3.5.2.2JumpConditions ....................................57 3.5.2.3TheNormalVelocityat ˝ "t ............................58 3.5.3Sharpinterfacelimit:TrivialMullins-SekerkaandCurvatureDrivenFlow .......60 3.5.4Equilibriaestimatefortimescale ˝ "t ............................61 Chapter4GeometricEvolutionofPoresin R 3 ............................ 64 4.1Thewhiskeredcoordinatesystemandinner-expansions .......................65 4.2MatchingConditions ............................................65 4.3ExpansionoftheChemicalPotential ..................................67 4.3.1OuterExpansionoftheChemicalPotential ..........................67 4.3.2InnerExpansionoftheChemicalPotential ..........................67 4.4Timescale ˝ " 2 t .............................................68 4.4.1Outerexpansion ..........................................68 4.4.2Innerexpansion ..........................................68 4.5TimeScale ˝ t :SharpInterfaceLimit ................................69 4.5.1OuterExpansion ..........................................69 v 4.5.2InnerExpansion ..........................................70 4.5.3JumpConditions .........................................70 4.5.4TheNormalVelocity .......................................71 4.5.5SharpInterfaceLimit .......................................74 4.6Timescale ˝ "t :CurvatureDrivenFlow ..............................75 4.6.1Outerexpansion ..........................................75 4.6.2InnerExpansion ..........................................76 4.6.2.1Solvingequation( 4.93 )for~ u 1 ............................76 4.6.2.2JumpConditions ....................................77 4.6.2.3Thenormalvelocity ..................................78 4.6.3Sharpinterfacelimit .......................................80 4.6.4Equilibriaestimatefortimescale ˝ "t ............................81 4.7CompetitiveGeometricEvolutionofBilayersandPores ......................83 Chapter5ThePearlingEigenvalueProblem,Co-Dimension1 .................. 86 5.1Overview ...................................................87 5.2Eigenvaluesof M L b ........................................90 5.2.1Bounding ~ M ............................................91 5.2.2Bounding M 0 ...........................................92 5.3BoundingtheOperators .................................96 5.3.1Bounding ~ L b v .........................................97 5.3.2Bounding ~ L 2 b v .........................................100 5.3.3Bounding ~ L b ..........................................109 5.4RelatingtheEigenvaluesof L b and L b ...............................111 5.4.1Perturbationestimate .......................................112 5.4.2Semi-groupestimates .......................................113 5.5Connectingthepearlingeigenvaluesof L b andthoseof L b ....................117 Chapter6ThePearlingEigenvalueProblem,Co-Dimension2 .................. 125 6.1Overview ...................................................126 6.2Eigenvaluesof M L p ........................................128 6.2.1Bounding M 0 ...........................................129 6.3BoundingtheOperators .................................131 6.3.1Bounding ~ L p v .........................................132 6.3.2Bounding ~ L 2 p v .........................................135 6.3.3Bounding ~ L p ..........................................142 Chapter7AnalysisofNetworkBifurcations .............................. 145 7.1Introduction .................................................145 7.2MeanderingEquilibria ...........................................146 7.3PearlingStability ..............................................147 7.4NumericalEvaluationofBifurcationRegions .............................148 7.5ComparisontoExperimentandFullFCHsimulations ........................153 7.6VerifyingtheNumericalResults .....................................155 APPENDICES ................................................... 157 AppendixA:CoordinatesSystem .......................................158 AppendixB:GeometricEvolutionCo-dimension1 .............................161 AppendixC:GeometricEvolutionCo-dimension2 .............................167 AppendixD:PearlingCo-dimension1 ....................................171 AppendixE:PearlingCo-dimension2 ....................................177 vi LISTOFTABLES Here Table7.1:Numericalevaluationsoftheshapefactorofthebilayers S b andtheshapefactorof thepores S p asafunctionofthetiltofthedoublewellpotential W ˘ ‹ u “ . .......150 Table7.2:Numericalevaluationsofthekeyparameterscomparedtotheiralgebraicvalues,for thedoublewellpotential W 3 ‹ u “ . ..............................156 vii LISTOFFIGURES Here Figure1.1:Morphologicalphasesandvesicletransformationsindilutesolutions.Thecoloredre- gionsbetweensphereandrodphasesandbetweenrodandvesiclephasescorrespondto coexistenceregions,thevertical-axisrepresenttheconcentrationofpolymerbyweight andthehorizontal-axisisthepercentofwater.From[ DischerandEisenberg,2002 ]. ReprintedwithpermissionfromAAAS. ...........................2 Figure1.2:MorphologydiagramofPB-PEOinwaterasafunctionofmolecularsizeandcom- positiondescribesthetregionsofbilayers,poresandmicelles.Reprinted (adapted)withpermissionfrom[ JainandBates,2004 ].Copyright2004American ChemicalSociety. ........................................3 Figure1.3:Szostak'smechanismfordivisionofprimitivecellmembrane:(left)raisingtheback- groundconcentrationoflipidsinducesthevesicletogrowworm-like(co-dimension two)protrusionsovera74nano-secondtimeperiod[ BudinandSzostak,2011 ],(right) changingthedensityofchargedgroupsonthesurfaceviaaphotochemicallyinduced redoxreactionincitestheporetopearlandbreakintomicelles[ Zhuetal.,2012 ]. ReprintpermissiongrantedbyProceedingsofNationalAcademyofScience. .....4 Figure1.4:Aporousmembraneassembledfromcholormethylatedpolysufone(CPSF)withpyri- dinegraphedvianucleophilicsubstitution(ammoniumagent).A500foldincrease infroma1microntoa20nanometerlengthscaleshowsaFCH-like nanoscalenetworkmorphologyembeddedwithinthedomainwallsofamicron-scale Cahn-Hilliard-likephaseseparation.Themixtureiselectroneutralonthemicron scale,buthaschargeseparationonthenanometerscale.Reproduced(Adapted) from,[ Zhangetal.,2013 ]withpermissionofTheRoyalSocietyofChemistry. DOI . 8 Figure1.5:Thewhiskeredcoordinatesystemofageneric,admissible,co-dimensiononeinterface. 11 Figure1.6:Single-layer(left)andBilayer(right)dressingsofthesameco-dimensiononeinterface (solidblackline).Thedressingfunctionisaone-DsolutionoftheCHEuler- Lagrangeequation.Forthesingle-layersolutionseparatesregions u b from u b ,whilethebilayersolutioncorrespondsto u b oneithersideofthebilayer, withabriefexcursion u A b near .............................13 Figure1.7:(left)Acomparisonofco-dimension 1 ; 2 ; and3computedfrom( 1.27 ), ( 1.39 ),and( 1.41 )respectively.Therelativewidthstheismostsensitiveto thederenceindepthsofthetwowells: W ‹ b “ W ‹ b “ A 0.(right)Atableof experimentaldataindicatingradiiofbilayer,pore,andmicellemorphologiesobtained byvaryingthehydrophiliclengthofpolymerinPEO-PBamphiphilicdi-blockswith hydrophobic(core)moleculeweight, M core n ,asindicated.Reprinted(adapted) withpermissionfrom[ JainandBates,2004 ].Copyright2004AmericanChemical Society. ..............................................17 Figure1.8:Depictionofbilayer(left,source: academic.brooklyn.cuny.edu ),pore(center),and micelle(right)morphologiesoflipids.Theco-dimensionassociatedtothemorphology isthebetweenthespacedimensionandthenumberoftangentdirections oftheminimalmanifoldwhosenormalbundlelocallyfoliatesthemorphology.In R 3 bilayersareco-dimensionone,poresareco-dimensiontwo,andmicellesareco- dimensionlthree. .........................................19 viii Figure1.9:Timeevolutionofacircular,co-dimensiononebilayerundertheFCHgradientw ( 1.17 )forvales " 0 : 1and 1 2 2.Thetimesdepictedcorrespondto t 0 ; t 114 ; and t 804andshowtheonsetofthepearlingbifurcation. ...........21 Figure1.10:Competitionfortheamphiphilicphasebetweenasphericalbilayer(beachball)and circularsolidpore(hulahoop)asafunctionofthewelltilt W ‹ b “ W ‹ b “ .The imageshows t 100endstatesoftheFCHgradientw( 1.17 )fromidenticalinitial databutwithincreasingvaluesofthewelltilt.Smalltiltprefersbilayers,largertilt prefersporesbyincreasing ⁄ b andthepearlingthreshold, P ⁄ b ,whichdrivesbilayers topearl.ImagescourtesyofAndrewChristliebandJaylanJones. ...........25 Figure2.1:Figure(a)isthesharpinterfacereductionandthebasepointisagivenpoint x > ˆ b ‹ s “ . ThewhiteareainFigure(b)isthereachoftheinterface, b;` ,andforthewhiskered coordinates,thebasepointistheintersectionpointofthewhiskerwiththeinterface 33 Figure2.2:(a)depictsthebilayer U b ‹ z “ whichconvergesto b as z ª .Sub- ‹ b “ describesthedressingoftheinterfacewiththebilayer U b marked inred.Theblueregionsrepresentthebackgroundstateandthewhiteregionisthe neighborhoodoftheinterface b;` . ‹ c “ depictsthespectrumof L b; 0 ,with theverticalaxisrepresentingtherealline. ..........................34 Figure2.3:Thestructureoftherealspectrumof L b ,inequation( 1.54 ),plottedverses Laplace-Beltramiwavenumber n .(left)TheSturm-Liouvilleoperator L b; 0 ,in ( 1.29 ),hasonepositivegroundstateeigenvalue, b; 0 A 0andaonedimensionalkernel, denoted b; 1 .(center)Theextensionof L b; 0 to L b L b; 0 "H@ z " 2 s addsside- bandsin n ,theLaplace-Beltramiindexwhichbendbacknegativelyattherate ‹ b; 0 " 2 k “ 2 .(right)Thespectrumoftheoperator L b L 2 b O ‹ " “ ,(minussignchosen topreserveorientationofimages)is,to O ‹ " “ ,thenegativesquareofthespectrum of L b .Theside-bandassociatedto b; 0 hasaquadratictangencyatleadingorder, whichmayberaisedorloweredbythefunctionalterms, 1 and 2 ,thecrossingof thisspectrumthroughzeroisthemechanismofthepearlinginstability.Springerand theoriginalpublisher[ HayrapetyanandPromislow,2014 ],originalcopyrightnotice isgiventothepublicationinwhichthematerialwasoriginallypublished,byadding; withkindpermissionfromSpringerScienceandBusinessMedia ............36 Figure2.4:Theeigenvaluesof L b withthelimitofthemeanderingeigenmodes, N 3 ,andthelimits ofthepearlingeigenmodes N 2 ;N 3 . ..............................36 Figure2.5:Co-dimension2whiskeredcoordinatesin R 3 ........................38 Figure2.6:FullOperatorsSpectrum ....................................43 Figure2.7:Thespectrumofthesub-operators L p;m for m 0 ; 1 ; 2withtherealaxisvertical. ..43 Figure3.1:Thegeometricevolutionofageneric,admissible,co-dimensiononeinterface, b ‹ t 0 “ istheinitialinterfaceand b ‹ t 1 “ describestheinterfaceatalatertime t 1 A t 0 . ....47 Figure7.1:Thebilayer(left)andthepore(right)correspondingtothetilted double-wellpotential W ....................................146 ix Figure7.2:Themeanderingequilibrialines:Theblue(solid)lineisforthebilayersystem, ⁄ b , andthered(dashed)lineisfortheporesystem ⁄ p ,asafunctionof d ,where 1 1 andadouble-wellpotential. ...........................147 Figure7.3:Thetilteddouble-wellpotential W ‹ u “ . ............................149 Figure7.4:Thegroundstateeigenfunctions: b; 0 (left)and p; 0 (right). ..............149 Figure7.5:The L 1 1functions: b; 1 (left)and p; 1 (right). .....................150 Figure7.6:Pearlingbifurcationlinesasafunctionof d (topleft),PearlingandEquilibrialines (topright),theco-existenceequilibriaismarkedbyagreencircle.Zoomingontothe blackcircleinthegureontheright(bottomcenter). ..................151 Figure7.7:Thetwoequilibrialinesandthetwobifurcationlinesforadoubletiltedwellwith ˘ 0 : 7(top)andwith ˘ 0 : 5(bottom). ............................152 Figure7.8:(left)Increasingthebackgroundstate 1 movestheblackpointfromitsequilibria, whichresultsingrowthofbothmorphologies.(right)Szostak'sexperiment:raising thebackgroundconcentrationoflipidsinducesthevesicletogrowworm-like(co- dimensiontwo)protrusionsovera74nano-secondtimeperiod[ BudinandSzostak,2011 ] ...................................................153 Figure7.9:(left)Increasing d movestheblackpointtoatpearlingstabilityarea,which quicklyleadstopearlinginthebilayers.(right)Changingthedensityofcharged groupsonthesurfaceviaaphotochemicallyinducedredoxreactionincitesthepore topearlandbreakintomicelles[ Zhuetal.,2012 ]. .....................154 Figure7.10:Pearlingandmeanderingstabilityregionsforttiltsofthepotential W ˘ .For thetilt ˘ 0 : 7,thegreendotisbothmeanderingandprealingstable(left). For ˘ 0 : 9,thegreendotislocatedinthebilayerunstablepearlingregion. .....155 Figure7.11:Atilteddouble-wellpotentialoftheform( 7.11 )for p 3(blue)andoftheform( 7.10 ) for ˘ 0 : 9(red). ........................................156 x KEYTOABBREVIATIONS Algorithm SymbolMeaningDetailedin CHCahn-HilliardChapter 1 FCHFunctionalizedCahn-HilliardChapter 1 W k;p ‹ A “ Thesubsetoffunctions f > L p ‹ A “ suchthatthefunction f anditsweakderivativesuptosomeorderkhaveaite L p norm,forgiven1 B p B ª Standard R n n dimensionalEuclideanspaceStandard C k ‹ “ Thesetoffunctionshavingallderivativesoforder B k con- tinuousin Standard C k 0 ‹ “ Thesetoffunctionsin C k ‹ “ withcompactsupportinStandard C k ‹ “ Theset C k ‹ “ ` C k ‹ “ consistsoffunctionswhosek-th orderpartialderivativesarelocallyoldercontinuouswith exponent in Standard C k 0 ‹ “ Thesetoffunctionsin C k ‹ “ withcompactsupportinStandard CartesianrepresentationoftheLaplaceoperatorStandard E Cahn-HilliardfreeenergyEquation( 1.3 ) F FCHfreeenergyEquation( 1.14 ) ` R d ,boundeddomainChapter 1 u ThevolumefractionofonecomponentofthebinarymixtureChapter 1 W ‹ u “ TilteddoublewellpotentialChapter 1 " rationofamphiphilicmoleculetodomainsizeChapter 1 1 1 A 0, 1 > R Chapter 1 2 2 > R Chapter 1 d d 1 2 Chapter 1 N Quasi-minimizernetworkmorphologyof F Chapter 1 Q C Quasi-stationarynetworkmorphologyof F ,fora C A 0Chapter 1 TheEuler-LagrangemultiplierEquation( 1.18 ) ^ ^ ~ " thescaledLagrangemultiplierEquation( 1.20 ) b Anadmissible,co-dimensiononemanifoldChapter 2 b ‹ s;z “ Themappingtothewhiskeredcoordinatesforco-dimension one Equation( 2.1 ) b;` Thereachof b Equation( 1.23 ) H Theextendedcurvaturesofco-dimensiononemanifoldEquation( 2.10 ) xi J b TheJacobianofthechangeofvariablesinco-dimensionone manifold Equation( 2.6 ) H 0 Meancurvatureof b Standard s TheLaplace-BeltramioperatorEquation( 2.13 ) G TheextensionoftheLaplace-BeltramioperatorEquation( 2.12 ) U b ThebilayerEquation( 2.37 ) u b Thebilayerdressingof b with U b Chapter 1 u b; 1 The O ‹ " “ correctionto u b Equation( 1.32 ) L b; 0 Thelinearizationof( 1.27 )about U b Equation( 2.39 ) 1 ThechemicalpotentialEquation( 1.33 ) Thewellcoercivity, W œœ ‹ b “ Equation( 1.34 ) ~ b;` ~ b;` … b;` Chapter 1 ˙ b Thebilayer`surfacetension'Equation( 1.46 ) m b ThemassofamphiphilicmaterialperunitlengthofbilayerEquation( 1.50 ) ⁄ b TheoptimalvalueofamphiphilicmaterialinthebulkregionEquation( 1.52 ) p Anadmissible,co-dimensiontwomanifoldChapter 2 p Themappingtothewhiskeredcoordinatesforco-dimension twomorphologyin R 3 Equation( 2.58 ) @ 2 G AnextensionofthelinediusionoperatorEquation( 2.67 ) U p TheporeEquation( 2.75 ) u p Theporedressingof p with U p Chapter 1 u p; 1 The O ‹ " “ correctionto u p Chapter 1 b TheconstantvalueChapter 1 U m Therotationalsymmetricsolutionfortheco-dimension threeODE Equation( 1.41 ) u c AcriticalpointofCHfreeenergyChapter 1 L b Thelinearizationof F about U b Equation( 1.54 ) b; 0 ;n Thepearlingeigenfunctionsof L b Chapter 2 b; 0 ;n Thepearlingeigenvaluesof L b Chapter 2 b; 1 ;n Themeandereigenfunctionsof L b Chapter 2 b;j Theeigenfunctionsof L b 0 Chapter 2 b;j Theeigenvaluesof L b 0 Chapter 2 S b TheshapefactorofthebilayerEquation( 5.41 ) P ⁄ b ThebilayerpearlingconditionEquation( 1.59 ) xii b;j satisfy L j b; 0 b;j 1for j 1 ; 2,Equation( 2.40 ) p;j satisfy L j p; 0 p;j 1for j 1 ; 2,Equation( 2.86 ) xiii Chapter1 Introduction 1.1AmphiphilicMaterials Traditionallytheterm`amphiphilicmolecule'denotesasmallmoleculewhichdsanenergeticallyfavorable interactionattheinterfaceoftwodisparatesuchassoapinanoil-water-soapmixture.Indeed,early studiesofamphiphilicmaterialsconcernedemulsionsformedfromtwoimmisciblecombinedwithan amphiphilicsurfactant.Lipids,formedofahydrophilicheadgroupandahydrophobictailbelongtothis classofamphiphilicmolecules.Morerecently,developmentsinsyntheticchemistry,suchasatomtransfer radicalpolymerization,havetheprocessofattachingchargegroupstopolymers,greatlyexpand- ingthepossibleclassesofamphiphilicpolymersthatcanreadilybesynthesized,see[ Matyjaszewski,2012 ] and[ Charleuxetal.,2012 ].Amphiphilicblendstypicallyphaseseparate;howeverthepropensityofthe amphiphilicmoleculestoformmonolayersleadstoanenergeticpreferenceforthininterfaces.Asaresult, amphiphilicmixturestypicallyformnetworkmorphologieswhichsupportasymptoticallylargeinterfacesof variousco-dimension.Theseincludeco-dimensiononebilayers,orco-dimensiontwoporestructures.To maketheideaofanetworkmoreprecise,wethefollowingmotivation: Givenasmallparameter0 @ " 0 P 1,wesaythatafamilyofclosedsubdomains Ÿ D " š 0 @ " B " 0 ` R d ;d C 2,isa networkmorphologyifthesetsarenested,thatis D " 1 ` D " 2 for " 1 @ " 2 B " 0 ,andeachconstituentpointof D " lieswithin O ‹ " “ ofitscompliment.If D 0 istheintersectionof D " ,thenthe(local)co-dimensionofthe networkisthebetweenthedimension, d ,oftheambientspaceandthe(local)dimension of D 0 : Intuitively, D 0 isspandthesets D " canbethoughtofasthepointsthatliewithin " of D 0 ,{where " playstheroleofthemolecularwidth.Networkshavesigntvalue:theydescribethearrangements 1 ofamphiphilicmolecules,whichselfassembleintonano-scalestructureswithhugedensitiesofsolvent- accessiblesurfacearea.Theresultingnetworkmorphologiesaretypicallycharge-lined,renderingthem tcharge-selectiveionicconductors.Duetothesetraits,amphiphilicmaterialshavefoundusein manytypesofenergyconversiondevices,formingtheionomermembranesinfuelcells,thephoto-active collectingmatrixinbulk-heterojunctionsolarcells,andtheseparatormembraneinLithiumionbatteries, [ Anderson,1975 , WilsonandGottesfeld,1992 , Peetetal.,2009 ]. Thecastingofblendsofamphiphilicmixturesandsolventpresentsaricharrayofdistinctmorphologies, howevercontroloftheend-statemorphologyisexperimentallychallengingduetothedelicaterolesplayedby solventtype,saltconcentrationandcounter-iontype,di-blockcompositionandpolydispersity,temperature, andpH.Ithasbeenshown,[ DischerandEisenberg,2002 ],thatchangingtheconcentrationofwaterina water-dioxanesolventblendinducesbifurcationsinamphiphilicdi-blocksyieldingmicelle,micelle-pore,pore (rod),pore-bilayers,andbilayernetworkmorphologies,seeFigure 1.1 . Figure1.1: Morphologicalphasesandvesicletransformationsindilutesolutions.Thecoloredregions betweensphereandrodphasesandbetweenrodandvesiclephasescorrespondtocoexistenceregions,the vertical-axisrepresenttheconcentrationofpolymerbyweightandthehorizontal-axisisthepercentofwater. From[ DischerandEisenberg,2002 ].ReprintedwithpermissionfromAAAS. SimilarbifurcationwereobtainedinPEO-PBamphiphilicdi-blocksbychangingthedensityofchargegroups inthehydrophilicportion,[ JainandBates,2004 ].Figure 1.2 depictsthemorphologydiagramofPB-PEO diblockinwaterasafunctionofmolecularsizeandcomposition.Theaxis,N PB andW PEO ,denotethemolec- ularweightofthePBportionofthechainandtheweightfractionofthePEOportion,respectively.Thefour mainstructuresobservedarebilayervesicles(B),cylinders(pores)(C),andspheres(S).Asthehydrophilic content(W PEO )isincreased,asequenceofstructuralelementsisdocumented:startingwithbilayers,fol- 2 Figure1.2: MorphologydiagramofPB-PEOinwaterasafunctionofmolecularsizeandcomposi- tiondescribesthentregionsofbilayers,poresandmicelles.Reprinted(adapted)withpermission from[ JainandBates,2004 ].Copyright2004AmericanChemicalSociety. lowedbypores,andthenmicellesseparatedbycompositionwindowscontainingmixedmorphologies,such as,bilayers-poresandpores-micelles.Increasingthehydrophilicweightfractioninducesgreaterinterfacial surfacearea,andincreasestheaspectratioofthediblock,asPEOisasofterchainthanPB,andformsmore ofaball-likestructure.Morphologicalcanalsobeachievedthroughvaryingtemperature, [ Zareetal.,2012 ]and[ Gomezetal.,2005 ],andconcentrationsofcounter-ions[ ZhulinaandBorisov,2012 ]. Wepayparticularattentiontotheexperimentalinvestigationin[ BudinandSzostak,2011 ]and[ Zhuetal.,2012 ] addressingthedivisionofprimitivelipidmembranes.Szostak'sgroupderivedaparticularlysimplemethod toinducethebilayertomicellemorphologicalchange;theyformedasuspensionofsphericalvesicles of10%phospholipidandfoundthatincreasingtheconcentrationoffreeoleo-lipidsdispersedinthebulk solventinducedainstabilityinsphericalphospholipidvesicles;thistransformationisdepictedin thethreehorizontallyarrangedpanelsontheleftsideofFigure 1.3 ,theendstateofwhichconsistsoflong, co-dimensiontwoporemorphologies.Inasubsequentexperiment,thechargedensityonthesurfaceofcylin- dricalporeswassuddenlyincreasedthroughaphoto-oxidationprocess;thejumpinchargedensityinduces apearlingbifurcationcausingtheporestructurestobreakintoindividualmicelles,asdepictinthethree verticallyarrangedpanelsontherightsideofFigure 1.3 .Oneaimofthisthesisistopresentananalysisof relatedbifurcationswithinthecontextoftheFunctionalizedCahn-Hilliardfreeenergy,whichweintroduce inthefollowingsection. 3 Figure1.3: Szostak'smechanismfordivisionofprimitivecellmembrane:(left)raisingthebackground concentrationoflipidsinducesthevesicletogrowworm-like(co-dimensiontwo)protrusionsovera74 nano-secondtimeperiod[ BudinandSzostak,2011 ],(right)changingthedensityofchargedgroupsonthe surfaceviaaphotochemicallyinducedredoxreactionincitestheporetopearlandbreakintomicelles [ Zhuetal.,2012 ].ReprintpermissiongrantedbyProceedingsofNationalAcademyofScience. 1.2TheFunctionalizedCahn-HilliardFreeEnergy ThesteptowardstheintroductionoftheFCHistorecallthederivationoftheCahn-Hillard(CH) freeenergy,[ CahnandHilliard,1958 ],whichdescribesthespinodaldecompositionofanimmisciblebinary mixture.Foradomain, ` R 3 ,aphasefunction u > H 1 ‹ “ describesthevolumefractionofone componentofthebinarymixture,andthefreeenergyismodeledbyafunctionofthedensity u weakly perturbedbythespatiallyisotropicgradients E ‹ u “ S f ‹ u;" 2 S © u S 2 ;" 2 u “ dx: (1.1) Expandingthefreeenergyinordersof " andkeepingtermsupto O ‹ " 2 “ ,yieldsanexpressionoftheform E ‹ u “ S f ‹ u; 0 ; 0 “ " 2 A ‹ u “S © u S 2 " 2 B ‹ u “ u dx: (1.2) Toobtainagenericnormalformforthefreeenergy,CahnandHilliardintegratedbypartsthelastterm in( 1.2 ),settheresultingcotof S © u S 2 to 1 2 ,andrelabeledthepotential f ‹ u; 0 ; 0 “ as W ‹ u “ .Theresult istheCahn-Hilliardfreeenergy E ‹ u “ S " 2 2 S © u S 2 W ‹ u dx: (1.3) 4 Thecorresponding H 1 gradientw,theCahn-Hilliardequation,takestheform u t E u ‹ " 2 u W œ ‹ u ““ : (1.4) Subjecttoboundaryconditions, u n 0 ; (1.5) n 0 ; (1.6) where n istheouternormalto @ theCahn-Hilliardequationpreservesthetotalmass d dt S u ‹ x;t “ dx 0 ; (1.7) anddissipatestheCahn-Hilliardfreeenergy d dt E ‹ u “ c u t ; E u h L 2 \ © E u \ 2 L 2 B 0 : (1.8) ItisknownthattheminimizersoftheCHfreeenergyoverthespace H 1 ‹ “ subjecttoamassconstraint areachieved.Moreover,theseminimizerssatisfytheEuler-Lagrangeequationexpressedintermsofthe variationalderivativeof E E u " 2 u W œ ‹ u “ (1.9) where isaLagrangemultiplierassociatedtoatotalmassconstraint. Tomodelamphiphilicmixtures,suchasemulsionsformedbyaddingaminorityfractionofanoilandsoap mixturetowater,[ TeubnerandStrey,1987 ]and[ GompperandSchick,1990 ]weremotivatedbysmall-angle X-rayscattering(SAXS)datatoincludeahigher-ordertermintheusualCahn-Hilliardexpansion.Inspired bytheirwork,weaddthenextordertermto( 1.2 ), ~ F ‹ u “ S < @ @ @ @ @ > f ‹ u; 0 ; 0 “ " 2 A ‹ u “S © u S 2 " 2 B ‹ u “ u C 0 C ‹ u “‹ " 2 u “ 2 = A A A A A ? dx: (1.10) Thefullformofthissystemsupportstoomanypossibleregimestopermitasystematicstudy.Itisimportant tothesimplestmathematicalframeworkthatsupportsthenetworkmorphologiestypicalofamphiphilic mixtures;weneedanewnormalform.Withthisgoalweshiftallthetialtermstopowersofthe Laplacian.Sp,letting A denotetheprimitiveof A ,wereplace A ‹ u “ © u with © A ‹ u “ and,assuming 5 appropriateboundaryconditions,weintegratetheterm © A © u bypartstoobtain ~ F ‹ u “ S f ‹ u; 0 ; 0 “ ‹ B ‹ u “ A ‹ u ““ " 2 u C ‹ u “‹ " 2 u “ 2 dx: (1.11) Theenergydensityisaquadraticpolynomialin " 2 u ,whichsuggeststhatwecompletethesquare ~ F ‹ u “ S C ‹ u “ < @ @ @ @ > „ " 2 u A B 2 C ‚ 2 f ‹ u; 0 ; 0 “ ‹ A B “ 2 4 C ‹ u “ = A A A A ? dx: (1.12) Forsimplicitywereplace C ‹ u “ with 1 2 ,andrelabelthepotentialwithinandoutsidethesquaredtermby W œ ‹ u “ and P ‹ u “ ; respectively.Thekeypointisthatthetermisthesquareofthevariationalderivative ofaCahn-Hilliardtypefreeenergy,consequentlythecase P 0,whentheenergyisaperfectsquare,hasthe specialpropertythatitsglobalminimizersarepreciselythe criticalpoints ofthecorrespondingCahn-Hilliard energy.Indeed,avariantofthiscasewasproposedasatargetfor convergenceanalysisbyDeGiorgi,see [ ogerandScatzle,2006 ].Ourgeneralformofthenetworkisobtainedbyperturbingtheperfectsquare withanasymptoticallysmallpotential, ~ F ‹ u “ S 1 2 › " 2 u W œ ‹ u “ ” 2 P ‹ u dx; (1.13) where P 1.Thefunction W ‹ u “ isassumedtobeadouble-wellpotentialwithtwominimaat u b whose unequaldepthsarenormalizedsothat W ‹ b “ 0 A W ‹ b “ .Typically b 0 ; howeveritishelpfultogive thisvalueaspname.Thus u b isassociatedtoabulksolventphase,whilethesizeof u b A 0 isproportionaltothedensityoftheamphiphilicphase.Thesmallparameter " P 1,associatedtothe amphiphilicmolecularwidth,determinestheinterfacialwidthandcorrespondstotheratioofthetypical lengthofanamphiphilicmoleculetothedomainsize. TheFunctionalizedCahn-Hilliardfreeenergycorrespondstoaclassoftwodistinguishedlimitsanda particularchoicefor P , F ‹ u “ S 1 2 › " 2 u W œ ‹ u “ ” 2 " p „ " 2 1 2 S © u S 2 2 W ‹ u dx: (1.14) Thefunctionalizationterms,parameterizedby 1 A 0and 2 > R ,areanalogoustothesurfaceandvolume energiestypicalofmodelsofchargedsolutesineddomains,see[ Scherlisetal.,2006 ]andparticularly equation(67)of[ Andreussietal.,2012 ].Theminussigninfrontof 1 isofconsiderable{it incorporatesthepropensityoftheamphiphilicsurfactantphasetodrivethecreationofinterface.Indeed, 6 experimentaltuningofsolventqualityshowsthatmorphologicalinstabilityinamphiphilicmixturesisasso- ciatedto(small)negativevaluesofsurfacetension,[ Zhuetal.,2009 ]and[ ZhuandHayward,2012 ].Inthe FCHenergythegradientterm, 1 S © u S 2 @ 0,islocalizedoninterfaces,associatedtosingle-layersofsurfactant molecules,whosegrowthlowersoverallsystemenergy{howevertheis perturbative andunrestricted growthisarrestedbythepenaltynatureofthesquaretermwhichkeeps u closetothecriticalpointsof E .Therearetwonaturaldistinguishedlimitscorrespondingtotchoicesfortheexponent p inthe functionalizationterms.IntheStrongFunctionalization, p 1,thefunctionaltermsdominatetheWillmore correctionsfromthesquaredvariationalterm.TheWeakFunctionalization,correspondingto p 2,isthe naturalscalingfortheasthecurvature-typeWillmoretermsappearatthesameasymptoticorder asthefunctionalterms. Thewell-posednessoftheminimizationproblemfortheFCH,includingtheexistenceofglobalminimizers forvaluesof " A 0wasestablishedin[ PromislowandZhang,2013 ]foramoregeneralfunctionalform overvariousnaturalfunctionspaces.Dependingupontheapplication,thevolume-type 2 functionalization perturbationincorporatestheimpactofcounter-ionentropy(PEMfuelcells),capillarypressure,orentropic fromconstraintoftailgroups(lipidbilayers),[ Gavishetal.,2012 ].Theform 2 W ‹ u “ ischosen primarilyforconvenience,asintegralsof W ‹ u “ evaluatedatcriticalpointsof E CH growincreasinglynegative withincreasinginterfacialco-dimension.Weremarkthatthesurfaceterm 1 S © u S 2 isequivalenttoan 1 uW œ ‹ u “ functional-formsinceanintegrationbypartson 1 S © u S 2 yields 1 u u whichcanbeabsorbed intothesquaredvariationwithaperturbedformof W . Thegoalofthisstudyistopresentananalysisofthestabilityanddynamicsofclassesofquasi-stationary networkmorphologies N of F ,whichwetobefunctions u > H 2 ‹ “ whichhaveanasymptotically smallminorityofamphiphilicphase,satisfyassignedboundaryconditions,andrenderthedrivingforceof thefreeenergyasymptoticallysmall.Spforeach C A 0wethesetofquasi-stationary networkmorphologies Q C œ u > H 2 ‹ “U S S u b S dx B C" and VV 0 F u VV L 2 ‹ “ B C" p 3 2 ¡ ; (1.15) where p takesthesamevalueasinequation( 1.14 )whichtheFCHfreeenergy.Theexponent term, p 3 2 ,intheboundontheresidualcorrespondstotemporaldynamicsonthe " p timescale.Wealso introducethezero-massprojection 0 f f 1 S f ‹ x “ dx: (1.16) Ouranalysishingesontheconstructionofquasi-stationaryfunctionswhoseproperlychosenlevelsetsform 7 locallyco-dimensiononeandtwonetworkmorphologiesinthesenseof ?? . ItisimportanttoemphasizethebetweentheCHfreeenergyandtheFCHfreeenergy.TheCH freeenergydescribesthespinodaldecompositionofhydrophobicmaterials.TheFCHfreeenergymodels networkformationinamphiphilicmaterials.Inexperimentalsettings,amphilicitydrivesthesystemtophase separateonamolecularlengthscale.Figure 1.4 (a)resemblesanearlystageofCHspinodaldecomposition. Figure1.4: Aporousmembraneassembledfromcholormethylatedpolysufone(CPSF)withpyridine graphedvianucleophilicsubstitution(ammoniumagent).A500foldincreaseinfroma1mi- crontoa20nanometerlengthscaleshowsaFCH-likenanoscalenetworkmorphologyembeddedwithinthedo- mainwallsofamicron-scaleCahn-Hilliard-likephaseseparation.Themixtureiselectroneutralonthemicron scale,buthaschargeseparationonthenanometerscale.Reproduced(Adapted)from,[ Zhangetal.,2013 ] withpermissionofTheRoyalSocietyofChemistry. DOI . Zoominginwheretheredcircleis,aftera500-foldthephaseseparatednetworkmorphology isvisiblewithintheCH-cellwalls,seeFigure 1.4 (d).Averagedoveramicronlengthscalethesystemis electroneutral,andthephaseseparationisgovernedbyaCHdynamic.Onthenanometerlengthscalethe systemisnotelectroneutral,andthephaseseparationisgovernedbytheFCHwiththeassociatednetwork morphologies. 1.3OverviewofMainResults TheoverdampeddynamicsofamphiphilicpolymersuspensionscanbereceivedfromtheFunctionalized Cahn-Hilliardfreeenergyviaitsgradientwswhoseevolutionpreservesthevolumefractionofthecon- stituentspeciesandlowersthefreeenergy.SimilartotheCahn-Hilliardgradientwgivenin( 1.4 ),the simplestmasspreservinggradientwofthe strongFCH isgeneratedbythe H 1 gradient, u t F u " 2 W œœ ‹ u “ 1 ”› " 2 u W œ ‹ u “ ” " ‹ 1 2 “ W œ ‹ u “ : (1.17) 8 Thisresearchincludesaformalderivativeofthegeometricevolutionofco-dimensiononeandco-dimension twomorphologiesunderthestrongFCHequation,followedbyarigourousanalysisofthepearlingeigenvalues formorphologiesofeithercodimension. Westartbyformalasymptoticreduction:Inchapter 3 wederivethegeometricmotionofacollectionof disjoint,farfromself-intersecting,closed,co-dimensiononemorphologies,referredtoasbilayermorphologies, in R d .Thekeyresultsarethatthechemicalpotentialofthepurebilayersystem, 1 ,isspatiallyconstant atleadingorderinthefarandthe H 1 gradientwdrivespurebilayerinterfacesbyaquenched mean-curvaturew,seeequation( 1.63 ).Moreover,forabilayermorphologythechemicalpotential willconvergetemporallytoaprescribedconstantvalue, ⁄ b .Inchapter 4 weinvestigatedthegeometric evolutionofafamilyofco-dimensiontwomorphologies,referredtoasporemorphologies,in R 3 .Away fromtheinterfaces,thechemicalpotentialisspatiallyconstant,andthe H 1 gradientwdrivespurepore manifoldsbyacurvaturew,wherethevectornormalvelocityiscoupledtothechemicalpotential,see equation( 1.65 ).Moreover,thechemicalpotentialwilldecaytoaprescribedconstant ⁄ p . Note1. Thetwoequilibriapoints ⁄ b and ⁄ p aredeterminedonlybythetilteddouble-wellpotential W and thevaluesofthefunctionalizationparameters 1 and 2 . Dependinguponthevalueofthechemicalpotentialthegeometricwcanbemotionbycurvature ormotionagainstcurvature;thelaterinducesastronggeometricinstability,akintoabackwardsheat equationinstabilityforthecurvatures,seeequation( 1.64 ),whichmanifestsinexperimentsasa instability,asshowninFigure 1.3 .Forthebilayersystem,if 1 @ ⁄ b thenthebilayerwillshrinkas 1 grows, whileif 1 A ⁄ b ,thebilayerwillgrow,whichmayinducegoftheinterface b .Asimilarinstability mechanismholdsforporemorphologies. Tocompletetheformalanalysisweconsiderthegeometricevolutionofaco-existingsystemcomprised ofafamilyofdisjoint,farfromself-intersecting,closed,co-dimensiononeandtwostructures,in R 3 .In 2014,[ DaiandPromislow,2015 ]haveshownthatfortheweakFCHthetwomorphologiescanco-exist. However,weconcludethat,generically,the strong FCHequationdoesnotsupportco-existence.Morpholo- giesofdistinctco-dimensionwillcompeteviathecommonvalueofthefarchemicalpotential,and dependingupontheinitialandthevaluesofthefunctionalizationparameters, 1 and 2 ,the structureswillcompeteforsurfactantphaseviathecommonvalueofthefarchemicalpotential, 1 , withvariouspossibleoutcomesincludingtheextinctionofonephase,apearlingbifurcationofoneorboth phases,orabifurcation.Wealsonon-genericvaluesof 1 and 2 inwhichco-dimensionone andco-dimensiontwomorphologiescanco-exist. Thegeometricevolutionresultsareformal,inparticulartheyassumethattheunderlyingbilayerandpore 9 morphologiesarestable.Thevulnerabilityofthematchedasymptoticsmethodisthatitignoresanypossible instabilities.Inchapter 2 wereviewthespectrumofthelinearizedoperators,seeFigure 2.3 ,andshowthat boththeco-dimensiononeandco-dimensiontwomorphologieshavepotentialinstabilitiesassociatedto periodic,high-frequencymodulationsoftheinterfacialwidth,calledapearlinginstabilty. ArigourousanalysisoftheeigenvalueproblemcorrespondingtothestrongFCHforthebilayerandthe poremorphologiesispresentedinChapters 5 and 6 ,respectively.WeshowthatinthestrongFCHscaling theleadingorderbehaviorofthepearlingeigenvaluesisindependentoftheshapeoftheunderlyingco- dimensiononeortwomorphology,whichallowstheofassociatedpearling-stabilityregionsin parameterspace.Inchapter 7 weanalyzethecombinedbilayer-poreevolution.Underthe H 1 gradient wthepearlinginstabilitymanifestsitselfonatimescalethatis O ‹ " 2 “ fasterthenthanthegeometric evolution,andhencecanbetakentobeinstantaneousonthegeometrictimescale.Conversely,theg instabilityoccursonthesametimescaleasthegeometricw,andmaynotnecessarilyimmediatelymanifest itselfonthegeometrictimescales. 1.4Quasi-StationarysolutionsofthestrongFunctionalizedCahn- HilliardFreeEnergy Forsimplicity,wefocusonthestrongFCH,whosecriticalpoints,subjecttoatotalmassconstraint,arethe solutionsoftheassociatedEuler-Lagrangeequation F u › " 2 W œœ ‹ u “ ”› " 2 u W œ ‹ u “ ” " › " 2 1 u 2 W œ ‹ u “ ” (1.18) where > R istheLagrangemultiplier,andtheboxedterm ‹ " 2 u W œ ‹ u ““ isthevariationalderivative ofaCHfreeenergy,oftheformpresentedinEquation( 1.3 ).Intuitively,approximatesolutionsoftheCH Euler-Lagrangeequation E u " 2 u W œ ‹ u “ O ‹ " “ : (1.19) arenaturalstartingplacesforaperturbativeconstructionofsolutionsoftheFCHEuler-Lagrangeequation. ForsuchapproximatecriticalpointsoftheCHfreeenergyitisnaturalthattheLagrangemultiplier in (1.14)shouldscalewith ,thatis " ^ .WithinthisscalingwemayrewritetheFCHEuler-Lagrange 10 equation,( 1.18 ),astwo,coupledsecondordersystems " 2 u W œ ‹ u “ "v; › " 2 W œœ ‹ u “ ” v › " 2 1 u 2 W œ ‹ u “ ” ^ (1.20) ThesingularlyperturbednatureoftheFCHEuler-Lagrangesystemmakesitamenabletodimensional reduction,yieldinglocalizedsolutionsbuilduponimmersionsin R d oftco-dimensions. 1.4.1Constructionofco-dimension1Quasi-stationarysolutionsofFCH Wereviewsomebasicfromelementarytialgeometry.Let b ` R d beasmooth, co-dim1interface,whichdividesintotwodisjointsets 8 ,seeFigure 1.5 .Let ˆ b ‹ s “ beitslocal parametrization, ˆ b Q ` R d 1 R d ,and s ‹ s 1 ;:::;s d 1 “ > Q ` R d 1 ,andlet r bethesigneddistance (unscaled)from b .Forsimplicitywechoosetheparameterizationsothe s i correspondtoarclengthalong Figure1.5: Thewhiskeredcoordinatesystemofageneric,admissible,co-dimensiononeinterface. the i th coordinatecurveandthecoordinatecurvesarelinesofcurvature.Inthissetting,thevectors T i ‹ T i 1 ;:::T i d 1 “ by T i @ˆ @s i ;i 1 ;::;d 1 ; (1.21) formanorthonormalbasisforthetangentspaceto b at ˆ b ‹ s;t “ .Denotingtheouternormalvectorof b pointingtowards by n ‹ s;t “ ‹ N 1 ;:::;N d “ ,wehavetherelations @ T i @s i k i n ; @ n @s i k i T i ;i 1 ;::;n 1 ; (1.22) where k i aretheprinciplecurvaturesof b . 1.1. Ford K;` A 0 thefamily, G K;` ,of\admissibleco-dimensiononeinterfaces"iscomprised 11 ofclosed(compactandwithoutboundary),oriented 2 dimensionalmanifolds b embeddedin R d ,whichare farfromself-intersectionandwithasmoothsecondfundamentalform.Moreprecisely, (i)The W 4 ; ª ‹ Q “ normofthe2ndFundamentalformof b anditsprincipalcurvaturesareboundedby K . (ii)Thewhiskersoflength 3 ` ,intheunscaleddistance,dforeach s 0 > Q by, w s 0 Ÿ x s ‹ x “ s 0 ; S z ‹ x “S @ 3 ` ~ " š ,neitherintersecteach-othernor @ (exceptwhenconsideringperiodicboundaryconditions). (iii)Thesurfacearea, S b S ,of b isboundedby K . Assume b > G K;` .Theset b;` by b;` ı b ‹ s;z “ > R d U s > S; ` ~ " B z B ` ~ " ` ; (1.23) willbecalledthe reach of b ,whereweemphasizethat ` isindependentof " andof b > G K;` . Each x > b;` canbeuniquelyexpressedviathe whiskeredcoordinates suchthat x b ‹ s;z “ ˆ b ‹ s “ "z n ‹ s “ ; (1.24) where z > l ~ ";l ~ " isthescaledsigneddistanceto b , z r " .Thelinesegments Ÿ ˆ b ‹ s “ `;` U s > Q š arethe whiskers oflength2 ` of b ,andthepair ‹ s;z “ formthelocalwhiskeredcoordinatesystem. Figure 1.5 presentsthewhiskeredcoordinatesystem.BytheImplicitFunctionTheoremthismapislocally andsmoothlyinvertible.Inparticular,thefunctions s s ‹ x “ and z z ‹ x “ whichrelatethewhiskered coordinatestothecartesianonesandtheassociatedchangeofvariables,areall C 4 onthe reach, b;l ,of b .ThewhiteregioninFigure 1.6 (right)depictsthereachoftheassociatedimmersion b : 1.2. Given b > G K;` andafunction f R R whichtendstoconstantvalues f ª atO(1) exponentialratesas r ª ,wesaythatwe dresstheinterface b with f ,obtainingthe b -extension f b ‹ x “ f ‹ r ‹ x ““ ˜ ‹S r ‹ x “S~ l “ f ª ‹ 1 ˜ ‹S r ‹ x “S~ l ““ f ª ‹ 1 ˜ ‹S r ‹ x “S~ l ““ ; (1.25) where r ‹ x “ isthe(unscaled)distancefrom b and ˜ R R isad,smoothfunctionwhichisone on ; 1 ,while ˜ ‹ s “ 0 for s C 2 .Byabuseofnotationwewilldropthe b subscriptinthe b -extension whendoingsocreatesnoconfusion. Thestepintheconstructionofthequasi-stationarysolutionsistobuildthebilayerwhichisa 1-dimensionequilibriumofequation( 1.20 ).InthewhiskeredcoordinatestheCartesianLaplaciantakesthe form " 2 @ 2 z "H ‹ s;z “ @ z " 2 G ; (1.26) 12 Figure1.6: Single-layer(left)andBilayer(right)dressingsofthesameco-dimensiononeinterface(solid blackline).Thedressingfunctionisaone-DsolutionoftheCHEuler-Lagrangeequation.Forthesingle- layersolutionseparatesregions u b from u b ,whilethebilayersolutioncorrespondsto u b on eithersideofthebilayer,withabriefexcursion u A b near where H istheextendedcurvature,intermsoftheJacobian, J b ,ofthechangeofvariables, inEquation( 1.24 ).Inparticular,atleadingorder H ‹ s;z “ H 0 ‹ s “ O ‹ "z “ where H 0 isthemeancurvature of b at b ‹ s; 0 “ and G s O ‹ "z “ where s istheusualLaplace-Beltramioperatoron b ,forfurther detailsseeSection 2.1 . Inthewhiskeredcoordinatestheequationof( 1.20 )reduces,atleadingorder,toasecond-orderODE in z ,fortheone-dimension ' ‹ z “ , @ 2 z ' ‹ z “ W œ ‹ ' “ ; (1.27) for z inthereach.Sincethedouble-well W isassumedtohaveunequaldepthwells0 W ‹ b “ A W ‹ b “ ,asimplephase-planeanalysisshowsthatthisequationsupportsauniquesolution U b whichis homoclinic to b ,thatis U b ‹ z “ b as z ,see[ HomburgandSandstede,2010 ]forageneraldiscussion ofhomoclinicorbits. Wetheleading-orderstructureofthebilayercriticalpoint, u b u b ‹ x ; b “ viathetwo-termexpansion, u b ‹ x “ U b ‹ z ‹ x ““ "u b; 1 ; (1.28) where U b isthebilayerdressingof b withinthereach b;` ,equaltoaconstantvalueon … b; 3 ` andsmoothly extendedtomatchintheintermediateregion,seeFigure 1.6 (right).Tothecorrectionterm u b; 1 we introducetheSturm-Liouvilleoperator L b; 0 L b; 0 @ 2 z W œœ ‹ U b “ ; (1.29) whichisthelinearizationof( 1.27 )about U b .Evaluatingequations( 1.20 )at u b andprojectingtheright-hand 13 sideontotherangeof L b; 0 yields L b; 0 u b; 1 v; (1.30) L b; 0 v 1 U œœ b 2 W œ ‹ U b “ ^ (1.31) Onthereach, b;` ,thecorrection u b; 1 ischosentosimplifytheresidualofequation( 1.20 )whenevaluated at u b ,isas u b; 1 L 2 b; 0 › 1 U œœ b 2 W œ ‹ U b “ ^ ” : (1.32) Remark1.1. Theinverseoperator L 1 b; 0 isnaturally L 2 ‹ R “ H 2 ‹ R “ ,byabuseofnotationweapply ittofunctionson ` ~ ";` ~ " whichhaveanaturalextensionto R byapplyingittotheextension,andthen restrictingtheresult. Wefurtherdecompose u b; 1 intoalocalterm~ u b; 1 whichdecaysexponentiallytozeroin z ,andissmoothly extendedtobezeroof b;` ,andaconstantterm 1 ^ 2 ; (1.33) wherewehaveintroducedthewellcoercivity W œœ ‹ b “ A 0 : (1.34) Theresulting u b isourqausi-steadysolution u b ‹ x “ U b ‹ z “ " ‹ 1 ~ u b; 1 ‹ z ““ ; (1.35) parameterizedby b > G K;` and 1 > R .Thelocalterm~ u b; 1 correctsthestructureof U b withinthereach, whilethespatialconstant 1 adjuststhebehaviorof u b ,whichisnow b b 1 . Intheregion u b takesthespatiallyconstantvalue u b ‹ x “ b b " 1 2 O ‹ " 2 “ ;x > ~ b;` ; (1.36) where 1 ,thechemicalpotential,istheleadingorder,non-zeroterminthevariationof F .Bymatching 14 theinnerandouterexpressionof u b ,givenin( 1.35 )and( 1.36 ),respectively,wethat 1 1 2 ; (1.37) i.e.,thevalueof 1 from 1 byafactorcorrespondingtothesquareofthebranchpoint, 2 ofthe essentialspectrumof L b; 0 : Thequantity 1 playsakeyroleintheevolutionandbifurcationofthequasi-steadyinterfaces.Weasso- ciate 1 withthedensityofamphiphilicmolecule{preciselythequantitythatSzostak'tweaked'by addingoleo-lipidstothebulksolventphaseinhisexperiment[ BudinandSzostak,2011 ],seeFigure 1.3 . Remark1.2. Therearecriticalpointsof F forwhich is O ‹ 1 “ ,inparticularthe single-layer solutions, whichcorrespondtoheteroclinicorbitsof( 1.27 )thatconnecttwoequilibriumvalues,seeFigure 1.6 (left).For theCahn-Hilliardfreeenergysingle-layersformthedominantglobalminimizers,howevertheyaregenerically saddlepointsoftheFCH,andaresusceptibletomeanderinstabilitiesinthegradientw.Itisimportant toemphasizethatsingle-layersandbilayersaredistinctmorphologies{single-layersseparatephaseAfrom phaseBwhilebilayersseparatephaseAintotworegionsbyathinlayerofphaseB,seeFigure 1.6 .In particularbilayerscanrupture,re-unitingthetworegionsofphaseA,aswhenalipidbilayeropensapore, ortears.Inaddition,theinterfacialcomponentisaconservedquantityforbilayers,andwhenthebilayeris stretchedtheinterfacemustthin,whichnaturallyincreasesitsfreeenergyasitdeformsfromitsequilibrium U b {bilayerscansupportnon-zerotangentialstresses. 1.4.2Constructionofco-dimension2andco-dimension3Quasi-stationaryso- lutionsoftheFCH TheFCHEuler-Lagrangeequation,( 1.20 ),alsopossessesco-dimensiontwoandco-dimensionthreesolutions in ` R 3 .Weconsiderco-dimensiontwosolutions.Thesearebaseduponafoliationofaneighborhood ofasmooth,closed,non-selfintersectingone-dimensionalmanifold p immersedinTheco-dimension twowhiskeredcoordinatesystem,introducedinChapter 2 ,isusingthemapping x p ‹ s;z “ ,and theideasofadmissibility,reach,anddressingextendnaturallyfromtheco-dimensiononecase. Withinthereach, p;` ,of p theLaplacianadmitsthelocalform " 2 z " Ñ 1 " Ñ z Ñ © z " 2 @ 2 G ; (1.38) where z istheCartesianLaplaceoperatorinthescalednormaldistances Ñ z ‹ z 1 ;z 2 “ ,thevector Ñ ‹ 1 ; 2 “ T isthecurvaturesvectorof p at p ‹ s; Ñ 0 “ ,and @ 2 G reducestothelineoperator, @ 2 s ,on p 15 when Ñ z 0 ; seeSection 2.2 fordetails.Assumingradialsymmetry,theleadingorderporeassociated totheEuler-Lagrangeequation( 1.18 )sco-dimensiontwocriticalpointequation @ 2 R U p 1 R @ R U p W œ ‹ U p “ ; (1.39) subjectto @ R U p ‹ 0 “ 0and U p b b 1 O ‹ " 2 “ as R ª .Theleadingorderformforthepore quasi-stationarynetworkarisesfromtheporedressingofaco-dimensiontwointerface p , u p ‹ x “ U p ‹ R ‹ x ““ " ‹ 1 u p; 1 ‹ R ““ ; (1.40) wherethelocalterm~ u p; 1 correctsthestructureof U p withinthereach, p;` ,while 1 > R isaspatial constantthatadjuststhechemicalpotential.Itispossibletocombinequasi-stationarybilayer andporemorphologies,solongastheassociatedmanifoldshavenon-intersectingreaches, and the constant 1 takesacommonvalue.Indeed,thequasi-steadyevolutionbetweenco-existingco-dimensionone andco-dimensiontwointerfacesisdrivenbythecompetitionbetweenthiscommonvalue b ,which isgivenby b b 1 .Iftheoptimalvalues, ⁄ b and ⁄ p ,associatedtodistinctco-dimensional morphologiesthenthemorphologiescannotbothsimultaneouslybeinequilibrium,seesection 4.7 for details. Co-dimensionthreequasi-stationarysolutions,in R 3 ,aresphericallysymmetricmicellemorphologies.The associatedcoordinatesystemreducestotheusualsphericalvariablesandtheLaplacianreducestothe associatedsphericalform.Assumingrotationalsymmetry,theleadingordermicelleistheunique solutionof @ 2 R U m 2 R @ R U m W œ ‹ U m “ ; (1.41) subjectto @ R U m ‹ 0 “ 0and U m b as R ª .AnimmediatepredictionoftheFCHfreeenergyisthat bilayersmustbethinnerthanpores,whichinturnarethinnerthanmicelles.Thisobservationisbornout byexperimentaldata,Figure 1.7 (right). 1.4.3MinimizationofthestrongFCHfreeenergyoverco-dimension1quasi- stationary Itisconstructivetoexaminetheminimizersof F overaclassofco-dimension1quasi-stationarysolutions. Momentarilysettingasidethemassconstraint,therearetwoclassesoffreeparametersinourconstruction of u b ,thespatiallyconstantbackgroundcorrection, 1 ,andtheinterfaceshape b .Wewillshow,in 16 M core n (g/mol) 2500 40 5850 204 bilayer 8 : 7 1 : 2 15 : 8 2 : 8 pore 14 : 3 1 : 6 25 : 4 3 : 3 micelle(nm) 18 : 4 2 : 6 38 : 8 10 : 2 Figure1.7: (left)Acomparisonofco-dimension 1 ; 2 ; and3computedfrom( 1.27 ),( 1.39 ), and( 1.41 )respectively.Therelativewidthstheismostsensitivetotheindepthsofthe twowells: W ‹ b “ W ‹ b “ A 0.(right)Atableofexperimentaldataindicatingradiiofbilayer,pore,and micellemorphologiesobtainedbyvaryingthehydrophiliclengthofpolymerinPEO-PBamphiphilicdi-blocks withhydrophobic(core)moleculeweight, M core n ,asindicated.Reprinted(adapted)withpermission from[ JainandBates,2004 ].Copyright2004AmericanChemicalSociety. Equation( 1.52 ),thatforthe strongFCH freeenergy,theoptimalvalueofamphiphilicmaterialinthe bulkregionisdeterminebythedouble-wellpotential W andthefunctionalizationterms 1 and 2 .We evaluatethefreeenergy,at u b ,whichtakestheform F ‹ u b “ S 1 2 › " 2 u b W œ ‹ u b “ ” 2 " „ " 2 1 2 S © u b S 2 2 W ‹ u b “‚ dx; (1.42) andbreaktheintegraloverthe b;` and ~ b;` … b;` : Denotingtheintegralby F ` ‹ u b “ wechangetolocalcoordinates F ` ‹ u b “ S b;` › " 2 u b W œ ‹ u b “ ” 2 " „ " 2 1 2 S © u b S 2 2 W ‹ u b “‚ dx; S b S ` ~ " ` ~ " 1 2 › @ 2 z U b W œ ‹ U b “ "H 0 ‹ s “ U œ b ” 2 " ‰ 1 2 S U œ b S 2 2 W ‹ U b “’ J b ‹ s;z “ dzds; (1.43) wheretheJacobian,in( 2.6 ),admitstheexpansion J b " " 2 zH 0 ‹ s “ O ‹ " 3 z 2 “ .Expandingthe Jacobianandkeepingonlyleadingordertermswe F ` ‹ u b “ " S b S ` ~ " ` ~ " " 2 2 „ L b; 0 „ 1 2 ~ u b; 1 ‚ H 0 ‹ s “ U œ b ‚ 2 " ‰ 1 2 S U œ b S 2 2 W ‹ U b “’ dsdz: (1.44) Thelocalizedfunctionsinthesquaredtermwillyield O ‹ " 3 “ integralswhicharenegligible.However,the correctioninthesquaredterm L b; 0 1 2 W œœ ‹ U b “ 1 2 1 yieldsanasymptoticallyrelevant contribution.Moreoverintegrating( 1.27 )weseethat ‹ U œ b “ 2 2 W ‹ U b “ .Togetherthesetwoobservations 17 allowustorewritethelocalizedcomponentofthefreeenergyas, F ` ‹ u b “ " 2 S b S„ ` 2 1 2 1 2 2 ˙ b ‚ ; (1.45) whereweintroducedthebilayer'surfacetension' ˙ b Y U œ b Y 2 L 2 ‹ R “ : (1.46) Thevalueof u b intheregionisgivenin( 1.36 ),and,byTaylorexpansion,wenotethat W œ ‹ b “ " 1 O ‹ " 2 “ ,and W ‹ b “ O ‹ " 2 “ .Denotingtheintegralby ~ F ` ‹ u b “ wethatitscontributionto theenergyreducestotheleadingorderexpression, ~ F ` ‹ u b “ " 2 ‹S S 2 ` S b S“ 1 2 2 1 2 O ‹ " 3 “ : (1.47) Combiningthenear-andexpressions,thetotalenergytakestheform F ‹ u b “ " 2 „ S S 2 2 1 2 S b S 1 2 2 ˙ b ‚ O ‹ " 3 “ : (1.48) Asimilardecompositionappliedtothetheintegralsyieldstheexpressionforthetotal massofamphiphilicmaterial. M S u b ‹ x “ b dx S ~ b;` " 1 2 dx S b S ` ~ " ` ~ " ‹ U b " 1 2 “ J b dzds " S S 1 2 " S b S m b ; (1.49) where m b S R U b ‹ z “ b dz A 0 ; (1.50) isthemassofamphiphilicmaterialperunitlengthofbilayer.Typicallytheamphiphiliccomponentisscarce withinthebulk,sothat M " ^ M (don'tputtoomuchsoapinthewashingmachine!),andsince b is admissibleitsinterfacialarea S b S is O ‹ 1 “ .Theseassumptionsrender u b aquasi-stationarywithrespectto F ,moreoveraprescribedvalueof ^ M and 1 determinesthearea, S b S ,ofthebilayerinterface.Consequently, wesolveequation( 1.49 )for S b S ,andplugtheresultintoequation( 1.48 )whichyields F ‹ u b “ " 2 ™ Œ fl S S 2 2 1 2 ‹ ^ M S S 1 2 “‹ 1 2 “ ˙ b 2 m b fi Š Ł : (1.51) 18 Figure1.8: Depictionofbilayer(left,source: academic.brooklyn.cuny.edu ),pore(center),andmicelle (right)morphologiesoflipids.Theco-dimensionassociatedtothemorphologyisthebetweenthe spacedimensionandthenumberoftangentdirectionsoftheminimalmanifoldwhosenormalbundlelocally foliatesthemorphology.In R 3 bilayersareco-dimensionone,poresareco-dimensiontwo,andmicellesare co-dimensionlthree. Theminimizationof F ‹ u b “ over b and 1 ,subjecttothemassconstraintreducestotheoptimizationofa quadraticpolynomialin 1 ,andtheoptimalvalueofamphiphilicmaterialinthebulkregiontakestheform ⁄ b 1 2 2 ˙ b m b : (1.52) Forthestrongfunctionalizationonlytheareaofanadmissibleco-dimensiononeinterface,andnotits curvature,enterintotheleading-orderdeterminationofthefreeenergyofitsbilayerdressing.Moreover bilayerspreferanoptimalvalueoflipid, ⁄ b whichisindependentofthescaledmassconstraint ^ M andhencetheareaofthebilayer{itisauniversalpropertyofthesystemasdeterminedbytheshapeof thewell W through m b , ˙ b ,and andthroughthefunctionalizationparameters 1 and 2 .Fortheweak functionalizationtheWillmoreterm,theintegralofthesquareofthemeancurvatureover b ,entersinto thefreeenergyatleadingorder,andtheoptimizationismoresubtle. 1.5NetworkBifurcationintheFCH Thequasi-stationarynetworkmorphologiesdevelopedinSection 1.4 are,atleadingorder,criticalpointsof theCahn-Hilliard,howeverthesestructuresarenotperturbationsoflocalminimabutratherapproximate saddlepointsoftheCHfreeenergy.AnessentialfeatureofthefunctionalformoftheFCHisitsfacilityto buildcompetitorsforitslocalminimaoutofthesaddlepointsofthesimplerCHfreeenergy.Thisprocessis bestunderstoodbyexaminingthesecondvariationalderivativeoftheFCHfreeenergyatasmoothcritical point, u c oftheCahn-Hilliardfreeenergy.Fortracelessboundaryconditions,suchasperiodicboundary conditions,see[ PromislowandZhang,2013 ]foradetaileddiscussionofappropriateboundaryconditions, 19 thesecondvariationtakestheform L u c 2 F u 2 ‹ u c “ › " 2 W œœ ‹ u c “ ” 2 " p › 1 " 2 2 W œœ ‹ u c “ ” : (1.53) Foraquasi-steadybilayer, u b ,associatedtoanadmissible,co-dimensiononeinterface b ,thesecondvaria- tionalderivative L b L u b ,takesaformwhenactingonfunctions u > H 4 ‹ “ whosesupportlies withinthereach, b;` ,of b .Onthissubspacetheoperatoradmitstheasymptoticexpansion L b › L b; 0 "H@ z " 2 G ” 2 " p › 1 @ 2 z 2 W œœ ‹ U b “ ” O ‹ " p 1 “ ; (1.54) Aninvestigationofthespectrumoftheoperator L b ispresentedinChapter 5 .Indeed,itwasshownby [ HayrapetyanandPromislow,2014 ]thatthereexists U A 0,independentof " suchthattheeigenfunctions associatedto L b correspondingtoeigenvalues b @ U comprisetwosets,the pearlingeigenmodes Ÿ b; 0 ;n š N 2 n N 1 andthe meandereigenmodes Ÿ b; 1 ;n š N 3 n 0 ,seedetailsinChapter 2 .Inchapter 5 wecharacterizethepearling eigenmodes,showingthattheyareindependentof b > G K;` andconsequentlydetermineparametricregions ofpearlingstabilityandinstability,forthestrongFCH.For b anadmissible,generic,co-dimensionone interfaceweconsidertheeigenvalueproblem L b b; 0 ;n b; 0 ;n b; 0 ;n ; (1.55) associatedtothesecondvariationof F aboutthebilayerdressing u b .Thespectrumof L b cannotbelocalized byaregularperturbationexpansionsincetheeigenvaluesareasymptoticallyclosetogether. Theexpressionforthepearlingstabilityconditionofbilayerinterfaceswithconstantcurvatureswasestab- lishedin[ Doelmanetal.,2014 ],seeFigure 1.9 ,wheretheeigenvaluesassociatedtothebilayerdressingof suchaninterfaceareuncoupled.Inthisthesisweextendthisresulttothelinearizationaboutadressing ofgenericadmissibleco-dimensiononeandco-dimensiontwomanifolds.Themainyarisesfromthe couplingamongtheeigenvaluesthroughthederivativesofthecurvatures.Theanalysisrequiresboundson thespectrumthatareuniformin " P 1.Tothisendweintroducethe L 2 ‹ “ orthogonalprojectiononto thespaceofthepearlingeigenmodesanditscomplementaryprojectiondenoted ~ I Adecomposition oftheoperator L b intoa2 2blockformusingtheprojectionstakestheform ~ L b < @ @ @ @ @ @ @ @ @ @ > L b L b ~ ~ L b ~ L b ~ = A A A A A A A A A A ? : (1.56) 20 Figure1.9: Timeevolutionofacircular,co-dimensiononebilayerundertheFCHgradientw( 1.17 )for vales " 0 : 1and 1 2 2.Thetimesdepictedcorrespondto t 0 ;t 114 ; and t 804andshowtheonset ofthepearlingbifurcation. InChapter 5 ,Section 5.3 ,weprovethattheoperatorsaresmall,innorm,independentof " . Thespectrumofthefullyitedimensionalpiece, ~ L b ~ isboundedfrombelowbyaconstant U A 0 independentof " ,[ HayrapetyanandPromislow,2014 ].Theupper-leftelement L b canbereducedtoa largematrix M > R N N where N " 3 ~ 2 d . Thespectrumof L b iscontrolledbythespectrumofthematrix M andthesingularscalingisrctedinthegrowthof N as " 0.Caremustbetakentodistinguish betweenthesizeoftheentriesof M andthesizeof M asanoperatorfrom l 2 ‹ R N “ to l 2 ‹ R N “ ,asthelatter genericallyscaleslike º N timesthe l ª normoftheentries.Forsimplicitywefocusonlyonthepearling modes j 0,neglectingthemeandertermsassociatedto j 1.InChapter 5 weobservethatthematrix M admitsanasymptoticdecomposition M M 0 diag "A; (1.57) where M 0 diag isadiagonalmatrixand A isuniformlyboundedasanoperatoron l 2 ‹ R N “ aslongasthe curvaturesaretlysmooth.Therefore,atleadingorder,theeigenvaluesof M arethediagonalentries of M 0 diag whichtaketheform b; 0 ;n ‹ b; 0 " 2 n “ 2 " ‹ 1 2 S b b; 0 ‹ 1 2 “Y b; 0 Y 2 2 “ ; (1.58) where b; 0 isthegroundstateeigenvaluesofthelinearoperator L b; 0 withthecorrespondingeigenfunc- tion b; 0 ,and k isaneigenvalueoftheLaplace-Beltramioperator s ,correspondingtotheeigenfunc- tion k .Thecot S b isthe\bilayershapefactor",inequation( 5.41 ),whosesigndetermines ifthepearlingbifurcationabsorbsorreleasesamphiphilicmaterialfromthebulk. Thepositivequadraticterminthepearlingeigenvalueexpression( 1.58 )isdominantexceptwhentheLaplace- 21 Beltramieigenvalue n isapproximatelyequalto b; 0 " 2 .BytheWeylasymptoticformulafortheLaplace- Beltramieigenmodes,theresidualofthedominanttermis O ‹ " “ foranasymptoticallylargevalue, N of indices n .Thenatureofthebilayerpearlingbifurcationdependssensitivelyuponthesignof S b .For S b @ 0,whichholdsforagenericclassofdouble-wellpotentials W ,seesection5of[ Doelmanetal.,2014 ], thespectrumof M willbestrictlypositiveifandonlyif 1 the pearlingstabilitycondition P ⁄ b b; 0 ‹ 1 2 “Y b; 0 Y 2 L 2 2 S b A 1 : (1.59) Wealsoidentifyaclassofwellsforwhich S b A 0,inwhichcasethethedirectionoftheinequalityin( 1.59 ) isreversed. InChapter 5 ,Section 5.4 ,weconnectthespectrumof M tothatof L b ,showingthattheeigenvaluesof M areinfactasmallperturbationofthesmalleigenvaluesof L b ,andweobtainaperturbationestimate.We alsoexaminethesolutionofthelinearwgeneratedby L b .Assumingtheeigenvaluesof M arestableunder pearling,intermsofEquation( 1.59 ),wewillshowthatthesemi-groupsgeneratedby L decayexponentially fastanddescribetheresultingexponentialdichotomy. Asimilaranalysiscanbeperformedforco-dimensiontwoporestructures,parametrizedbytheone-dimensional immersion p ,seeChapter 6 .Assuminganegativevalueofthe\poreshapefactor" S p ,in( 6.33 ) weshowthattheporestructurewillremainpearlingstableifandonlyif 1 the pearlingstability condition P ⁄ p d − TT œ p; 0 TT 2 L R p; 0 SS p; 0 SS 2 L R ‘ S p A 1 ; (1.60) where p; 0 isthegroundstateeigenvalueof L p withthecorrespondingeigenfunction p; 0 . ThisanalysisisconsistentwithSzostak'sexperiment,see[ Zhuetal.,2012 ],inwhichaphoto-inducedincrease inchargeonthelipidheadsinducedapearlingbifurcationwhichdroveporestomicelles,seeFigure 1.3 (right). Theincreaseinchargecorresponds,withintheFCH,toaninstantaneousincreasein 1 ;atlylarge increase,foravalueof 1 ,willtriggerthebilayerpearlingcondition( 1.59 )aswellastheporepearling condition( 1.60 ).Figure 7.9 depictsthepearlingasaresultofinstantaneouslyincreasein 1 . InSection 5.5 werelatethesmalleigenvaluesof L tothoseof L andthatthepearlingeigenvalues of L whicharetwoordersof " largerthanthepearlingeigenvaluesof L . 22 1.6CompetitiveGeometricEvolutionofBilayersandPores Inthisthesiswemodeltheover-dampeddynamicsofamphiphilicpolymersuspensionsviathemass- preserving H 1 gradientwgiveninequation( 1.17 ).Thequasi-stationarynetworkmorphologiescon- structedinsection 1.4 arenotstationarysolutionsoftheFCHgradientw,butgenerateslowdynamics whichmaybelocallyparameterizedbytheinterfacialsub-manifoldsofbilayersandpores,respectively b and p .Indeed,whenthebilayerpearlingstabilityconditionholds,thenthebilayersmeandereigen- modes Ÿ b; 1 ;n š N 3 n 0 ,depictinFigure 2.3 ,comprisethepotentiallynegativeeigenspaceoftheassociated linearization.Thewoftheunderlyinginterfacialstructurecanbeobtainedbyprojectingtheresidual F u ‹ u b “ ofthecriticalpointequation( 1.18 )ontothiseigenspace.Themethodofmatchedasymptoticexpan- sionprovidesanaccessible,butformalmethodtoderivetheinterfacialmotion.Forabilayermorphology, theansatz( 1.35 )for u b isaugmentedbytakingthesigneddistance z totheinterface b andthebackground state 1 tobefunctionsoftheslowscaledtime ˝ t ~ " ,andthegradientwissolvedbymatching particularlyacrosstheinterfaciallayers.Forsingle-layermorphologies,undertheCahn-Hilliardgradient wthisresultsinaMullins-Sekerkaproblemfortheinterface,see[ Pego,1989 ].For " P 1itwasshown thattheleadingordernormalvelocityoftheinterfaceofthespinodaldomainsisdeterminedbythejump inthenormalderivativeofthechemicalpotentialacrosstheinterface,separatethecomplementary domains.MorerigorousderivationsofPego'sresultsquicklyfollowed,particularly[ Alikakosetal.,1994 ] and[ DeMottoniandSchatzman,1995 ]. FortheFCHgradientw,( 1.17 )reduces,atleadingorder,to "U œ b ‹ z “ @z @˝ " 1 d˝ F u ‹ u b “ " H 0 ‹ s “ U œ b ‹ z “ O ‹ " 2 “ : (1.61) Theleadingorderresidualarisesfromthemean-curvaturetermwhichwasneglectedintheconstructionof thebilayer, u b .Thistermnowbecomesadrivingforcefortheevolutionoftheinterface b throughthetime derivativeinthesigneddistancefunction.Indeed,thequantity V b ‹ s “ @z @˝ ; (1.62) isthenormalvelocityoftheinterface b .TheasymptoticreductiondoesleadtoaMullins-Sekerkaprob- lemforthechemicalpotential,howeveritsdrivingforceisgivenbytheinterfacialmeancurvature timesthejumpofthebilayereacrosstheinterface, H 0 ‹ s “ J U b K .Sincethebilayerisahomoclinicorbit itsjump J U b K 0,andtheMullins-Sekerkaproblemistrivial.Theouterchemicalpotentialreducesto aspatialconstant,andtheischaracterizedbyamphiphilicdensity, 1 ‹ ˝ “ ,whosevalueindeter- 23 minedbyconservationoftotalmass,see[ DaiandPromislow,2013 ]fordetailsforbilayersundertheweak functionalization.Forthestrongfunctionalizationtheresultingsystemtakestheform V b b ‹ 1 ⁄ b “ H 0 ; 1 dt 1 b m b ‹ 1 ⁄ b “ S b H 2 0 dS; (1.63) where H 0 isthemeancurvature, b m b R R ‹ U b b “ 2 dz A 0and ⁄ b istheoptimalamphiphilicdensity, thesamequantityderivedbytheoptimizationprocessin( 1.52 ).The H 1 gradientwdrivespurebilayer interfacesbyaquenchedmean-curvaturew.Whilethewdrives 1 toitsoptimalvalue ⁄ b ,thesignof the 1 ⁄ b isconsequential.Indeed,intwospacedimension,moduloreparameterizationofthe evolvinginterface,thecurvaturedrivenwcanberecastasanevolutionequationofthesinglecurvature H 0 , @H 0 @t 1 ‹ @ 2 s H 2 0 “ V b b ‹ 1 ⁄ b “‹ @ 2 s H 2 0 “ H 0 ; (1.64) seesection3.3of[ Gavishetal.,2011 ]fordetails.If 1 A ⁄ b ,thatisifthebulkvalueofamphiphilicmaterial isinexcessthenthecurvaturedrivenwisabackwards-heatequationinthecurvatures.Thisisthenature oftheinstabilityinducedin[ BudinandSzostak,2011 ]whenoleo-lipidswereaddedtothebulkof thesphericalbilayersuspension.Thengeringinstabilityinitiatesasabackwardheatwinthecurvature. Theresultingsingularityisassociatedtothedevelopmentoftheporetypegrowthemanatingfromthebilayer surface.Moreover,in[ Doelmanetal.,2014 ]thecondition 1 A ⁄ b wasidenasthepointofbifurcation tolinearinstabilityofthemeandereigenvaluesassociatedtosphericalbilayers.For 1 @ ⁄ b thecurvature drivenwislocallywell-posedbutissubjecttoblow-upduetothecubicdrivingforce, H 3 0 .This isthefamiliarextinctionofdropletsundercurvaturedrivenw.However,forthequenchedw ( 1.63 )therelaxationof 1 toitsequilibriumvalueprecludestheblow-upiftheinitialcurvaturesarenottoo large. Asimilarreductioncanbeperformedforco-dimensiontwoporestructures,parametrizedbytheone- dimensionalimmersion p .Theresultisasimilarquenchedcurvaturedrivenwforthevectorvalued normalvelocity Ñ V p ‹ @z 1 @˝ ; @z 2 @˝ “ T ; Ñ V p p ‹ 1 ⁄ p “ Ñ ‹ s “ ; 1 d˝ "m p ‹ 1 ⁄ p “ S p S Ñ S 2 ds; (1.65) where p m p ˇ R ª 0 ‹ U œ p “ 2 RdR A 0, Ñ isthevectorcurvatureof p , m p 2 ˇ R ª 0 ‹ U p b “ RdR isthemassof 24 Figure1.10: Competitionfortheamphiphilicphasebetweenasphericalbilayer(beachball)andcircular solidpore(hulahoop)asafunctionofthewelltilt W ‹ b “ W ‹ b “ .Theimageshows t 100endstates oftheFCHgradientw( 1.17 )fromidenticalinitialdatabutwithincreasingvaluesofthewelltilt.Small tiltprefersbilayers,largertiltprefersporesbyincreasing ⁄ b andthepearlingthreshold, P ⁄ b ,whichdrives bilayerstopearl.ImagescourtesyofAndrewChristliebandJaylanJones. amphiphilicmaterialperunitlengthofporestructureandtheequilibriumvalue ⁄ p 1 2 R ª 0 ‹ U œ p “ 2 RdR R ª 0 ‹ U p b “ 2 RdR ; (1.66) isagainindependentof p .Mostintriguingly,initialdatacorrespondingtospatiallyseparatedporesand bilayersyieldsacompetitiveevolutionthatcanbeunderstoodasatforsurfactant,mediatedthroughthe commonvalueofthebulkamphiphilicdensity 1 ,whoseevolutionisdeterminedtoimposetheconservation oftotalmass, V n b ‹ 1 ⁄ b “ H Ñ V p p ‹ 1 ⁄ p “ Ñ 1 dt 1 b m b ‹ 1 ⁄ b “ S b H 2 0 dS p m p ‹ 1 ⁄ p “ S p S Ñ S 2 ds; (1.67) Thecompetitiveevolutionofthebilayersandporescouplesthroughcurvature-weightedsurfacearea.How- ever,thetwomorphologiesseekequilibriavalues,whichgenericallysatisfy ⁄ b x ⁄ p ,makingcoex- istenceofbilayersandporesimpossibleunderthestrongfunctionalization,unlessoneofthestructuresis sincezerocurvatureinterfacesareatequilibriumindependentofbulkvalueofamphiphile.Forcurved interfaces,therange 1 > ⁄ p ; ⁄ b isinvariantunderthew,andonce 1 entersthisrangethebilayers willshrink,whiletheporemorphologieswillgrow.Moreover,ifthepearlingthreshold P ⁄ b lieswithinthe invariantrange ⁄ p ; ⁄ b thenthevalueof 1 maytransientlydecreasethroughthepearlingthresholdfor 25 bilayers,( 1.59 ),causingthebilayerstopearlastheyshrink.Figure1.9depictsvarious t 100end-statesof theFCHgradientwforadoublewellpotential W withincreasingvalueofwelltilt W ‹ b “ W ‹ b “ .Inall casestheinitialdataconsistsofasphericalbilayerandtwocircularporesplacedwithantipodalsymmetry. Increasingwelltiltleadstoaporeendstatewithalargerradiusandtopearlingofthebilayer.Adetailed analysisofthebifurcationstructureofthebilayer-porenetworkmorphologiesisgiveninChapter 7 . 26 Chapter2 CoordinateSystem,and Notation Onemaingoalofthisthesisistodescribethegeometricevolutionofthefunctionalizedpolymer-solvent bilayerandporemorphologies.Weintroducethe whiskeredcoordinatesystem whichdescribesthe tangentialandnormalcoordinatesinaneighborhoodofanadmissibleinterface.InSection 2.1 weaddressthe co-dimensiononemorphologyin R d , ‹ d C 2 “ ,andestablishnecessaryfromelementarytial geometry,andinSection 2.2 werepeattheprocessforco-dimensiontwomorphologyin R 3 . Note2. Throughoutthisthesiswewillusesubscript b or p todistinguishbetweenquantitiesassociatedwith thebilayersstructuresandandthoseassociatedwiththeporestructures,respectively. DerivativeNotation: Givenafunctionofasinglevariable,suchas f ‹ x “ ,weuse ‹ “ œ notationtoindicate itsderivative.e.g., f œ @ x f .Ifafunctioninvolvesmorethenonevariable,wespwritewithrespect towhichvariableswetiatetoavoidambiguity. 2.1Co-dimensionOneMorphologyin R d , d C 2 Admissibleco-dimensiononeinterfacesarein 1.1 .We K and ` andlet b > G K;` bean admissible,co-dim1initialinterface,whichdividesintotwodisjointsets,seeFigure 1.5 .Thereachofthe interface, b;` isinequation( 1.23 ),andaccordingtoequation( 1.24 ),each x > b;` canbeuniquely expressedusingthe whiskeredcoordinates suchthat x b ‹ s;z “ ˆ b ‹ s “ "z n ‹ s “ ; (2.1) 27 where ˆ p isaparameterizationof b , z > l ~ ";l ~ " isthescaledsigneddistanceto b and n istheouter normal. 2.1. We Ñ k b ‹ k b; 1 ;:::k b;d 1 “ tobethevectoroftheprinciplecurvaturesof b . Let g thematrixrepresentationofthefundamentalformof b ,whoseentriesaregivenby g ij d @ˆ b @s i ; @ˆ b @s j i L 2 ‹ R d “ ; (2.2) andtherepresentationofthesecondfundamentalformof b givenby h ij d @ @s i ; @ˆ b @s j i L 2 ‹ R d “ ; (2.3) where istheGaussmapassociatedto b .Then,theJacobian, J b ,ofthetransformation x ‹ s;z “ takes theform J b ‰ @ˆ b @s 1 ;:::; @ˆ b @s d 1 ; n ’ ™ Œ Œ Œ Œ Œ fl I d 1 "zh j i 0 0 " fi Š Š Š Š Š Ł ; (2.4) where I d 1 isthe ‹ d 1 “ ‹ d 1 “ identitymatrix,and h j i isrelatedtotheandsecondfundamentalforms of b viathefollowingequation h j i d 1 Q m 1 h im g mj : (2.5) ThedeterminantoftheJacobianmatrix, J b det ‹ J b “ , J b ‹ s;z “ "J 0 ‹ s “ ~ J b J 0 ‹ " " 2 zH 0 “ O ‹ " 3 “ : (2.6) where J 0 isby J 0 ‹ s “ » det g ; (2.7) isrelatedtothematrixrepresentationofthefundamentalform,in( 2.2 ),and ~ J b isgivenby ~ J b ‹ s;z “ d 1 M i 1 ‹ 1 "zk b;i “ ; (2.8) where k b;i aretheprinciplecurvaturesof b .Formoredetailssee[ HayrapetyanandPromislow,2014 ,Ap- pendix6]. 28 Onthereach b;` ,inthewhiskeredcoordinates,theLaplaceoperatortakestheform " 2 x @ 2 z "H@ z " 2 G ; (2.9) where H istheextendedcurvatureterm,givenby H d 1 Q j 1 k b;j 1 "zk b;j ª Q j 0 H j ‹ s “ " j z j H 0 ‹ s “ "H 1 ‹ s “ z O ‹ " 2 “ ; (2.10) k b;j aretheprinciplecurvaturesof b ,and H j arerelatedtothesumofthe j th powerofthecurvatures.To understandthethirdtermontherighthandsideofequation( 2.9 )weconsiderthematrix G J T b J b . Withthismatrix,thegeneralizedLaplaceterm G takestheform G J 1 b d 1 Q i 1 d 1 Q j 1 @ @s i G ij J b @ @s j ; (2.11) and,accordingto[ HayrapetyanandPromislow,2014 ,Proposition6.6],thegeneralizedLaplaciantermcan bewrittenas G s "zD s; 2 : (2.12) Here s isthe Laplace-Beltrami operator,by s J 1 0 d 1 Q i 1 d 1 Q j 1 @ @s i g ij J 0 @ @s j ; (2.13) where g isthefundamentalformof b ,introducedinequation( 2.2 ),theelements g ij aretheentries of g 1 ,and D s; 2 isa2 nd orderoperator,relativelyboundedperturbationof s ,givenby D s; 2 d 1 Q i;j 1 d i;j ‹ s;z “ @ 2 @s i @s j d 1 Q j 1 d j ‹ s;z “ @ @s j : (2.14) Foradmissible b thecots Ÿ d i;j š and d j satisfy max i;j −SS @ m z d i;j SS L ª ‹ l “ ; SS @ m z d j SS L ª ‹ l “ ‘ B C" m ; for m 0 ; 1 ; 2(2.15) forsome C A 0independentof " . Lemma2.1. Let ` R d beaboundeddomainandconsiderthesubspace H 2 c ‹ b;` “ wherethesubscript c denotescompactsupportwithin b;` .Then D s; 2 isarelativelyboundedperturbationof s on H 2 c ‹ b;` “ . 29 TheproofofLemma 2.1 followsfromolder'sEstimatesforthesecondderivatives,giveninthefollowing theorem- Theorem2.1.1 older'sEstimatesforthesecondderivatives,[ GilbargandTrudinger,2001 ]) . Let u > C 2 0 ‹ R n “ , f > C 0 ‹ R n “ ,satisfyPoisson'sequation u f in R n .Then u > C 2 0 ‹ R n “ and,if B B R ‹ X 0 “ is anyballcontainingthesupportof u ,wehave S D 2 u S œ 0 ; B B C S f S œ 0 ; B ; (2.16) where C C ‹ n; “ ,theoldercontinuousexponentsati 0 B B 1 ,andthenormisdby S f S œ 0 ; B sup x;y > B; x x y ‹ f ‹ x “ f ‹ y ““ d sup x;y > B; x x y f ‹ x “ f ‹ y “ S x y S (2.17) and d diam ‹ B “ . Note3. For 1 ,the S S œ 0 ; 1; B normisthe W 1 ; ª ‹ B “ norm. ProofofLemma 2.1 . Fix f > C 2 0 ‹ “ with supp ‹ f “ ` b;` ,and ⁄ > ˆ ‹ s “ .WLOG,take ⁄ 0.We thefunction u 1 s f .Then,thefollowingcalculationshowsthatwecanboundthe L 2 -normof D s; 2 using the L 2 -normof D 2 u S D s; 2 u S œ 0 ; R R R R R R R R R R R ™ fl d 1 Q i;j 1 ‹ d i;j ‹ s;z ““ @ 2 @s i @s j d 1 Q j 1 ‹ d j ‹ s;z ““ @ @s j fi Ł u R R R R R R R R R R R œ 0 ; (2.18) B SS d i;j ‹ s;z “SS L ª ‹ l “ R R R R R R R R R R R d 1 Q i;j 1 @ 2 @s i @s j u R R R R R R R R R R R œ 0 ; SS d j ‹ s;z “SS L ª ‹ l “ R R R R R R R R R R R d 1 Q j 1 @ @s j u R R R R R R R R R R R œ 0 ; B max i;j −SS d i;j ‹ s;z “SS L ª ‹ l “ ; SS d j ‹ s;z “SS L ª ‹ l “ ‘ ™ fl T D 2 u T œ 0 ; R R R R R R R R R R R d 1 Q j 1 @ @s j u R R R R R R R R R R R œ 0 ; fi Ł B c 1 ™ fl T D 2 u T œ 0 ; R R R R R R R R R R R d 1 Q j 1 @ @s j u R R R R R R R R R R R œ 0 ; fi Ł B c 1 − T D 2 u T œ 0 ; c 2 T D 2 u T œ 0 ; ‘ B C T D 2 u T œ 0 ; ; wherethethirdinequalityfollowsfrom( 2.15 ),thefourthinequalityfollowsfromPoincareinequalityand theconstants c 1 ;c 2 and C areindependentof " .Considerthe S S œ 0 ; B normoftheoperator D 2 ‹ s “ 1 acting on u andapply( 2.16 )to u toobtain S D 2 u S œ 0 B C S s u S œ 0 C S f S œ 0 ; (2.19) 30 wheretheequalityfollowsfromreplacing u 1 f .Combiningequation( 2.19 )and( 2.18 )weobtain S D s; 2 1 s f S œ 0 B C S f S œ 0 : (2.20) Inparticular,for 1wehave TT D s; 2 1 s f TT W 1 ; ª B C SS f SS W 1 ; ª : (2.21) Wewanttoshowthatthefollowinginequalityholds TT D s; 2 1 s f TT L 2 ‹ “ B C SS f SS L 2 ‹ “ : (2.22) Assuminginequality( 2.22 )doesnotholds,i.e., TT D s; 2 1 s f TT L 2 ‹ “ A C SS f SS L 2 ‹ “ ; (2.23) thenthereexistasequence Ÿ f n š suchthat SS f n SS L 2 ‹ “ 0 ; (2.24) TT D s; 2 1 s f n TT L 2 ‹ “ 1 : (2.25) Sincethe W 1 ; ª normisboundedbelowbythe L 2 norm,equation( 2.25 )impliesthat 1 B TT D s; 2 1 s f n TT W 1 ; ª B C SS f n SS W 1 ; ª : (2.26) However,bytheSobolevEmbeddingTheoremweknowthat W 1 ; ª `` L p ; (2.27) where 1 p 1 d ,and d isthespacedimension.Theembedding( 2.27 )impliesthatthereisasubsequence Ÿ f n k š suchthat SS f n k f SS L p 0,and SS f SS L p A 1.However,thiscontradicts( 2.25 ),andweconcludethat TT D s; 2 1 s f TT L 2 ‹ “ B C SS f SS L 2 ‹ “ : (2.28) Ì TobetterunderstandthegeneralizedLaplacianoperator, G ,wereviewsomebasicfactsabouttheLaplace- 31 Beltramioperator s :Theeigenvalues, Ÿ k š ª k 0 ,ofthe s ,andthecorrespondingeigenfunctions, Ÿ k š ª k 0 , satisfythefollowingproperties: ‹ s k k k , ‹ 0 0and k A 0for k A 0. ‹ TheeigenfunctionofLaplace-Beltramiareorthonormalinthe b innerproduct, ‹ k ; j “ b S b k j J 0 ‹ s “ dS k;j ; (2.29) where J 0 isin( 2.7 ). ‹ AccordingtoWeyl'sasymptoticformula,[ Chavel,1984 ],thenumberofeigenvalues B , N ‹ n B “ , includingmultiplicity,satisfy N ‹ n B “ C 1 ‹ d 1 “~ 2 : (2.30) Inparticular, n C 2 n 2 ~‹ d 1 “ ,where C 1 ;C 2 > R constants. 2.2. Let b beanadmissibleinterface.Wesaythat f > L 1 ‹ “ islocalizedon b ifthereexist constants M; A 0 ,independentof " A 0 ,suchthat S f ‹ x ‹ s;z ““S B Me S z S ; (2.31) forall x > b;` . 2.3. Givenafunction f f ‹ s;z “ localizedon b wethejumpof f acrossagivenwhisker by J f K ‹ s “ lim z ª f ‹ s;z “ lim z f ‹ s;z “ : (2.32) Giventwofunctions f;g > L 2 ‹ “ with supp ‹ f “ ;supp ‹ g “ ` b;` wemaychangethe L 2 ‹ “ -innerproductto thewhiskeredcoordinates ‹ f;g “ L 2 ‹ “ S f ‹ x “ g ‹ x “ dx S b S l … " l … " f ‹ s;z “ g ‹ s;z “ J b ‹ s;z “ dzds; (2.33) wheretheJacobian, J b ,wasin( 2.6 ).Moreover,integrationofalocalizedfunctionyields S fdx S b S ` ~ " ` ~ " f ‹ x ‹ s;z ““ J b ‹ s;z “ dzds O ‹ " l ~ " “ : (2.34) 32 Weintroducethe J 0 innerproduct,as ‹ f;g “ J 0 S b S ` ~ " ` ~ " f ‹ s;z “ g ‹ s;z “ J 0 dzds: (2.35) 2.4. Foradwhisker w ,wethepoint ‹ b ‹ s “ ; 0 “ tobeits basepoint (seeFigure 2.1 ). (a) BasePointin (b) BasePointinthe whiskered coordinates Figure2.1: Figure(a)isthesharpinterfacereductionandthebasepointisagivenpoint x > ˆ b ‹ s “ .The whiteareainFigure(b)isthereachoftheinterface, b;` ,andforthewhiskeredcoordinates,thebasepoint istheintersectionpointofthewhiskerwiththeinterface Lemma2.2. Thecurvelengthevolvesaccordingto d S b S dt S b VH ‹ s “ ds: (2.36) Weconsiderthedressing,asin 1.2 ,ofanadmissibleinterface, b > G K;` ,withthebilayer U b ,whichsolves ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ @ 2 z U b W œ ‹ U b “ ; U b ‹ “ b : (2.37) Figure( 2.2 )depictsthebilayersolution(left)andthedressingoftheinterface(middle).Observethat U b is translationinvariant,i.e., U b ‹ z “ Ð U b ‹ z p “ alsosolves( 2.37 ).Takingthederivativeof( 2.37 )withrespect to z yields L b; 0 U œ b 0 ; (2.38) 33 where L b; 0 isthelinearoperator,inequation( 1.29 ),givenby L b; 0 @ 2 z W œœ ‹ U b “ : (2.39) FromSturm-Liouvilletheory,see[ Titchmarsh,1946 ],weknowthattheeigenvalueproblem L b; 0 ^ b;j b;j ^ b;j (a) BilayerSolution U b (b) Dressing b with U b (c) Spectrumof L b; 0 Figure2.2: (a)depictsthebilayer U b ‹ z “ whichconvergesto b as z ª . ‹ b “ describesthedressingoftheinterfacewiththebilayer U b markedinred.Theblueregionsrepresent thebackgroundstateandthewhiteregionistheneighborhoodoftheinterface b;` .e ‹ c “ depicts thespectrumof L b; 0 ,withtheverticalaxisrepresentingtherealline. hasanumberofsimpleeigenvalues Ÿ b;j š ,seeFigure 2.2 (right).From( 2.38 )weknowthat U œ b isan eigenfunctionof L b; 0 ,andsinceithasonenode,itistheeigenfunction ^ b; 1 U œ b ,i.e., U œ b istheexcited- stateeigenfunctioncorrespondingtotheexcited-stateeigenvalue b; 1 0.Thegroundstateeigenfunction ^ b; 0 correspondstothegroundstateeigenvalue b; 0 A 0.By Weyl'sessentialspectrumtheorem ,see[ Kato,1976 , Theorem5.35],thereminderofthespectrumisreal,negativeand O ‹ 1 “ distanceto0.Forfurtherdetails seeAppendix A.2 . Weintroduce theco-dimensionone, L j 1 functions b;j > L ª ‹ R “ for j 1 ; 2whicharethesolutionsof L j b; 0 b;j 1 ; (2.40) andareorthogonaltothekernelof L b; 0 .Thefunction b; 1 takestheform b; 1 ^ b; 1 1 ; (2.41) where ^ b; 1 isthesolutionof L b; 0 ^ b; 1 W œœ ‹ U b “ ; (2.42) and isthewellcoercivityintroducedinequation( 1.34 ).Since U b Ð b atanexponentialrateas z Ð ª , theright-handsideof( 2.42 )isin L 2 ‹ R “ ,andevenabout z 0,henceorthogonaltoker L b; 0 U œ b .The existenceof b; 2 followsfromasimilarargument. 34 2.5. Wethe scaledeigenfunctions b;k ˜ ‹ z “ ~ J 1 ~ 2 ^ b;k ,where ^ b;k isthe k th eigen- functionof L b; 0 and ˜ ‹ z “ isacutfunction, ˜ ‹ z “ ¢ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¤ 0 if S z S C l ~ "; 1 if S z S @ l ~ 2 "; monotoneinbetween : (2.43) Thescaledeigenfunctionsareorthonormalinthe L 2 ‹ “ -innerproduct SS b;k SS 2 L 2 ‹ “ S b S l … " l … " ‹ b;k “ 2 J 0 ‹ s “ ~ J b dzds S b S l … " l … " ‹ ^ b;k “ 2 ˜ 2 ‹ z “ ~ J 1 b J 0 ‹ s “ ~ J b dzds (2.44) S b J 0 ‹ s “ ds S l … " l … " ‹ ^ b;k “ 2 ˜ 2 ‹ z “ dzds 1 : 2.6. Thefulloperatorisdby L b L b; 0 "H@ z " 2 G ; (2.45) where H and G aregiveninequations( 2.10 ),( 2.11 ),respectively.Thefulloperator, L b ,isself-adjointin the L 2 ‹ “ -innerproduct,formoredetailsseeappendix( A.3 ). Accordingto[ HayrapetyanandPromislow,2014 ],thereexists C A 0sothateigenmodescorrespondingto eigenvaluesfromtheset ˙ ‹ L b “ 9 C;C admittheleadingorderexpansion b;j;n b;j ‹ z “ n ‹ s “ O ‹ " “ ; (2.46) for j 0or1.Heretheerrorisinthe L 2 ‹ “ -norm,andweemphasisthattheeigenvalues b;j aresmoothly extendedovertheentiredomainsee 2.5 .Here b;j arethescaledeigenmodesof L b; 0 introduced in 2.5 ,theterm n isaLaplace-Beltramieigenfunctionin( 2.29 ),andthecorresponding eigenvaluestaketheform b;j;n ‹ b;j " 2 n “ O ‹ " “ ; (2.47) where b;j aretheeigenvaluesof L b; 0 correspondingtothescaledeigenfunctions b;j ,and n aretheeigen- valuesof s ,seeFigure 2.3 (center)foradepictionofthespectrumof L b . Tounderstandthegeneralstructureofthespectrumofthesecondvariationof F atthebilayerdressingofan admissibleinterface,werecallthattheleadingorderstructureofthesecondvariationof F , L b ,introduced 35 Figure2.3: Thestructureoftherealspectrumof L b ,edinequation( 1.54 ),plottedversesLaplace- Beltramiwavenumber n .(left)TheSturm-Liouvilleoperator L b; 0 ,in( 1.29 ),hasonepositiveground stateeigenvalue, b; 0 A 0andaonedimensionalkernel,denoted b; 1 .(center)Theextensionof L b; 0 to L b L b; 0 "H@ z " 2 s addsside-bandsin n ,theLaplace-Beltramiindexwhichbendbacknegativelyatthe rate ‹ b; 0 " 2 k “ 2 .(right)Thespectrumoftheoperator L b L 2 b O ‹ " “ ,(minussignchosentopreserve orientationofimages)is,to O ‹ " “ ,thenegativesquareofthespectrumof L b .Theside-bandassociatedto b; 0 hasaquadratictangencyatleadingorder,whichmayberaisedorloweredbythefunctionalterms, 1 and 2 ,thecrossingofthisspectrumthroughzeroisthemechanismofthepearlinginstability.Springerandthe originalpublisher[ HayrapetyanandPromislow,2014 ],originalcopyrightnoticeisgiventothepublication inwhichthematerialwasoriginallypublished,byadding;withkindpermissionfromSpringerScienceand BusinessMedia in( 1.54 ),iscontrolledby L 2 b .Theremainingpartsof L b arerelativelyboundedandasymptoticallysmallin comparisonto L 2 b .Thespectralmappingtheoremimpliesthattheeigenvaluesof L b areapproximatelythe squareoftheeigenvaluesof L b .Figure 2.3 (right)depictstheeigenvaluesoftheoperator L b . Theeigenfunctionsassociatedto L b correspondingtoeigenvalues b;j;n @ U ,with j;n C 0,comprisetwo sets,the pearlingeigenmodes Ÿ b; 0 ;n š N 2 n N 1 andthe meandereigenmodes Ÿ b; 1 ;n š N 3 n 0 ,wheretheindex N 3 isthebiggestindexwhich b; 1 ;N 3 @ U ,andtheindices N 1 ;N 2 arechosensothat N 1 isthe indexsatisfying b; 0 ;N 1 @ U and N 2 isthebiggestindexwhich b; 0 ;N 2 @ U ,seeFigure 2.4 .The indices N i ;i 1 ; 2 ; 3areindependentof " . Figure2.4: Theeigenvaluesof L b withthelimitofthemeanderingeigenmodes, N 3 ,andthelimitsofthe pearlingeigenmodes N 2 ;N 3 . 36 For j 0 ; 1weintroduce b;j ,thesetofindices n forwhich L b actingon j n issmall,i.e., b;j Ÿ n S‹ b;j " 2 n “ O ‹ º " “š : (2.48) Weyl'sasymptoticsdeterminethesizeoftheset b; 0 ,which S b; 0 S O ‹ " 3 ~ 2 d “ Q 1.Weintroduce theco-dimensionone meandereigenspace Y b;me span Ÿ b; 1 ;n š N 3 n 0 ; (2.49) andtheco-dimensiontwo pearlingeigenspace Y b;pe span Ÿ b; 0 ;n š N 2 n N 1 : (2.50) Theco-dimensiononemorphologiesareapproximatecriticalpointsoftheFCH,howevertheymay frombothlow-frequency(meanderororhigh-frequency(pearling)instabilities.Wecharacterize themeandertypemotionthroughthebilayergeometricwinChapter 3 ,whilethepearlinginstabilityof bilayersischaracterizedinChapter 5 . 2.2Co-dimensionTwoMorphologyin R 3 Let ` R 3 beaboundeddomainandlet p ` R 3 beasmooth,closedcurve,parameterizedby ˆ p p Ÿ ˆ p ‹ s “ 0 ;L ‹ t R 3 U ˆ p ‹ 0 “ ˆ p ‹ L ‹ t ““š ; (2.51) where s denotesarc-lengthand L isthetotalcurvelength. Atagivenpointon p ,theunittangentvector T ,theprinciplenormalvector N andthebinormalvector B by T @ˆ p @s ; (2.52) N VV @ T @s VV 1 @ T @s ; (2.53) B T N ; (2.54) formtheFrenet-Serretframe.weintroducethevectors Ÿ T ; N 1 ; N 2 š which,ateachpoint ˆ p ‹ s “ onthe 37 curve p ,formanorthonormalbasisforthenormalplaneandaregivenby @ N i @s i T ;i 1 ; 2 ; (2.55) where Ñ ‹ s;t “ ‹ 1 ; 2 “ t (2.56) isthenormalcurvaturevectorwithrespectto Ÿ N 1 ; N 2 š .Thelocal T;N 1 ;N 2 coordinatesystemgivesamore naturalexpressionfortheresultinggeometricw(See[ DaiandPromislow,2015 ]forfurtherdetails). 2.7. Ford K;` A 0 thefamily G p K;` of\admissibleco-dimensiontwointerfaces"iscomprised ofsmooth,closedcurve,1-dimensionalmanifolds p embeddedin R 3 ,whicharefarfromself-intersection andhaveasmoothsecondfundamentalform. Theset p;` by p;` ı p ‹ s;z “ > R 3 U s > S; 0 B z B ` ~ " ` ; (2.57) willbecalledthe reach of p ,whereweemphasizethat ` isindependentof " . Assume p > G p;K;` isaco-dimensiontwoadmissibleinterface.Then,bytheImplicitFunctionTheoremeach point x > p;` isuniquelyexpressedusingthe whiskeredcoordinates x ‹ s;z “ ˆ p ‹ s;t “ "z 1 N 1 ‹ s;t “ "z 2 N 2 ‹ s;t “ ; (2.58) where z ‹ z 1 ;z 2 “ t isthescaledsigneddistancevectorand t > 0 ; ª representtime,seeFigure 2.5 . Figure2.5: Co-dimension2whiskeredcoordinatesin R 3 38 2.8. Let x > p;` ‹ t “ beapointonagivenwhisker w .Wethepoint b p ‹ x “ ˆ p ‹ s ‹ x;t “ ;t “ ; (2.59) tobethewhisker's basepoint ,orthebasepointassociatedto x . 2.9. Foratime-dependentfamilyofadmissiblesurfacesparameterizedby ˆ p ‹ ;t “ ,thenormal velocity V ‹ V 1 ;V 2 “ ofapoint ˆ p ‹ s;t “ on p isdby V i N i @ˆ p @t ‹ s;t “ i 1 ; 2 : (2.60) Lemma2.3 ([ DaiandPromislow,2015 ]) . The Ÿ T ; N 1 ; N 2 š coordinatesystem @ T @s 1 N 1 2 N 2 ; (2.61) whilethecurvelengthevolvesaccordingto d S p S dt S p V Ñ ds: (2.62) Lemma2.4 ([ DaiandPromislow,2015 ]) . Fix p > G p;K;` andassumethat ` iscientlysmall,so that SS Ñ SS L ª ‹ p “ @ 1 .Then,on p;` ,inthewhiskeredcoordinates,theJacobian, J p ,ofthetransforma- tion x ‹ s;z “ takestheform J p ‹ s;z “ " 2 ~ J p ; (2.63) where ~ J p ‹ 1 "z Ñ “ ; (2.64) and Ñ isdinequation( 2.56 ).Moreover,TheLaplaceoperatortakestheform x " 2 z " 1 D z @ 2 G (2.65) whereweintroducetheoperators D z Ñ ~ J © z ; (2.66) @ 2 G @ s ‰ 1 ~ J 2 @ s ’ 1 ‹ 1 "z Ñ “ 2 @ 2 s " z @ s Ñ ‹ 1 "z Ñ “ 3 @ s ; (2.67) 39 andthenormalvelocity V takestheform V 1 " @z 1 @t "z 2 N 2 @ N 1 @t ; (2.68) V 2 " @z 2 @t "z 1 N 1 @ N 2 @t : (2.69) Forfuturecalculation,wealsointroduceamorecompactversionfortheLaplacianexpansion: x " 2 z " 1 Ñ © z @ 2 s ‹ z Ñ “ Ñ © z O ‹ " “ : (2.70) Weintroducethe\co-dimeaniontwoLaplacian"operator, @ 2 s ,andassumethatitseigenvalues, Ÿ k š ª k 0 ,and itscorrespondingeigenfunctions, Ÿ k š ª k 0 ,satisfythefollowing, ‹ @ 2 s k k k , ‹ 0 0and k A 0for k A 0, ‹ theeigenfunctionsofco-dimeaniontwoLaplacianareorthonormalinthe p -innerproduct ‹ k ; j “ p S p k j dS k;j : (2.71) ‹ Weyl'sasymptoticformula,introducedin( 2.30 ),isvalidalsofortheco-dimensiontwocase.Forthis chapterwe d 3andasaresult,thenumberofeigenvalues B , N ‹ n B “ ,includingmultiplicity, satisfy N ‹ n B “ C 1 ,and n C 2 n . Note4. Byabuseofnotationwewilldropthebarsignsfrom k and k whendoingsocreatesnoconfusion with k and k introducedfortheco-dimensiononeinterfaces. 2.10. Foraradialfunction f R R whichtendstoconstantvalue f ª atanO(1)exponential rateas R ª ,wesaythatwe dresstheinterface p with f ,obtainingthe p -extendedfunction f p ‹ x “ f ‹ z ‹ x ““ ˜ ‹S r ‹ x “S~ ` “ f ª ‹ 1 ˜ ‹S r ‹ x “S~ ` ““ ; (2.72) where ` A 0 istheminimal(unscaled)distanceof p tothecompliment c p;` ofitsneighborhood p;` and ˜ R R isad,smoothfunctionwhichisoneon 0 ; 1 ,while ˜ ‹ s “ 0 for s C 2 .Byabuseof notationwewilldropthe p subscriptinthe p -extensionwhendoingsocreatesnoconfusion. Giventwofunctions f;g > L 2 ‹ “ with supp ‹ f “ ;supp ‹ g “ ` p;` wemaychangethe L 2 ‹ “ -innerproductto 40 thewhiskeredcoordinates ‹ f;g “ L 2 ‹ “ S f ‹ x “ g ‹ x “ dx S p S ` ~ " 0 f ‹ s;z “ g ‹ s;z “ J p ‹ s;z “ dzds; (2.73) wheretheJacobian, J p ,isin( 2.63 ).Fortwofunctions f;g > L 2 ‹ R 2 “ wemaychangetopolar coordinatesanddenotethecorrespondingR-weightedinnerproductby ‹ f;g “ L R S 2 ˇ 0 S ª 0 fgRdR: (2.74) Weconsiderthedressing,asedin 1.2 ,ofanadmissibleco-dimensiontwointerface, p > G p K;` , withthepore U p ,whichsolves ¢ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¤ − @ 2 @R 2 1 R @ @R ‘ U p W œ ‹ U p “ ; U p ‹ ª “ b ; @U p @R ‹ 0 “ 0 : (2.75) Tounderstandthegeneralstructureofthespectrumof F weconsiderthesecondvariationof F at U p L p L 2 p O ‹ " “ ; (2.76) whereweintroducethefulloperator L p L p "D z " 2 @ 2 G : (2.77) Tounderstandthespectrumof L p weinvestigatethespectrumofthelinearoperator L p @ 2 R 1 R @ R 1 R 2 @ 2 W œœ ‹ U p “ ; (2.78) Wethespaces Z m by Z m Ÿ f ‹ R “ cos ‹ “ g ‹ R “ sin ‹ “ T f;g > C ª ‹ 0 ; ª “ ;m > N š : (2.79) Thesespacesareinvariantundertheoperator L p ,andmutuallyorthogonalin L 2 ‹ “ .Moreover,onthese spaces L p reducesto L p ‹ f ‹ R “ cos ‹ “ g ‹ R “ sin ‹ ““ cos ‹ “ L p;m f sin ‹ “ L p;m g; (2.80) 41 where L p;m @ 2 @R 2 1 R @ @R m 2 R 2 W œœ ‹ U p “ : (2.81) Eachoperator L p;m isself-adjointinthe R -weightedinnerproduct,andtheoperator L p; 1 ,introducedin( ?? ), hasa1-dimensionalkernelspannedby @ R U p .For m A 1weobservethat ‹ L p;m f;f “ L R @ ‹ L p; 1 f;f “ L R and since L p; 1 B 0wededucethat L p;m @ 0.Inparticulartheoperator L p;m isboundedlyinvertibleforall m x 1. Theoperator L p; 0 L p; 0 p; 0 ;j p; 0 ;j p; 0 ;j : (2.82) Wedenotetheeigenfunctionsandeigenvaluesof L p;m by Ÿ p;m;j š ª j 0 and Ÿ p;m;j š ª j 0 ,respectively. Wetiateequation( 2.75 )withrespectto R toobtain L p; 1 U œ p 0 : (2.83) Equation( 2.83 )impliesthatthefunctions @ z 1 U p ;@ z 2 U p lieinker L p . Assumption2.2.1. Theoperator L p; 0 hasnokernelandithasa1-dimensionalpositiveeigenspace, i.e., p; 0 ; 0 A 0 and p; 0 ;j @ 0 forevery j C 1 . ItfollowsfromAssumption 2.2.1 that ker ‹ L p “ span Ÿ @ z 1 U p ;@ z 2 U p š span Ÿ @ R U p cos ;@ R U p sin š : (2.84) Undertheseassumptions,wecanwrite L p initsblock-matrixform L p ™ Œ Œ Œ Œ Œ Œ Œ Œ Œ Œ Œ fl L p; 0 0 0 L p; 1 fi Š Š Š Š Š Š Š Š Š Š Š Ł (2.85) wherethespectrumofeachoperator L p;m , m C 0isdescribesinFigure 2.7 .Theeigenvaluesof L p , in( 2.77 ),atleadingorder,aredescribedinFigure 2.6 ,wherethepearlingeigenvaluesarethesmalleigenvalues oftheoperator ‹ L p; 0 " 2 @ 2 s “ 2 ,seeFigure 2.6 (d). Weintroduce theco-dimensiontwo, L j 1 functions p;j ;j 1 ; 2whichsolves L j p p;j 1 ; (2.86) 42 (a) ˙ ‹ L p; 0 " 2 @ 2 s “ (b) ˙ ‹ L p; 1 " 2 @ 2 s “ (c) ˙ ‹ L p;m " 2 @ 2 s “ , m C 2 (d) ˙ ‹ L p; 0 " 2 @ 2 s “ 2 Figure2.6: FullOperatorsSpectrum (a) Spectrumof L p; 0 (b) Spectrumof L p; 1 (c) Spectrumof L p;m for m C 2 Figure2.7: Thespectrumofthesub-operators L p;m for m 0 ; 1 ; 2withtherealaxisvertical. andconvergeexponentiallytoasymptoticvalue j as R ª suchthat p;j j > ‹ ker L p “ Œ ,and is thewellcoercivity,in( 1.34 ). 2.11. Wethe scaledeigenfunctions p;k ˜ ‹ z “ J 1 ~ 2 p; 0 ;k ,where p; 0 ;k isthe k th eigenfunctionof L p; 0 and ˜ ‹ z “ isthecutfunction,din( 2.43 ) Thescaledeigenfunctionsareorthonormalinthe L 2 ‹ “ -innerproduct ‹ p;k ; p;j “ L 2 ‹ “ S p S ` … " 0 p; 0 ;k p; 0 ;j ˜ 2 ‹ z “ dzds 0 : (2.87) Note5. Any f > L 2 ‹ R 2 “ admitstheFourierexpansion f f 0 ‹ R “ ª Q m 1 ‹ f m ‹ R “ cos ‹ “ g m ‹ R “ sin ‹ ““ ; (2.88) andaslongas Ÿ f 1 ;g 1 š Œ ker L p; 1 ,wehavetheinverseformulation L 1 p f L 1 p; 0 f 0 ª Q m 1 ‹‹ L 1 p;m f m ‹ R ““ cos ‹ “ ‹ L 1 p;m g m ‹ R ““ sin ‹ ““ ; (2.89) 43 Assumption2.2.2. Weassumethattheresultsfrom[ HayrapetyanandPromislow,2014 ]holdfortheco- dimension 2 morphology. Assumption 2.2.2 ,impliesthatthereexists C A 0suchthat ˙ ‹ L p “ 9 C;C havetheleadingorderexpansion p;j;n ‹ p;j ‹ z “ n ‹ s ““ 2 O ‹ " “ (2.90) where p;j arethescaledeigenfunctionsof L p ,introducedin 2.11 ,and n aretheeigenfunctions ofco-dimeaniontwoLaplacian,andthecorrespondingeigenvaluestaketheform p;j;n ‹ p;j " 2 n “ 2 O ‹ " “ ; (2.91) where p;j aretheeigenvaluesof L p and n aretheeigenvaluesof @ 2 s . Similarlytotheco-dimensiononecase,seeFigure 2.3 (right),theeigenfunctionsassociatedto L p correspond- ingtoeigenvalues p;j;n @ U ,with j;n C 0,comprisetwosets,the pearlingeigenmodes Ÿ p; 0 ;n š N 2 n N 1 andthe meandereigenmodes Ÿ p; 1 ;n š N 3 n 0 ,wheretheindex N 3 isthebiggestindexwhich p; 1 ;N 3 @ U ,and theindices N 1 ;N 2 arechosensothat N 1 istheindexsatisfying p; 0 ;N 1 @ U and N 2 isthebiggestindex which p; 0 ;N 2 @ U ,andtheindices N i ;i 1 ; 2 ; 3areindependentof " .For j 0 ; 1weintroduce p;j ,thesetofindices n forwhich L p actingon j n issmall,i.e., p;j Ÿ n S‹ p;j " 2 n “ O ‹ º " “š : (2.92) Thesizeoftheset p; 0 followsFromWeyl'sasymptoticformula,whichimpliesthat S 0 S O ‹ " 3 ~ 2 d “ Q 1. 2.12. Thespace, X ,correspondingtothesmalleigenvaluesof L isdas X Ÿ 0 k S k > š ; (2.93) The L 2 -orthogonalprojection,onto X isgivenby f Q k > ‹ f; 0 k “ L 2 ‹ “ SS 0 k SS 2 L 2 ‹ “ 0 k Q k > ‹ f; 0 k “ L 2 ‹ “ 0 k ; (2.94) anditscomplementaryprojectionis ~ I Assumption2.2.3. Weassumethattherestrictedoperator ~ L p ~ isuniformlycoerciveon X Œ andits spectrumisboundedfrombelowby A 0 whichmaybechosenindependentofssmall " A 0 . 44 Assumption 2.2.3 impliesthat ˙ ‹ L p “~ X isstrictlypositive. Weintroducetheco-dimensiontwo meandereigenspace Y p;me span Ÿ p; 1 ;n š N 3 n 0 ; (2.95) andtheco-dimensiontwo pearlingeigenspace Y p;pe span Ÿ p; 0 ;n š N 2 n N 1 : (2.96) 45 Chapter3 GeometricEvolutionofBilayersin R d Inthischapterwederivethegeometricevolutionofadmissibleco-dimensiononeinterfacesin R d under the H 1 gradientwofthe strongFCH .Incontrasttoanalysisofsingle-layerinterfaces,multi-scale analysisshowsthattheStefanandMullins-Sekerkaproblemsforbilayersaretrivial,andthesharpinterface limityieldsasimple,quenchedmeancurvature-drivennormalvelocityatleadingorder.Toobtainthew oftheunderlyinginterfacialstructureweprojecttheresidual F u ‹ u b “ ofthecriticalpointequation,( 1.18 ), ontothe meanderingeigenspace ,in( 2.49 ). Note6. Byabuseofnotationwewilldropthe b subscriptinthe u b criticalpointwhendoingsocreatesno confusion. RecallthestrongFCHfreeenergywhichcorrespondstothechoice p 1in( 1.14 ), F S 1 2 ‹ " 2 u W œ ‹ u ““ 2 " „ " 2 1 2 S © u S 2 2 W ‹ u “‚ dx; (3.1) where ` R d ;d C 2isaboundeddomain, W ‹ u “ isatilteddouble-wellpotentialwithtwominimaat b , u R isthedensityofoneoftheamphiphilicspecies, " P 1controlsthewidthoftheboundarylayerand 1 and 2 arethefunctionalizationconstants. Thechemicalpotential, ,isasthevariationof F , F u ‹ u “ ‹ " 2 W œœ ‹ u “ 1 “‹ " 2 u W œ ‹ u ““ d W œ ‹ u “ ; (3.2) where d 1 2 .InthischapterwepresentaformalreductionofthestrongFCHequation, u t ‹ " 2 W œœ ‹ u “ 1 “‹ " 2 u W œ ‹ u ““ d W œ ‹ u “ ; (3.3) 46 forfunctions u thatareclosetoabilayerdressingofanadmissibleinterfaceinsubjecttoperiodicor boundaryconditions.WemayrewritethestrongFCHequationusingtheofthechemical potential,givenin( 3.2 ), u t (3.4) 3.1InnerandOuterExpansions Assuminganadmissibleinitialco-dimensiononeinterface b ‹ t 0 “ > G K;` .Wedescribethegeometricevolution oftheinterfaceasawintime t ,yieldingthecurve b ‹ t “ ,seeFigure 3.1 ,byperformingamulti-scaleanalysis Figure3.1: Thegeometricevolutionofageneric,admissible,co-dimensiononeinterface, b ‹ t 0 “ isthe initialinterfaceand b ‹ t 1 “ describestheinterfaceatalatertime t 1 A t 0 . ofthesolution u andthechemicalpotential .Awayfromtheinterface b ,inthe ~ b;` ,theouter solution u andtheouterchemicalpotential havetheexpansions u ‹ x;t “ u 0 ‹ x;t “ "u 1 ‹ x;t “ " 2 u 2 ‹ x;t “ O ‹ " 3 “ ; (3.5) ‹ x;t “ 0 ‹ x;t “ 1 ‹ x;t “ " 2 2 ‹ x;t “ O ‹ " 3 “ : (3.6) Inthereach b;` ,atatime-scale ˝ ,wehavetheinnerspatialexpansions u ‹ x;t “ ~ u ‹ s;z;˝ “ ~ u 0 ‹ s;z;˝ “ " ~ u 1 ‹ s;z;˝ “ " 2 ~ u 2 ‹ s;z;˝ “ O ‹ " 3 “ ; (3.7) ‹ x;t “ ~ ‹ s;z;˝ “ ~ 0 ‹ s;z;˝ “ " ~ 1 ‹ s;z;˝ “ " 2 ~ 2 ‹ s;z;˝ “ O ‹ " 3 “ : (3.8) Thenormalvelocity V n of b atapoint s ‹ t “ isby V ˝ ‹ s “ @r @t " 1 @z @t : (3.9) 47 where r isthesigneddistanceawayfrom b and z r " isthescaleddistance.Wedevelopanexpressionfor thetimederivativeoftheouterdensityfunction~ u ,in( 3.7 ),usingthewhiskeredcoordinates.Tothis end,wetreat s and z asfunctionsof t andapplythechainruletoobtain @ ~ u @t * 0 @s @t © s ~ u @ ~ u @z @z @t @ ~ u @˝ @˝ @t ; (3.10) wherethetermontheright-handsideof( 3.10 )iszerosincewemayreparameterizetheevolvedcurve locally. Pluggingthenormalvelocity,( 3.9 ),into( 3.10 )yields @ ~ u @t " 1 V ˝ ‹ s “ @ ~ u @z @ ~ u @˝ @˝ @t : (3.11) 3.2MatchingConditions Weconnecttheinnerandoutersolutionsviamatchingconditionsacrosstheinner-outerboundary.We formallyexpandtheoutersolution u ‹ x;t “ givenin( 3.5 )andtheinnersolution~ u ‹ s;z;˝ “ givenin( 3.7 ).Fix awhisker, w ,andlet x > b beitsbasepoint,see 2.4 ,suchthat x hn > w .Then,thematching conditioncanbewrittenas lim h 0 u ‹ x hn;t “ lim z ª ~ u ‹ s;z;˝ “ ; (3.12) seeFigure 2.1 .Anexpansionofthelefthandsideofequation( 3.12 )around x ,as h 0 ,isgivenby u 0 ‹ x;t “ " ‹ u 1 ‹ x;t “ z@ n u 0 ‹ x;t ““ " 2 ‹ u 2 ‹ x;t “ z@ n u 1 ‹ x;t “ z 2 @ 2 n u 0 ‹ x;t ““ O ‹ " 3 “ ; (3.13) where @ n isthederivativeinthenormaldirectionof b ,and u i areas u i lim h 0 u i ‹ x hn;t “ ; (3.14) forall i C 0.Wecanobtainsimilarexpressionas h 0 .Using( 3.7 )toexpandtherighthandsideof equation( 3.12 )andmatchingittothelefthandside,( 3.13 ),yieldsthefollowingmatchingconditions u 0 lim z ~ u 0 ; (3.15) u 1 z@ n u 0 lim z ~ u 1 : (3.16) 48 Similarly,wecanobtainmatchingconditionsforthechemicalpotential 0 lim z ~ 0 ; (3.17) 1 z@ n 0 lim z ~ 1 ; (3.18) 2 z@ n 1 1 2 z 2 @ 2 n 0 lim z ~ 2 ; (3.19) 3 z@ n 2 1 2 z 2 @ 2 n 1 1 6 z 3 @ 2 n 0 lim z ~ 3 : (3.20) 3.3Expansionofthechemicalpotential Wewillalsohaverecoursetotheinnerandouterexpansionsofthechemicalpotential › " 2 W œœ ‹ u “ 1 ”› " 2 u W œ ‹ u “ ” d W œ ‹ u “ : (3.21) 3.3.1OuterExpansionoftheChemicalPotential Atagiventimescale ˝ ,theouterexpansionforthedensityfunction u ‹ x;t “ isgivenbyequation( 3.5 ). Plugging( 3.5 )into( 3.21 )andrewritingthechemicalpotential inordersof " yields ‹ x;t “ 0 ‹ x;˝ “ 1 ‹ x;˝ “ " 2 2 ‹ x;˝ “ :::; (3.22) where 0 W œœ ‹ u 0 “ W œ ‹ u 0 “ ; (3.23) 1 ‹ W œœœ ‹ u 0 “ u 1 1 “ W œ ‹ u 0 “ ‹ W œœ ‹ u 0 ““ 2 u 1 d W œ ‹ u 0 “ ; (3.24) 2 ‰ W œœœ ‹ u 0 “ u 2 1 2 W ‹ 4 “ ‹ u 0 “ u 1 ’ W œ ‹ u 0 “ ‹ W œœœ ‹ u 0 “ u 1 1 “ W œœ ‹ u 0 “ u 1 (3.25) W œœ ‹ u 0 “‰ u 0 W œœ ‹ u 0 “ u 2 1 2 W œœœ ‹ u 0 “ u 2 1 ’ d W œœ ‹ u 0 “ u 1 : SeeAppendix B.1 forfurthercalculationdetails. 49 3.3.2InnerExpansionoftheChemicalPotential Atagiventimescale ˝ ,theinnerexpansionforthedensityfunction u ‹ x;t “ isgivenbyequation( 3.7 ),and inlocalcoordinates,theLaplacianoperator,see( 2.9 ),takestheform " 2 x @ 2 z "H 0 @ z " 2 zH 1 @ z " 2 G O ‹ " 3 “ ; (3.26) where H i arein( 2.10 ).Plugging( 3.7 )and( 3.26 )into( 3.21 ),wecanrewritethechemicalpotential inordersof " ‹ x;t “ ~ 0 ‹ s;z;˝ “ " ~ 1 ‹ s;z;˝ “ " 2 ~ 2 ‹ s;z;˝ “ " 3 ~ 3 ‹ s;z;˝ “ O ‹ " 4 “ ; (3.27) where ~ 0 ‹ @ 2 z W œœ ‹ ~ u 0 ““‹ @ 2 z ~ u 0 W œ ‹ ~ u 0 ““ ; (3.28) ~ 1 ‹ @ 2 z W œœ ‹ ~ u 0 ““‹ H 0 @ z ~ u 0 @ 2 z ~ u 1 W œœ ‹ ~ u 0 “ ~ u 1 “ (3.29) ‹ H 0 @ z W œœœ ‹ ~ u 0 “ ~ u 1 1 “‹ @ 2 z ~ u 0 W œ ‹ ~ u 0 ““ d W œ ‹ ~ u 0 “ ; ~ 2 ‹ @ 2 z W œœ ‹ ~ u 0 ““‹ @ 2 z ~ u 2 zH 1 @ z ~ u 0 H 0 @ z ~ u 1 s ~ u 0 W œœ ‹ ~ u 0 “ ~ u 2 1 2 W œœœ ‹ ~ u 0 “ ~ u 2 1 “ (3.30) ‹ H 0 @ z W œœœ ‹ ~ u 0 “ ~ u 1 1 “‹ @ 2 z ~ u 1 H 0 @ z ~ u 0 W œœ ‹ ~ u 0 “ ~ u 1 “ ‹ zH 1 @ z s W œœœ ‹ ~ u 0 “ ~ u 2 1 2 W ‹ 4 “ ‹ ~ u 0 “ ~ u 2 1 “‹ @ 2 z ~ u 0 W œ ‹ ~ u 0 ““ d W œœ ‹ ~ u 0 “ ~ u 1 ; ~ 3 ‹ @ 2 z W œœ ‹ ~ u 0 ““‰ L ~ u 3 H 0 @ z ~ u 2 zH 1 @ z ~ u 1 s ~ u 1 1 ~ u 0 W œœœ ‹ ~ u 0 “ ~ u 1 ~ u 2 1 6 W ‹ 4 “ ‹ ~ u 0 “ ~ u 3 1 ’ (3.31) ‹ H 0 @ z W œœœ ‹ ~ u 0 “ ~ u 1 1 “‰ L ~ u 2 H 0 @ z ~ u 1 zH 1 @ z ~ u 0 s ~ u 0 1 2 W œœœ ‹ ~ u 0 “ ~ u 2 1 ’ ‰ zH 1 @ z s W œœœ ‹ ~ u 0 “ ~ u 2 1 2 W ‹ 4 “ ‹ ~ u 0 “ ~ u 2 1 ’‹ L ~ u 1 H 0 @ z ~ u 0 “ ‰ 1 W œœœ ‹ ~ u 0 “ ~ u 3 W ‹ 4 “ ‹ ~ u 0 “ ~ u 1 ~ u 2 1 6 W ‹ 5 “ ‹ ~ u 0 “ ~ u 3 1 ’‹ @ 2 z ~ u 0 W œ ‹ ~ u 0 ““ d ‰ W œœ ‹ ~ u 0 “ ~ u 2 1 2 W œœœ ‹ ~ u 0 “ ~ u 2 1 ’ : SeeAppendix B.3 forfurthercalculationdetails. 50 3.4Timescale ˝ t :aGradientFlow WestartbylookingforapproximationsofthesolutionsofthestrongFCHequation u t x ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ ‹ " 2 W œœ ‹ u “ 1 “‹ " 2 u W œ ‹ u ““ d W œ ‹ u “ in ; (3.32) forthetimescale ˝ t . 3.4.1Outerexpansion Awayfromtheinterface,pluggingtheouterexpansionforthedensityfunction u ‹ x “ andtheouterexpansion ofthechemicalpotential ,givenin( 3.5 )and( 3.22 ),respectively,intothestrongFCHequation,( 3.32 ), yields,atleadingorder, O ‹ 1 “ , @u 0 @˝ ‹ W œœ ‹ u 0 “ W œ ‹ u 0 ““ in 8 : (3.33) Thissecondorderproblemhasboundaryconditionsonbuttosolveitwealsoneedboundaryconditions on b .Thisleadsustotheinnerexpansion. 3.4.2Innerexpansion Weexpresseachofthetermsin( 3.32 )usingthewhiskeredcoordinates.Pluggingtheinnerexpansionof u , givenin( 3.7 ),intotheleft-handsideofequation( 3.32 )yields u t " 1 V ˝ ‹ s “ @ ~ u 0 @z V ˝ ‹ s “ @ ~ u 1 @z @ ~ u 0 @˝ O ‹ " “ ; (3.34) seeAppendix B.2 forcalculationsdetails.AnexpandexpressionoftheLaplacianoperatorinthewhiskered coordinatesisgivenin( 3.26 )andanexpressionfortheinnerexpansionofthechemicalpotentialisgiven in( 3.27 ).Plugging( 3.34 ),( 3.26 )and( 3.27 )backintotheevolutionequation( 3.32 )andcomparingorders of " yields,atleadingorder, O ‹ " 2 “ , 0 @ 2 z ~ 0 in b;` ; (3.35) andatthenextorder, O ‹ " 1 “ ,wehave V ˝ ‹ s “ @ z ~ u 0 @ 2 z ~ 1 H 0 @ z ~ 0 in b;` : (3.36) 51 Considertheleadingorderequation,( 3.35 ),andrecallthat~ 0 isrelatedto~ u 0 through( 3.28 ).Then, equation( 3.35 )hasthesolution~ u 0 U b ‹ z “ where U b isthehomoclinicin( 2.37 ).Forthis choiceof~ u 0 itfollowsthat~ 0 0andthat~ 1 ,in( 3.29 ),takestheform ~ 1 L 2 b; 0 ~ u 1 d W œ ‹ U b “ ; (3.37) wherethelinearoperator L b; 0 isin( 2.39 ).Moreover,thenextorderequation,( 3.36 ),reducesto V ˝ ‹ s “ U œ b @ 2 z ~ 1 in b;` ;: (3.38) 3.4.3JumpConditionsontheOuterSolution:GradientFlow Anouterapproximationof( 3.32 )isgivenin( 3.33 )whichisoneachdomain and .Wewould liketosolve( 3.33 )andtoconnectthetwooutersolutiontoobtainasolutionovertheentiredomainTo thisend,weusetheinnerapproximationof( 3.32 )giveninequation( 3.38 )andthematchingconditionsfrom Section 3.2 toobtainsuitablejumpconditionsovertheinterface b . Motivatedby 2.3 oftheinterfacialjump,weintegrateequation( 3.38 )withrespectto z from to ª obtaining V ˝ ‹ s “ ™ Œ Œ fl 0 ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ U b ‹ ª “ U b ‹ “ fi Š Š Ł @ z ~ 1 U z ª @ z ~ 1 U z : (3.39) Since U b isahomoclinicorbit,equation( 3.39 )leadstothetwokeyidentities lim z ª ~ u 0 ‹ z “ lim z ~ u 0 ‹ z “ lim z ª U b ‹ z “ lim z U b ‹ z “ 0 ; (3.40) lim z ª @ z ~ 1 ‹ z “ lim z @ z ~ 1 ‹ z “ 0 : (3.41) tiatingthematchingcondition( 3.18 )withrespectto z yields lim z @ z ~ 1 ‹ z “ @ n 0 ; (3.42) andthecombinationofequation( 3.41 )andequation( 3.42 )impliesthatthenormalderivativeoftheouter chemicalpotentialiscontinuousacrosstheinterface b .Similarly,combiningthematchingcondition( 3.15 ) andequation( 3.40 )weconcludethattheouterdensityfunction, u 0 ,iscontinuousovertheinterface.we 52 summarizetheseresultsinthe jumpconditions ontheoutersolution J u 0 K 0 ; (3.43) J @ n 0 K 0 : (3.44) Combiningthejumpcondition( 3.43 )withtheouterequation( 3.33 )impliesthat u 0 isasolutionof( 3.33 ) overtheentiredomainandthat u 0 canbesolvedindependentlyof b atthisorder.Theresultingequation for u 0 is @u 0 @t ‹ W œœ ‹ u 0 “ W œ ‹ u 0 ““ in ; (3.45) subjecttotheboundaryconditions.Thisevolutionequationhasisamasspreserving H 1 gradientowon thereducedenergy F 0 ‹ u 0 “ S 1 2 ‹ W œ ‹ u 0 ““ 2 dx: (3.46) Considerinitialdataoftheform u 0 b v 0 where SS v 0 SS L 2 ‹ “ P 1,andtracktheevolutionof v ‹ t “ u ‹ t “ b . Plugging u b v intoequation( 3.45 )yieldsthelinearevolutionequationfor v v t 2 „ v W œœœ ‹ b “ v 2 O ‹ v 3 “‚ ; (3.47) where W œœ ‹ b “ isthewell-coercivityconstant.If SS v 0 SS L 2 ‹ “ istlysmall,thenaslongas SS v SS L 2 ‹ “ remainssmallitisplausiblethatthedynamicsofthenonlinearsystem( 3.45 )areprimarilygovernedbythose ofthelinearsystem v t 2 v; (3.48) anditisreasonabletoexpectthatfor u 0 closetotheequilibria. Forsimplicityofpresentation,weassumethatatleading-ordertheinitialvalue u 0 ‹ t 0 “ b ; (3.49) where b isthespatialconstant,whichisanequilibriatoequation( 3.45 ). Returnto( 3.38 )andnotethat U œ b ^ U œ b where ^ U b isgivenby ^ U b U b b ; (3.50) and ^ U b enjoystheproperty ^ U b Ð 0as z .Toobtainanexpressionforthenormalvelocity,we 53 integrate( 3.38 )twicew.r.t z from0to z andsolvefor~ 1 toobtain ~ 1 ‹ z “ V ˝ ‹ s “ S z 0 ^ U b ‹ w “ dw z ‰ V ˝ ‹ s “ ^ U b ‹ 0 “ @ z ~ 1 ‹ z “U z 0 ’ ~ 1 ‹ 0 “ : (3.51) Furthermore,integratingfrom z to z 0yieldstheexpression V ˝ ‹ s “ ^ U b ‹ 0 “ @ z ~ 1 ‹ z “U z 0 lim z @ z ~ 1 ‹ z “ @ z ~ 1 ‹ z “U z 0 @ n 0 ‹ z “ ; (3.52) for V ˝ ‹ s “ ^ U b ‹ 0 “ ,wherethesecondequalityfollowsfromthematchingcondition( 3.18 ).Since u 0 b wehave 0 W œœ ‹ b “ W œ ‹ b “ 0 ; (3.53) andequation( 3.52 )furtherreducesto V ˝ ‹ s “ ^ U b ‹ 0 “ @ z ~ 1 ‹ z “U z 0 : (3.54) Using( 3.54 )toreplace V ˝ ‹ s “ ^ U b ‹ 0 “ inequation( 3.51 )yieldsanexpressionfortheinnerchemicalpotential ~ 1 ‹ z “ V ˝ ‹ s “ S z 0 ^ U b ‹ z “ dz ~ 1 ‹ 0 “ : (3.55) Recallthatequation( 3.37 )relates~ 1 to~ u 1 .Plugging( 3.37 )into( 3.55 )andsolvingfor L 2 b; 0 ~ u 1 yields L 2 b; 0 ~ u 1 V ˝ ‹ s “ S z 0 ^ U b ‹ w “ dw ~ 1 ‹ 0 “ d W œ ‹ U b “ : (3.56) BytheFredholmAlternative,see[ Grisvard,1985 ],thisequationhasasolution~ u 1 > L 2 ‹ R “ ifandonlyifthe right-handsideisperpendiculartoker L b; 0 .Thesolvabilityconditionexpressedas S R ‰ V ˝ ‹ s “ S z 0 ^ U b ‹ w “ dw ~ 1 ‹ 0 “ d W œ ‹ U b “’ U œ b dz 0 : (3.57) Since U œ b isanoddfunctionitisorthogonaltoconstantswhichimpliesthattheintegralinvolving~ 1 ‹ 0 “ is zero.Forthe d term,weevaluatetheintegralto S R W œ ‹ U b “ U œ b dz S R ‹ W ‹ U b ““ œ dz W ‹ U b “U ª 0 ; (3.58) wherethelastinequalityfollowsfromthefactthat U b b as z .Finally,Integratingthesecond 54 integralinthe V ˝ ‹ s “ termin( 3.57 )bypartstoobtaintheequality V ˝ ‹ s “ S R S z 0 U b ‹ w “ dw ^ U œ b ‹ z “ dz V ˝ ‹ s “ TT ^ U b TT 2 L 2 ‹ R “ : (3.59) Thesecalculations,combinedwiththesolvabilitycondition( 3.57 ),yieldtheresult V ˝ ‹ s “ TT ^ U b TT 2 L 2 ‹ R “ 0 ; (3.60) andsince TT ^ U b TT 2 L 2 ‹ R “ x 0,itfollowsthatthenormalvelocity V ˝ 0atthistimescale. 3.5Timescale ˝ "t :MeanCurvatureDrivenFlow Usingtheinnerequationsweobtainjumpconditionsontheoutersolutionovertheinterfaceandanexpression forthenormalvelocityoftheinterface.WewillseethatthereducedsystemisatrivialMullins-Sekerka typesystemandthenormalvelocityisdrivenbyacurvature-typew.Finally,weusethemasspreserving propertyofthesystemtoobtainthecoupledsystemforthenormalvelocity, V ˝ ,andtheexternalchemical potential, 1 . 3.5.1OuterExpansion Awayfromtheinterface,theouterexpansionofthedensityfunction u isgivenin( 3.5 ).Atthistime scale, ˝ "t ,thetimederivative @ t expandsas u t "u 0 ;˝ " 2 u 1 ;˝ O ‹ " 3 “ : (3.61) Plugging( 3.61 )andtheouterexpansionofthechemicalpotential, ,givenin( 3.22 ),intothestrongFCH equation,( 3.32 ),andcomparingordersof " yields,atleadingorder, O ‹ 1 “ , 0 ‹ W œœ ‹ u 0 “ W œ ‹ u 0 ““ in 8 ; (3.62) andatthenextorder, O ‹ " “ , u 0 ;˝ › ‹ W œœœ ‹ u 0 “ u 1 1 “ W œ ‹ u 0 “ ‹ W œœ ‹ u 0 ““ 2 u 1 d W œ ‹ u 0 “ ” in 8 : (3.63) 55 Theequation,( 3.62 ),isconsistentwithourassumptionthat u 0 b andthesecondequation( 3.63 ) reducesto 0 2 u 1 in 8 ; (3.64) and A 0isthewell-coercivityin( 1.34 ).Thissecondorderproblemhasboundaryconditionson, whichwesupplementwithmatchingconditionsontheinnerboundary b thataredevelopedinthenext section. 3.5.2InnerExpansion Weexpresseachofthetermsin( 3.32 )ininnercoordinates.Pluggingtheinnerexpansionof u ,givenin( 3.7 ), intotheleft-handsideofequation( 3.32 )yields u t @r @˝ @ z ~ u 0 O ‹ " “ ; (3.65) seeAppendix B.2 forcalculationdetails.AnexpandexpressionoftheLaplacianinlocalcoordinatesis givenin( 3.26 )andanexpressionfortheinnerexpansionofthechemicalpotentialisgivenin( 3.27 ).Plug- ging( 3.65 ),( 3.26 )and( 3.27 )backintotheevolutionequation( 3.32 )andcomparingordersof " yields,at leadingorder, O ‹ " 2 “ , 0 @ 2 z ~ 0 in b;` ; (3.66) atthenextorder, O ‹ " 1 “ , 0 @ 2 z ~ 1 H 0 @ z ~ 0 in b;` ; (3.67) andat O ‹ 1 “ wehave V ˝ ‹ s “ @ z ~ u 0 @ 2 z ~ 2 H 0 @ z ~ 1 zH 1 @ z ~ 0 G ~ 0 in b;` ; (3.68) where~ 0 ,~ 1 and~ 2 arein( 3.28 ),( 3.29 )and( 3.30 ),respectively. Equation( 3.66 )isconsistentwithourchoice~ u 0 U b ,where U b isthedressingoftheinterfacewiththe bilayersolution,denedin( 2.37 ).Thischoiceimpliesthat~ 0 0.Moreover,forthischoiceof~ u 0 thenext ordersequationsreduceto 0 @ 2 z ~ 1 in b;` ; (3.69) V ˝ ‹ s “ @ z ~ u 0 @ 2 z ~ 2 H 0 @ z ~ 1 in b;` ; (3.70) 56 3.5.2.1SolvingEquation( 3.69 ) Equation( 3.69 )isasecondorderPDEwhichwewanttosolvefor~ 1 .Integratingequation( 3.69 )twice, w.r.t z ,yields ~ 1 ~ A 1 z ~ B 1 : (3.71) Thematchingcondition( 3.18 )impliesthat @ z ~ 1 @ n 0 0as z Ð 0,which,togetherwith( 3.69 )implies that ~ A 1 0andthat~ 1 isindependentof z ,i.e., ~ 1 ~ 1 ‹ s;t “ : (3.72) Since~ u 0 U b wecansimplifytheinnerexpressionfor~ 1 in( 3.29 )andsolvefor L 2 b; 0 ~ u 1 toobtain L 2 b; 0 ~ u 1 ~ 1 d W œ ‹ U b “ : (3.73) BytheFredholmAlternative,thisequationhasasolution~ u 1 > L 2 ‹ R “ ifandonlyiftheright-handsideisper- pendiculartoker L b; 0 .Recallthatker L b; 0 U œ b anditisoddaboutz=0,seethediscussionregarding ˙ ‹ L b; 0 “ inChapter 2.1 .Thefactthattheright-handsideofequation( 3.73 )inperpendicularto U œ b followsfromthe factsthat~ 1 isconstantin z ,and W œ ‹ U b “ iseven.Consequently,thereexistasolution~ u 1 denoted ~ u 1 ~ 1 b; 2 d L 2 b; 0 W œ ‹ U b “ ; (3.74) where b; 2 solves L 2 b; 0 b; 2 1,edin( 2.40 ).Since W œ ‹ U b “ @ 2 z U b ,usingidentity( B.43 ),wecan rewrite~ u 1 ~ u 1 ~ 1 b; 2 d L 1 b; 0 ‰ z 2 U œ b ’ ; (3.75) or,alternatively, L b; 0 ~ u 1 ~ 1 b; 1 d z 2 U œ b : (3.76) Notethatsince~ 1 isindependentof z ,thenextorderapproximationof( 3.32 ),equation( 3.70 ),reducesto V ˝ ‹ s “ @ z ~ u 0 @ 2 z ~ 2 in b;` : (3.77) 3.5.2.2JumpConditions Wearelookingforasolutionof( 3.32 )inAnouterapproximationof( 3.32 )isgivenin( 3.64 )whichis oneachdomain, and .Wewouldliketosolve( 3.64 )andtoconnectthetwooutersolution 57 toobtainasolutionovertheentiredomainWeobtainthejumpconditionoftheexternalchemical potential 1 overtheinterfacefrom( 3.100 )andthematchingcondition( 3.18 ):since~ 1 isindependentof z wehave lim z ~ 1 ~ 1 ‹ s;t “ 1 ; (3.78) andthejumpconditiontakestheform J 1 K 0 : (3.79) Toobtainasecondjumpconditionon 1 ,weturntothe 2.3 oftheinterfacialjump,weintegrate equation( 3.77 )withrespectto z from to ª toobtain V ˝ ‹ s “ ™ Œ Œ Œ fl 0 ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ ^ U b ‹ ª “ ^ U b ‹ “ fi Š Š Š Ł lim z ª @ z ~ 2 lim z @ z ~ 2 (3.80) Since U b isahomoclinicorbit,equation( 3.80 )leadstothekeyidentity lim z ª @ z ~ 2 ‹ z “ lim z @ z ~ 2 ‹ z “ 0 : (3.81) tiatingthematchingcondition( 3.19 )withrespectto z yields lim z @ z ~ 2 ‹ z “ @ n 1 ; (3.82) andthecombinationofequation( 3.81 )andequation( 3.82 )impliesthatthenormalderivativeoftheouter chemicalpotentialiscontinuousacrosstheinterface b .wesummarizetheseresultsinthe jumpconditions ontheoutersolution J 1 K 0 ; (3.83) J @ n 1 K 0 : (3.84) 3.5.2.3TheNormalVelocityat ˝ "t Wewouldliketodeterminetheevolutionoftheinterface b .Tothisend,recallequation( 3.77 ),which involvesthenormalvelocity V ˝ ,andtheinnerchemicalpotential~ 2 .Theof~ 2 isgivenin( 3.30 ), andsince~ u 0 U b ,( 3.30 )reducesto ~ 2 L 2 b; 0 ~ u 2 L b; 0 R ‹ H 0 @ z W œœœ ‹ U b “ ~ u 1 1 “‹ L b; 0 ~ u 1 H 0 U œ b “ d W œœ ‹ U b “ ~ u 1 ; (3.85) 58 where R zH 1 U œ b H 0 ~ u œ 1 1 2 W œœœ ‹ U b “ ~ u 2 1 : (3.86) Inordertogetanexpressionforthenormalvelocitywesolve( 3.77 )for~ 2 ,byintegrating( 3.77 )twicew.r.t z from0to z toobtain ~ 2 ‹ z “ ~ 2 ‹ 0 “ z ‹ @ z ~ 2 ‹ 0 “ V ˝ ‹ s “ ^ U b ‹ 0 “ “ V ˝ ‹ s “ S z 0 ^ U b ‹ w “ dw: (3.87) Furthermore,integratingfrom z to z 0andrecallingthat ^ U b 0as z ,yieldstheexpression V ˝ ‹ s “ ^ U b ‹ 0 “ lim z @ z ~ 2 @ z ~ 2 ‹ 0 “ @ n 1 @ z ~ 2 ‹ 0 “ : (3.88) for V ˝ ‹ s “ ^ U b ‹ 0 “ ,wherethesecondequalityfollowsfromthematchingcondition( 3.19 ).Using( 3.88 )to replace V ˝ ‹ s “ ^ U b ‹ 0 “ inequation( 3.87 )yields ~ 2 ‹ z “ ~ 2 ‹ 0 “ z@ n 1 V ˝ ‹ s “ S z 0 ^ U b ‹ w “ dw: (3.89) Replacing~ 2 in( 3.89 )withitsexpressionfrom( 3.85 )andsolvingfor L 2 b; 0 ~ u 2 yields L 2 b; 0 ~ u 2 L b; 0 R ‹ H 0 @ z W œœœ ‹ U b “ ~ u 1 1 “‹ L b; 0 ~ u 1 H 0 U œ b “ d W œœ ‹ U b “ ~ u 1 ~ 2 ‹ 0 “ z@ n 1 V ˝ ‹ s “ S z 0 ^ U b ‹ w “ dw (3.90) BytheFredholmAlternative,thisequationhasasolution~ u 2 > L 2 ‹ R “ ifandonlyiftheright-handsideis perpendiculartoker L b; 0 .Recallthatker L b; 0 U œ b andconsidertheinnerproductof U œ b withtheright-hand sideofequation( 3.90 ).Since~ u 1 ,in( 3.74 ),isevenandtheoperator L b; 0 preservessymmetry,parity considerationsshowthattheFredholmconditionreducesto ‰ H 0 @ z L b; 0 ~ u 1 H 0 W œœœ ‹ U b “ ~ u 1 U œ b 1 H 0 U œ b z@ n 1 V ˝ ‹ s “ S z 0 ^ U b ‹ w “ dw;U œ b ’ L 2 ‹ R “ 0 : (3.91) Simplifyingtheintegralsintheinnerproduct,andsolvingforthenormalvelocity V ˝ yieldstheexpression V ˝ ‹ s “ ‹ H 0 1 @ n 1 “ m b 1 2 H 0 ‹ 1 2 “ ˙ b B 1 ; (3.92) 59 where m b isthemassamphiphilicmaterialperunitlengthofbilayer,in( 1.50 ),andweintroduce theconstants B 1 TT ^ U b TT 2 L 2 ; (3.93) ˙ b TT ^ U œ b TT 2 L 2 : (3.94) Detailesofthecalculationsleadingtoequation( 3.92 )canbefoundinAppendix B.4 . 3.5.3Sharpinterfacelimit:TrivialMullins-SekerkaandCurvatureDriven Flow Theprecedingcalculationshowthat,inaneighborhoodofthedressedsolution,the ˝ "t timescaleevolution of( 3.32 )reducestoaMullins-Sekerkasystemfortheunknownexternalchemicalpotential, 1 . ‹ 3 : 64 “ 1 0in 8 ; (3.95) ‹ 3 : 83 “ J 1 K 0 ; (3.96) ‹ 3 : 84 “ J @ n 1 K 0 ; (3.97) ‹ 3 : 92 “ V ˝ ‹ s “ ‹ H 0 1 @ n 1 “ m 1 2 H 0 ‹ 1 2 “ ˙ b B 1 on : (3.98) TheMullins-Sekerkasystem( 3.95 - 3.97 )istrivialbecausethejumpinthenormalderivativeof 1 balances againstthejumpofthe z derivativeoftheinnerchemicalpotential~ 1 acrosstheinnerstructure.Thislater quantityiszeroastheunderlyingishomoclinic.Equations( 3.95 - 3.97 )implythat 1 0in ; (3.99) andsubjecttotheboundaryconditionson @ itfollowsfromthemaximumprinciple,[ Evans,2010 ],that 1 isspatiallyconstant,i.e., 1 ‹ x;˝ “ 1 ‹ ˝ “ ¦ x > : (3.100) Moreover,fromequation( 3.100 ),whichimpliesthat 1 isspatiallyconstant,andthejumpcondition( 3.97 ), weconcludethat 1 iscontinuousovertheinterface,and ~ 1 ‹ ˝ “ 1 ‹ ˝ “ : (3.101) 60 Combining( 3.99 )and( 3.100 ),weconcludethat @ n 1 0on b ,andtheexpressionforthenormalveloc- ity,( 3.98 ),reducestoacurvaturedrivenexpressioncoupledtothespatiallyconstantchemicalpotential 1 V ˝ ‹ s “ 1 m b 1 2 ‹ 1 2 “ ˙ b B 1 H 0 on b : (3.102) 3.5.4Equilibriaestimatefortimescale ˝ "t Theexternalchemicalpotential 1 ischaracterizedbythedensityfunction, u ,whosevalueis determinedbyconservationoftotalmass.Forthistimescale,wesummarizeourapproximationforthe densityfunctionineachregion.Intheouterregion ~ b;` … b;` ,ourassumptionthat u 0 b combined withequation( 3.24 )yields u ‹ x;t “ b " 1 2 O ‹ " 2 “ in ~ b;` ; (3.103) where W œœ ‹ b “ A 0.Inthereach, b;` ,ourchoice~ u 0 U b combinedwithequation( 3.74 )yields u ‹ x;t “ U b " ‹ 1 b; 2 d L 2 b; 0 W œ ‹ U b ““ O ‹ " 2 “ in b;` : (3.104) Weusemassbalancetodetermine 1 andtoobtainthecoupled 1 ;V ˝ systemevolution.Thetotalmassof thesystemisgivenby M S u ‹ x;t “ b dx S u ‹ x; 0 “ b dx; (3.105) whichisbytheinitialdata.Insertingtheexpressionsforthedensityfunction,givenin( 3.103 ) and( 3.104 ),intothetotalmassyields M " S ~ b;` 1 2 dx S b;` ^ U b " ‹ 1 b; 2 d L 2 b; 0 W œ ‹ U b ““ dx O ‹ " 2 “ : (3.106) Weassumethat S S O ‹ 1 “ ,andchangetothewhiskeredcoordinatesinthelocalizedintegraltoobtain M " „S S 1 2 S b S ` ~ " ` ~ " ^ U b dzds ‚ O ‹ " 2 “ : (3.107) Weexpand M " ^ M O ‹ " 2 “ andthesurfacearea S b S 0 1 O ‹ " 2 “ ,evaluatetheintegralsinequa- tion( 3.107 )andcomparingordersof " yields,atleadingorder, ^ M S S 1 2 0 m b ; (3.108) 61 where m b isin( 1.50 ).Moreover,solvingfor 1 yields 1 2 S S › ^ M 0 m ” : (3.109) Ontheotherhand,equation( 2.36 )impliesthat,subjecttothenormalvelocityattimescale ˝ "t ,the interfacialsurfaceareagrowthisgivenby @ S b S @˝ S V ˝ ‹ s “ H 0 ‹ s “ ds; (3.110) sothat,subjectto( 3.98 )theinterfacehastheleadingordergrowth d d˝ 0 1 m b 1 2 ‹ 1 2 “ ˙ b B 1 S H 2 0 ‹ s “ ds; (3.111) where B 1 and ˙ b arein( 3.93 )and( 3.94 ),respectively.Takingthetimederivative, d d˝ ,ofequa- tion( 3.109 ),solvingfor d d˝ 0 andpluggingtheresultexpressioninto( 3.111 )yields d d˝ ^ M S S 1 2 m b 1 m b 1 2 ‹ 1 2 “ ˙ b B 1 S H 2 0 ‹ s “ ds; (3.112) andwearriveattheODEforthechemicalpotential d d˝ 1 „ 1 m 2 b 2 S S B 1 1 2 ‹ 1 2 “ ˙ b m b 2 S S B 1 ‚ S H 2 0 ‹ s “ ds: (3.113) Theseresultsshowthattheevolutionoftheinterfaceisgovernedbythecoupledsystem ‹ 3 : 102 “ V ˝ ‹ s “ 1 m b 1 2 ‹ 1 2 “ ˙ b B 1 H 0 ; (3.114) ‹ 3 : 113 “ d d˝ 1 m b 2 S S B 1 ‰ 1 m b 1 2 2 ˙ b ’ S H 2 0 ‹ s “ ds: (3.115) The H 1 gradientwdrivespurebilayerinterfacesbyaquenchedmean-curvaturew.Whilethew drivestheexternalchemicalpotentialtoitsequilibriavalue 1 Ð 1 2 ‹ 1 2 “ ˙ b m b ; (3.116) thesignoftheright-handsideof( 3.114 ),determinedbyinitialdata,istial.Iftheright-handsideis positive,motion against meancurvatureleadstheinterfacialareatogrowuncontrollably,andthereduced 62 geometricwisill-posed.However,thesystemisalocallywell-posedmotionbymean-curvaturewfor right-handsideisnegative.Whilemean-curvaturedrivenwscanexhibitsingularities,inthe quenchedwthesingularitycanbearrestedbythedecayof 1 toitsequilibriavalue.Since 1 2 u 1 ,the densityfunctiondecaysto u 1 Ð 1 2 ‹ 1 2 “ ˙ b 2 ; (3.117) andthebehaviourofthedensity, u ,takestheform u b " 1 2 ‹ 1 2 “ ˙ b 2 O ‹ " 2 “ : (3.118) Assumingthesystemdecaytoanequilibriawithanadmissibleinterface b ,thentheanalysiscanbecontinued tothenexttime-scale,howeverourgoalistoinvestigatethecoupledbilayer-poreevolutionwhichoccursat thistimescale. 63 Chapter4 GeometricEvolutionofPoresin R 3 Inthischapterwederivethegeometricevolutionofadmissibleco-dimensiontwointerfacesin R 3 under the H 1 gradientwofthe strongFCH .Usingmulti-scaleanalysiswederiveanexpressionforthe curvature-drivennormalvelocityat O ‹ " 1 “ timescale.Wedescribethecompetitiveevolutionofdisjoint collectionsofbilayersandporeswhichcouplethroughcurvature-weightedsurfacearea,andshowthat, generically,thetwomorphologiesseektequilibriavalues,makingcoexistenceofbilayersandpores impossibleunderthe strongfunctionalization ,unlessoneofthestructuresissincezerocurvature interfacesareatequilibriumindependentofbulkvalueofamphiphile. RecallthestrongFCHfreeenergywhichcorrespondstothechoice p 1in( 1.14 ), F ‹ u “ S 1 2 ‹ " 2 u W œ ‹ u ““ 2 " „ " 2 1 2 S © u S 2 2 W ‹ u “‚ dx; (4.1) where ` R 3 isaboundeddomain, W ‹ u “ isatilteddouble-wellpotentialwithtwominimaat b , u R isthedensityofoneoftheamphiphilicspecies, " P 1controlsthewidthoftheboundarylayerand 1 and 2 arethefunctionalizationconstants. Note7. Byabuseofnotationwewilldropthe p subscriptinthe u p criticalpointwhendoingsocreatesno confusion. Thechemicalpotential, ,isasthevariationof F , F u ‹ u “ ‹ " 2 W œœ ‹ u “ 1 “‹ " 2 u W œ ‹ u ““ d W œ ‹ u “ ; (4.2) 64 where d 1 2 .InthischapterwepresentaformalreductionofthestrongFCHequation, u t ‹ " 2 W œœ ‹ u “ 1 “‹ " 2 u W œ ‹ u ““ d W œ ‹ u “ ; (4.3) forfunctions u thatareclosetoabilayerdressingofanadmissibleinterfaceinsubjecttoperiodicor boundaryconditions.WemayrewritethestrongFCHequationusingtheofthechemical potential,givenin( 4.2 ), u t (4.4) 4.1Thewhiskeredcoordinatesystemandinner-expansions Assuminganadmissibleinitialco-dimensiontwointerface p ‹ t 0 “ ,weperformamulti-scaleanalysisofthe solution u .Awayfromtheinterface,intheregion ~ p;` ,theoutersolution u hastheexpansion u ‹ x;t “ u 0 ‹ x;t “ "u 1 ‹ x;t “ " 2 u 2 ‹ x;t “ O ‹ " 3 “ : (4.5) Onthereach, p;` ,atagiventime-scale ˝ ,theoutersolution'sinnerexpansiontakestheform u ‹ x;t “ ~ u ‹ s;z;˝ “ ~ u 0 ‹ s;z;˝ “ " ~ u 1 ‹ s;z;˝ “ " 2 ~ u 2 ‹ s;z;˝ “ O ‹ " 3 “ : (4.6) 4.2MatchingConditions Fixawhisker, w ,andlet x > p beitsbasepoint.Wetaketwovectors n ; m > span Ÿ N 1 ; N 2 š inthenormal planeof p at x ,andspecifythat n cos ‹ “ N 1 sin ‹ “ N 2 : (4.7) Theusualdirectionalderivativealong n isdenoted @ n n © x cos ‹ “ N 1 © x sin ‹ “ N 2 © x ; (4.8) andfor f > C ª ‹ ~ p “ weintroducethe n ; m limit @ j n f m ‹ x “ lim h 0 ‹ n © x “ j f ‹ x h m ;t “ forall j C 0 ; (4.9) 65 andthelimitofthegradient © x f m ‹ x “ lim h 0 © x f ‹ x h m ;t “ ; (4.10) wherethelimitexists.If f > C 1 ‹ “ thenthenormalderivativeof f willsatisfy @ n f m @ n f m : (4.11) Thismotivatesthefollowingofthejumpcondition 4.1. Givenaradialfunction f f ‹ R “ localizedon p ,wethejumpof f acrossagiven whiskerby J @ n f m K p ‹ x “ @ n f m ‹ x “ @ n f m ‹ x “ (4.12) whichiszerowhen f hasasmoothextensionthrough p . Withthisnotationweexaminethematchingcondition u ‹ x h n ;t “ ~ u ‹ s;R;;˝ “ : (4.13) Anexpansionoftheleft-handsideofequation( 4.13 )around x ,as h 0 ,isgivenby u n ‹ x;t “ " ‹ u n 1 ‹ x;t “ z@ n u n 0 ‹ x;t ““ " 2 ‹ u n 2 ‹ x;t “ z@ n u n 1 ‹ x;t “ 1 2 z 2 @ 2 n u n 0 ‹ x;t ““ O ‹ " 3 “ ; (4.14) andequatingordersof " thematchingcondition( 4.13 )yields u n 0 lim R ª ~ u 0 ‹ s;R;;˝ “ ; (4.15) u n 1 R@ n u n 0 lim R ª ~ u 1 ‹ s;R;;˝ “ : (4.16) Similarly,wecanobtainmatchingconditionsforthechemicalpotential n 0 lim R ª ~ 0 ; (4.17) n 1 R@ n 0 lim R ª ~ 1 ; (4.18) n 2 R@ n n 1 1 2 R 2 @ 2 n n 0 lim R ª ~ 2 ; (4.19) n 3 R@ n n 2 1 2 R 2 @ 2 n n 1 1 6 R 3 @ 2 n n 0 lim R ª ~ 3 : (4.20) 66 4.3ExpansionoftheChemicalPotential Wewillhaverecoursetotheinnerandouterexpansionsofthechemicalpotential › " 2 W œœ ‹ u “ 1 ”› " 2 u W œ ‹ u “ ” d W œ ‹ u “ ; (4.21) 4.3.1OuterExpansionoftheChemicalPotential Atagiventimescale ˝ ,theouterexpansionforthedensityfunction u ‹ x;t “ isgivenbyequation( 4.5 ). Plugging( 4.5 )into( 4.21 )andrewritingthechemicalpotential inordersof " yields ‹ x;t “ 0 ‹ x;˝ “ 1 ‹ x;˝ “ " 2 2 ‹ x;˝ “ :::; (4.22) where 0 W œœ ‹ u 0 “ W œ ‹ u 0 “ ; (4.23) 1 ‹ W œœœ ‹ u 0 “ u 1 1 “ W œ ‹ u 0 “ ‹ W œœ ‹ u 0 ““ 2 u 1 d W œ ‹ u 0 “ ; (4.24) 2 ‰ W œœœ ‹ u 0 “ u 2 1 2 W ‹ 4 “ ‹ u 0 “ u 1 ’ W œ ‹ u 0 “ ‹ W œœœ ‹ u 0 “ u 1 1 “ W œœ ‹ u 0 “ u 1 (4.25) W œœ ‹ u 0 “‰ u 0 W œœ ‹ u 0 “ u 2 1 2 W œœœ ‹ u 0 “ u 2 1 ’ d W œœ ‹ u 0 “ u 1 : Notethattheouterexpansionofthechemicalpotentialisidenticalforbothco-dimensiononeandco- dimensiontwo. 4.3.2InnerExpansionoftheChemicalPotential Atagiventimescale ˝ ,theinnerexpansionforthedensityfunction u ‹ x;t “ isgivenbyequation( 4.6 ),and inlocalcoordinates,theLaplacianisgiveninequation( 2.70 ).Plugging( 4.6 )and( 2.70 )into( 4.21 ),wecan rewritethechemicalpotential inordersof " ‹ x;t “ ~ 0 ‹ s;z;˝ “ " ~ 1 ‹ s;z;˝ “ " 2 ~ 2 ‹ s;z;˝ “ O ‹ " 3 “ ; (4.26) where ~ 0 ‹ z W œœ ‹ ~ u 0 ““‹ z ~ u 0 W œ ‹ ~ u 0 ““ ; (4.27) ~ 1 ‹ z W œœ ‹ ~ u 0 ““‹ z ~ u 1 Ñ © z ~ u 0 W œœ ‹ ~ u 0 “ ~ u 1 “ (4.28) 67 ‹ Ñ © z W œœœ ‹ ~ u 0 “ ~ u 1 1 “‹ z ~ u 0 W œ ‹ ~ u 0 ““ d W œ ‹ ~ u 0 “ ~ 2 ‹ z W œœ ‹ ~ u 0 ““‹ z ~ u 2 Ñ © z ~ u 1 @ 2 s ~ u 0 ‹ z Ñ “ Ñ © ~ u 0 W œœ ‹ ~ u 0 “ ~ u 2 1 2 W œœœ ‹ ~ u 0 “ ~ u 2 1 “ (4.29) ‹ Ñ © z W œœœ ‹ ~ u 0 “ ~ u 1 1 “‹ z ~ u 1 Ñ © z ~ u 0 W œœ ‹ ~ u 0 “ ~ u 1 “ ‹ @ 2 s ‹ z Ñ “ Ñ © z W œœœ ‹ ~ u 0 “ ~ u 2 1 2 W ‹ 4 “ ‹ ~ u 0 “ ~ u 2 1 “‹ z ~ u 0 W œ ‹ ~ u 0 ““ d W œœ ‹ ~ u 0 “ ~ u 1 4.4Timescale ˝ " 2 t Onthefasttimescale,theinitialdataisexpectedtorelaxintoanequilibriasolution.Westartbylooking forapproximationsofthesolutionsofthestrongFCHequation u t x ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ ‹ " 2 W œœ ‹ u “ 1 “‹ " 2 u W œ ‹ u ““ d W œ ‹ u “ in ; (4.30) forthetimescale ˝ " 2 t . 4.4.1Outerexpansion Awayfromthepore p ,pluggingtheouterexpansionforthedensityfunction u ‹ x “ andtheouterexpansion ofthechemicalpotential ,givenin( 4.5 )and( 4.22 ),respectively,intothestrongFCHequation,( 4.30 ),and equatingordersof " yields O ‹ " 2 “ u 0 ;˝ 0 ; (4.31) O ‹ " 1 “ u 1 ;˝ 0 ; (4.32) O ‹ 1 “ u 2 ;˝ ‹ W œœ ‹ u 0 “ W œ ‹ u 0 ““ in ~ p;` : (4.33) Onthe ˝ " 2 t timescale,thesolution u isstationarytoandsecondorder.Equation( 4.33 )has boundaryconditionsonbuttosolveitwealsoneedboundaryconditionson p .Thisleadsustotheinner expansion. 4.4.2Innerexpansion Weexpresseachofthetermsin( 4.30 )inwhiskeredcoordinates.Pluggingtheinnerexpansionof u ,given in( 4.6 ),intotheleft-handsideofequation( 4.30 ),thetimederivativeof u takestheform u t " 2 ‰ © z ~ u @z @˝ @ s ~ u @s @˝ ~ u ˝ ’ : (4.34) 68 Inlightofthenormalvelocityrelationsgiveninequations( 2.68 )and( 2.69 ),equation( 4.34 )reducesto u t " 3 V © z ~ u 0 " 2 ‰‰ z 2 N 2 @N 1 @˝ ;z 1 N 1 @N 2 @˝ ’ © z ~ u @ s ~ u @s @˝ ~ u ˝ ’ O ‹ " 1 “ (4.35) AnexpressionoftheLaplacianinlocalcoordinatesisgivenin( 2.70 )andanexpressionfortheinnerexpansion ofthechemicalpotentialisgivenin( 4.26 ).Plugging( 4.35 ),( 2.70 )and( 4.26 )backintotheevolution equation( 4.30 )andcomparingordersof " yields, O ‹ " 3 “ V © z ~ u 0 0 ; (4.36) O ‹ " 2 “ V © z ~ u 1 @ s ~ u 0 @s @˝ ~ u 0 ;˝ z 2 N 2 @N 1 @˝ @ ~ u 0 @z 1 z 1 N 1 @N 2 @˝ @ ~ u 0 @z 2 z ~ 0 ; (4.37) where~ 0 isgivenin( 4.27 ).Weareinterestedinnon-trivialsolutionsbaseduponaquasi-stationaryradial consequentlyweassumethatthetransientdynamicsonthe ˝ timescalehaveequilibrated,that is V 0andall ˝ partialarezero,sothatthesystemofequationsreducesto 0 z ~ 0 : (4.38) Theseassumptionsareconsistentwithequilibriawhichatleadingorderhaveradiallysymmetricthat render~ 0 0. Thenexttimescale ˝ " 1 t yieldsthesameresultsandweskipthecalculations. 4.5TimeScale ˝ t :SharpInterfaceLimit RecallthatwearelookingforapproximationsforsolutionsofthestrongFCHequation u t x ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ ‹ " 2 W œœ ‹ u “ 1 “‹ " 2 u W œ ‹ u ““ d W œ ‹ u “ in : (4.39) Forthetimescale ˝ t .Wewillobtainanevolutionequationsfortheouterandinnerregions. 4.5.1OuterExpansion Awayfromtheinterface,theouterexpansionofthedensityfunction u isgivenin( 4.5 ),andtheouter expansionofthechemicalpotential, ,givenin( 4.22 ).Plugging( 4.5 )and( 4.22 )into( 4.39 )yieldsatleading 69 order, O ‹ 1 “ , u 0 ;˝ 0 in ~ p;` ; (4.40) where 0 isgivenin( 4.23 ). 4.5.2InnerExpansion Weexpresseachofthetermsin( 4.39 )ininnercoordinates.Pluggingtheinnerexpansionof u ,givenin( 4.6 ), intotheleft-handsideofequation( 4.39 )yields u t ~ u ˝ ~ u s @s @˝ @z @˝ © z ~ u " 1 V © z ~ u 0 O ‹ 1 “ : (4.41) AnexpandexpressionoftheLaplacianinlocalcoordinatesisgivenin( 2.70 )andanexpressionforthe innerexpansionofthechemicalpotentialisgivenin( 4.26 ).Plugging( 4.41 ),( 2.70 )and( 4.26 )backintothe evolutionequation( 4.39 )andcomparingordersof " yields, O ‹ " 2 “ 0 z ~ 0 in p;` ; (4.42) O ‹ " 1 “ V © z ~ u 0 z ~ 1 Ñ © z ~ 0 ; in p;` : (4.43) Equation( 4.43 )hasthesolution~ u 0 U p where U p istheporein( 2.75 ).Forthischoiceof~ u 0 itfollowsthat~ 0 0andthat~ 1 ,in( 4.28 ),takestheform ~ 1 L 2 p; 0 ~ u 1 d W œ ‹ U b “ ; (4.44) where L p; 0 isin( 2.81 ).Moreover,thenextorderequation,( 4.43 ),reducesto V © z ~ u 0 z ~ 1 ; in p;` : (4.45) 4.5.3JumpConditions Wewouldliketodeterminethenormalvelocityoftheinterface p forthetimescale ˝ t .Tothisend,we needtodetermineanexplicitsolutionfor~ 1 ,inequation( 4.45 ),subjecttothematchingconditionswith theoutersolution,giveninSection 4.2 . 70 Turningtopolarcoordinates,weusetheFouriermodeexpansion,( 2.88 ),toobtainanexpressfor~ 1 , ~ 1 A 1 ‹ s;R “ cos B 1 ‹ s;R “ sin C ‹ s;R “ ˘ ‹ s;R; “ ; (4.46) where ˘ ‹ s;R; “ ª Q m 2 ‹ A m ‹ s;R “ cos ‹ “ B m ‹ s;R “ sin ‹ ““ : (4.47) Fromthematchingcondition( 4.18 )weseethat~ 1 growsatmostlinearlyas R ª and lim R ª @ ~ 1 @R @ n n 0 : (4.48) Usingtheonofthedirectionalderivativealong n ,givenin( 4.8 ),werewriteequation( 4.48 )interms ofthesineandcosinefunctions lim R ª @ ~ 1 @R cos N 1 © z n 0 sin N 2 © z n 0 : (4.49) Takingthe R derivativeof( 4.46 )yields @ ~ 1 @R ‹ C 1 a œ ‹ R “ V 1 “ cos ‹ C 2 a œ ‹ R “ V 2 “ sin @˘ @R : (4.50) Comparing( 4.50 )with( 4.49 )weconcludethat ˘ ˘ ‹ s; “ .Usingbasictrigonometricidentities,( C.35 ),we notethat @ ~ 1 @R also @ ~ 1 @R ‹ s;R;;˝ “ @ ~ 1 @R ‹ s;R; ˇ;˝ “ : (4.51) Combining( 4.51 )and( 4.48 )weobtainthejumpconditionovertheinterface p J @ n n 0 K 0 ; (4.52) foranychoiceofnormalvector n . 4.5.4TheNormalVelocity Wewouldliketodeterminetheevolutionoftheinterface p .Tothisend,recallequation( 4.45 ),which involvethenormalvelocity V ,andtheinnerchemicalpotential~ 1 .Usingthepolarcoordinatesextensionof theLaplacian,givenin( A.3 ),andtheexpressionfor~ 1 givenin( 4.46 ),theright-handsideofequation( 4.45 ) 71 takestheform z ~ 1 ‹ C œœ 1 R C œ “ ‹ A œœ 1 1 R A œ 1 1 R 2 A 1 “ cos ‹ B œœ 1 1 R B œ 1 1 R 2 B 1 “ sin (4.53) ª Q m 2 m R 2 A m ‹ s “ cos ‹ “ m R 2 B m ‹ s “ sin ‹ “ : Usingthepolargradient,( A.4 ),theleft-handsideofequation( 4.45 )becomes V © z U p ‹ V 1 U œ p cos ;V 2 U œ p sin “ (4.54) Plugging( 4.53 )and( 4.54 )into( 4.45 )andmatchingcotsofcorrespondingtrigonometrictermsyields thesystem C œœ 1 R C œ 0 ; (4.55) A œœ 1 1 R A œ 1 1 R 2 A 1 V 1 U œ p ; (4.56) B œœ 1 1 R B œ 1 1 R 2 B 1 V 2 U œ p ; (4.57) m R 2 A m 0 ; (4.58) m R 2 B m 0 : (4.59) Fromequations( 4.58 )and( 4.59 )wededucethat A m B m 0,for m C 2.Equation( 4.55 )hasthesolution C C 0 ‹ s “ ; (4.60) andthenon-homogeneousequations,( 4.56 )and( 4.57 ),havethesolutions A ‹ s;R “ C 1 ‹ s “ R a ‹ R “ V 1 ‹ s “ ; (4.61) B ‹ s;R “ C 2 ‹ s “ R a ‹ R “ V 2 ‹ s “ ; (4.62) where a ‹ R “ isthesolutionofthenon-homogeneousODE a œœ 1 R a œ 1 R 2 a U œ ‹ R “ ; (4.63) andisgivenbytheexplicitformula a ‹ R “ 1 R S R 0 r ^ U p dr; (4.64) 72 whereweintroduce ^ U p U p b : (4.65) Notethat ^ U p ispositiveand ^ U p 0as R ª .Plugging( 4.60 ),( 4.61 )and( 4.62 )into( 4.46 )andtaking A m B m 0weseethat~ 1 takestheform ~ 1 C 0 ‹ s “ ‹ C 1 ‹ s “ R a ‹ R “ V 1 ‹ s ““ cos ‹ C 2 ‹ s “ R a ‹ R “ V 2 ‹ s ““ sin : (4.66) Recallthat~ 1 relatestothedensityfunction~ u throughequation( 4.28 ),whichforthechoice~ u 0 U p takes theform L 2 p ~ u 1 ~ 1 d W œ ‹ U p “ ; (4.67) wherethelinearoperator L p wasintroducedin( 2.78 ).BytheFredholmAlternative,thisequationhas asolution~ u 1 > L 2 ‹ R “ ifandonlyiftheright-handsideisperpendiculartoker L p .Recallthatker L p ‹ U œ p cos ;U œ p sin “ andconsidertheinnerproductof ‹ U œ p cos ;U œ p sin “ withtheright-handsideofequa- tion( 4.67 ).Weknowthat d W œ ‹ U p “ belongtothespace Z 0 ,in( 2.79 ),andhenceisperpendicular toker L p .Fromorthogonalityin ,theonlynon-trivialconditionisimposedonthesin andcos terms of~ 1 ,inequation( 4.66 ),andtheFredholmcondition( 4.67 )reducesto S ª 0 ‹ C i ‹ s “ R a ‹ R “ V i ‹ s ““ U œ p RdR 0 ; for i 1 ; 2 : (4.68) Pluggingthedenitionof a ‹ R “ ,givenin( 4.64 )into( 4.68 )andintegratingbypartsyieldstherelation C i V i S 2 2 S 1 (4.69) whereweintroducetheconstants S 1 S ª 0 ^ U p RdR (4.70) S 2 S ª 0 ^ U 2 p RdR: (4.71) Plugging( 4.69 )into( 4.66 )andtakingthe R derivativeof( 4.66 )yields @ ~ 1 @R V 1 ‹ S 2 2 S 1 a œ ‹ R ““ cos V 2 ‹ S 2 2 S 1 a œ ‹ R ““ sin : (4.72) 73 Equatingcotsofsin andcos inequations( 4.72 )and( 4.49 )yields N i © x N i 0 V i ‹ S 2 2 S 1 lim R ª a œ ‹ R ““ ; (4.73) andsince lim R ª a œ ‹ R “ lim R ª ‹ 1 R 2 S R 0 r ^ U œ p dr ^ U p “ 0 ; (4.74) wethatthenormalvelocity V i 2 S 1 S 2 N i © x N i 0 ; for i 1 ; 2 : (4.75) 4.5.5SharpInterfaceLimit Theprecedingcalculationshowsthat,inaneighborhoodofthedressedsolution,the ˝ t timescaleevolution of( 4.39 )reducestoasharpinterfacelimitproblemfortheevolutionof p ‹ 4 : 40 “ u 0 ;˝ 0 in ~ p;` ; (4.76) n © x 0 0on @ ; (4.77) 0 0on p ; (4.78) ‹ 4 : 52 “ J @ n n 0 K 0on p ; forallnormalvectors n of p ; (4.79) ‹ 4 : 75 “ V i 2 S 1 S 2 N i © x N i 0 ; forall x > p ‹ t “ ;i 1 ; 2 : (4.80) Wearefollowingtheargumentof[ DaiandPromislow,2015 ]andprovethefollowingLemma- Lemma4.1. Assumethattheco-dimensiontwointerface p ` hasonedimensional measure.Thentheonlyequilibriumsolutionof( 4.76 )-( 4.80 )isthetrivialsolution 0 0 ,howeverthe curve p canhavearbitraryshape. Proof. Atequilibiumwehave 0 0 ; in ~ p;` ; (4.81) n © x 0 0on @ ; (4.82) 0 0on p : (4.83) Since 0 isanalyticofasetofmeasure,then 0 hasananalyticextensionto 0 ,see[ ? ] 74 and[ Polking,1984 ],andwedropthebarnotation.Theextendedfunction 0 0 ; in ; (4.84) n © x 0 0on @ ; (4.85) then,bythe StrongMaximumprinciple impliesthat 0 isspatiallyconstant.Finally,since 0 0on p ,we concludethat 0 0. Ì Wesubsequentlyassumethesystemhasachievedequilibriumonthe ˝ t time-scale. 4.6Timescale ˝ "t :CurvatureDrivenFlow Weobtainevolutionequationsfortheouterandinnerregions.Usingtheinnerequationsweobtainajump conditionsoftheoutersolutionovertheinterfaceandanexpressionforthenormalvelocityoftheinterface. wewillseethatthenormalvelocityisdrivenbythecurvaturew.Finally,weusethemasspreserving propertyofthesystemtoobtainthecoupledsystemforthenormalvelocity, V ,andtheexternalchemical potential, 1 . 4.6.1Outerexpansion Awayfromtheinterface,theouterexpansionofthedensityfunction u isgivenin( 4.5 ),andtheouter expansionofthechemicalpotential, ,givenin( 4.22 ).Plugging( 4.5 )and( 4.22 )into( 4.39 )andequating ordersof " yields O ‹ 1 “ 0 x 0 ; in ~ p;` (4.86) O ‹ " “ u 0 ;˝ x 1 ; in ~ p;` ; (4.87) where 0 and 1 aregivenin( 4.23 )and( 4.24 ),respectively.Fromequations( 4.76 )-( 4.80 )weassumethat thesystemhasequilibratedto u 0 b in ~ p;` .Underthisassumption 0 0,whichequation( 4.86 ) andequation( 4.87 )reducesto x 1 0in ~ p;` : (4.88) 75 4.6.2InnerExpansion Weexpresseachofthetermsin( 4.39 )ininnercoordinates.Pluggingtheinnerexpansionof u ,givenin( 4.6 ), intotheleft-handsideofequation( 4.39 )yields u ˝ V © z ~ u O ‹ " “ : (4.89) AnexpandexpressionoftheLaplacianinlocalcoordinatesisgivenin( 2.70 )andanexpressionforthe innerexpansionofthechemicalpotentialisgivenin( 4.26 ).Plugging( 4.89 ),( 2.70 )and( 4.26 )backintothe evolutionequation( 4.39 )andcomparingordersof " yields, O ‹ " 2 “ 0 z ~ 0 ; (4.90) O ‹ " 1 “ 0 z ~ 1 Ñ © z ~ 0 (4.91) O ‹ 1 “ V t 1 © z ~ u 0 z ~ 2 Ñ © z ~ 1 ‹ @ 2 s ‹ z Ñ “ Ñ © z “ ~ 0 ; (4.92) where~ 0 ; ~ 1 and~ 2 areinequations( 4.27 ),( 4.28 )and( 4.29 ),respectively.Equation( 4.90 )is consistentwiththeassumptionthat~ u 0 U p whichimpliesthat~ 0 0.Since~ 0 0equations( 4.91 ) and( 4.92 )reducesto O ‹ " 1 “ 0 z ~ 1 ; (4.93) O ‹ 1 “ V t 1 © z ~ u 0 z ~ 2 Ñ © z ~ 1 : (4.94) 4.6.2.1Solvingequation( 4.93 )for ~ u 1 Tosolve( 4.93 )weusetheexplicitexpressionfor z ~ 1 ,givenin( 4.53 ).Plugging( 4.53 )into( 4.93 )and matchingcos,sintermsyields C œœ 1 R C œ 0 ; (4.95) A œœ 1 1 R A œ 1 1 R 2 A 1 0(4.96) B œœ 1 1 R B œ 1 1 R 2 B 1 0(4.97) m R 2 A m 0 ; (4.98) m R 2 B m 0 : (4.99) 76 Fromequations( 4.56 )-( 4.59 )wededucethat A i B i 0,for i C 1.Equation( 4.55 )hasthesolution C C 0 ‹ s “ ; (4.100) andwededucethat~ 1 isspatiallyconstant,i.e., ~ 1 ~ 1 ‹ s;t “ 1 : (4.101) Recallthat~ 1 isgivenin( 4.28 )andsince~ u 0 U p itreducesto L 2 p ~ u 1 ~ 1 d W œ ‹ U p “ : (4.102) BytheFredholmAlternative,thisequationhasasolution~ u 1 ifandonlyiftheright-handsideisperpendicular toker L p .Recallthatker L p ` ker L p; 1 andnotethat~ 1 ; d W œ ‹ U p “ > Z 0 .Sincethespaces Z m aremutually orthogonal,thereexistsasolution~ u 1 denoted ~ u 1 1 p; 2 d L 2 p W œ ‹ U p “ (4.103) where 1 isaspacialconstantand p; 2 L 2 p p; 2 1,in( 2.86 ). Todeterminetheinterfacenormalvelocitywecontinuetotheequation( 4.94 ).Sinceequation( 4.101 )implies that~ 1 isspatiallyconstantandequation( 4.94 )reducesto O ‹ 1 “ V © z ~ u 0 z ~ 2 : (4.104) 4.6.2.2JumpConditions Wewouldliketodeterminethenormalvelocityoftheinterface p forthetimescale ˝ "t .Tothisend,we needtodetermineanexplicitsolutionfor~ 2 ,inequation( 4.104 ),subjecttothematchingconditionswith theoutersolution,giveninSection 4.2 . Turningtopolarcoordinates,weusetheFourierexpansion,( 2.88 ),toobtainanexpressfor~ 2 , ~ 2 A 1 ‹ s;R “ cos B 1 ‹ s;R “ sin C ‹ s;R “ ˘ ‹ s;R; “ ; (4.105) where ˘ ‹ s;R; “ ª Q m 2 ‹ A m ‹ s;R “ cos ‹ “ B m ‹ s;R “ sin ‹ ““ : (4.106) 77 Fromthematchingcondition( 4.19 )weseethat~ 2 growsatmostlinearlyas R ª and lim R ª @ ~ 2 @R @ n n 1 : (4.107) Usingtheofthedirectionalderivativealong n ,givenin( 4.8 ),werewriteequation( 4.107 )interms ofsinandcos lim R ª @ ~ 2 @R cos N 1 © z n 1 sin N 2 © z n 1 : (4.108) Takingthe R derivativeof( 4.105 )yields @ ~ 2 @R ‹ C 1 a œ ‹ R “ V 1 “ cos ‹ C 2 a œ ‹ R “ V 2 “ sin @ ˘ @R : (4.109) Comparing( 4.109 )with( 4.108 )weconcludethat ˘ ˘ ‹ s; “ .Usingbasictrigonometricidentities,( C.35 ), wenotethat @ ~ 2 @R also @ ~ 2 @R ‹ s;R;;˝ “ @ ~ 2 @R ‹ s;R; ˇ;˝ “ : (4.110) Combining( 4.110 )and( 4.107 )weobtainthejumpconditionovertheinterface p J @ n n 1 K 0 ; (4.111) foranychoiceofnormalvector n .Moreover,plugging( 4.109 )into( 4.108 ),recallingthat a œ ‹ R “ 0as R ª , andcomparingcoientsofsinandcosyieldstherelation C i N i © z n 1 (4.112) 4.6.2.3Thenormalvelocity Wewouldliketodeterminetheevolutionoftheinterface p .Tothisend,recallequation( 4.104 ),which involvesthenormalvelocity, V ,andtheinnerchemicalpotential~ 2 .Theof~ 2 isgivenin( 4.29 ), andsince~ u 0 U p ( 4.29 )reducesto ~ 2 L 2 p ~ u 2 L p ‹ R “ ‹ © z W œœœ ‹ U p “ ~ u 1 1 “‹ L p ~ u 1 Ñ © z U p “ d W œœ ‹ U p “ ~ u 1 (4.113) where R © ~ u 1 ‹ z Ñ “ Ñ © z U p 1 2 W œœœ ‹ ~ U p “ ~ u 1 : (4.114) 78 Inordertogetanexpressionforthenormalvelocitywewanttosolve( 4.104 )for~ 2 .Followingthesamepro- cedureasinSection 4.5.4 ,spequations( 4.45 )-( 4.66 ),andconsideringthematchingcondition( 4.19 ) wededucethat~ 2 takestheform ~ 2 C 0 ‹ s “ ‹ C 1 ‹ s “ R a ‹ R “ V 1 ‹ s ““ cos ‹ C 2 ‹ s “ R a ‹ R “ V 2 ‹ s ““ sin ; (4.115) where a ‹ R “ , C 0 and C 1 ;C 2 aregivenin( 4.64 ),( 4.60 ),( 4.61 ),and( 4.62 ),respectively. Tosolveequation( 4.113 )for~ u 2 werewriteitinthefollowingform L 2 p ~ u 2 ~ 2 Q L p ‹ R “ ; (4.116) where Q ‹ Ñ © z W œœœ ‹ U p “ ~ u 1 1 “‹ L p ~ u 1 Ñ © z U p “ d W œœ ‹ U p “ ~ u 1 : (4.117) BytheFredholmAlternative,wecansolveequation( 4.116 )for~ u 2 ifandonlyiftheright-handsideis perpendiculartoker L p .Recallthatker L p span Ÿ U œ p cos ;U œ p sin š ` ker L p; 1 ,andthatthe Z m spacesare mutuallyorthogonal.Expanding Q ,givenin( 4.117 ),anddecomposingittoits Z m componentsyields Q Q 0 Q 1 Q 0 ; 2 ; (4.118) where Q 0 > Z 0 , Q 1 > Z 1 , Q 0 ; 2 > Z 0 Z 2 ,andaregivenby Q 0 W œœœ ‹ U p “ ~ u 1 L p ~ u 1 1 L p ~ u 1 d W œœ ‹ U p “ ~ u 1 ; (4.119) Q 1 Ñ © z L p ~ u 1 W œœœ ‹ U p “ ~ u 1 Ñ © z U p 1 Ñ © z U p ; (4.120) Q 0 ; 2 ‹ Ñ © z “ 2 U p : (4.121) Bytheorthogonalityofthe Z m spacesandsince L p ‹ R “ Œ ker L p theFredholmsolvabilityconditionofequa- tion( 4.116 )reducesto ‹ ~ 2 Q 1 ;@ z i U p “ L R 0 ; ª ““ 0 ; for i 1 ; 2 : (4.122) Inordertocalculatethesolvabilitycondition,givenin( 4.122 ),weexpand Q 1 usingtheexplicitexpression of~ u 1 givenin( 4.103 ),suchthat Q 1 1 Ñ © z p; 1 d Ñ © z L 1 p ‹ W œ ‹ U p ““ 1 W œœœ ‹ U p “ p; 2 Ñ © z U p (4.123) 79 d W œœœ ‹ U p “ L 2 p ‹ W œ ‹ U p ““ Ñ © z U p 1 Ñ © z U p ; andcalculatingtheinnerproduct(seeAppendix C.2 fordetails)yields ‹ Q 1 ;@ z i U “ L R 2 ˇ 1 i S 1 1 ˇ i S 4 ; (4.124) where S 1 isin( 4.70 )and S 4 S ª 0 ‹ U œ p “ 2 RdR: (4.125) Theinnerproductof @ z i U p with~ 2 ,where~ 2 isgiveninequation( 4.115 ),yields ‹ ~ 2 ;@ z i U “ 2 ˇ C i S 1 ˇV i S 2 2 ˇ N i © z N i 1 S 1 ˇV i S 2 ;; (4.126) where S 2 isin( 4.71 ),andthesecondequalityfollowsformthematchingconditions,seeequa- tion( 4.112 ).Returningto( 4.122 )andusing( 4.124 )and( 4.126 ),weconcludethatthenormalvelocityis givenby V i 2 1 S 1 1 S 4 S 2 i 2 S 1 S 2 N i © z N i 1 i 1 ; 2 : (4.127) 4.6.3Sharpinterfacelimit Onthetimescale ˝ "t ,theevolutionoftheinterface, p isgivenbythenormalvelocity ‹ 4 : 127 “ V i 2 1 S 1 1 S 4 S 2 i 2 S 1 S 2 N i © z N i 1 i 1 ; 2 : (4.128) where 1 isthesolutionofthesystem ‹ 4 : 88 “ x 1 0in ~ p;` ; (4.129) n © x 1 0on @ ; (4.130) ‹ 4 : 111 “ J @ n n 1 K 0 ; on p ; forallnormalvectors n of p : (4.131) Theinnerchemicalpotential 1 1 ‹ s;˝ “ on p .Sinceweassumedthat 1 > C 2 ‹ ~ p “ 9 C ‹ “ wemayuseLemma 4.1 toconcludethat 1 0ontheentiredomainApplyingtheStrongMaximum Principle(see[ Evans,2010 ])wededucethat 1 isspatiallyconstant. 80 Since 1 isspatiallyconstantwehave © x 1 0andthenormalvelocityreducesto V 2 1 S 1 1 S 4 S 2 Ñ (4.132) where S 1 and S 4 arein( 4.70 ),( 4.125 ),respectively. 4.6.4Equilibriaestimatefortimescale ˝ "t Theexternalchemicalpotential 1 ischaracterizedbythedensityfunction, u ,whosevalueis determinedbyconservationoftotalmass.Forthistimescale,wesummarizeourapproximationforthe densityfunctionineachregion.Intheouterregion ~ p;` ,ourassumptionthat u 0 b combinedwith equation( 4.24 )yields u ‹ x;t “ b " 1 2 O ‹ " 2 “ in ~ p;` ; (4.133) where isin( 1.34 ).Intheinnerregion, p;` ,ourchoice~ u 0 U p combinedwithequation( 4.103 ) yields u ‹ x;t “ U p " ‹ 1 p; 2 d L 2 p W œ ‹ U p ““ O ‹ " 2 “ in p;` : (4.134) Weusemassbalancetodetermine 1 andtoobtainthecoupled 1 ;V systemevolution.Thetotalmassof thesystemisgivenby M S u ‹ x;t “ b dx S u ‹ x; 0 “ b dx S … p;` ‹ u b “ dx S p;` ‹ u b “ dx (4.135) Using( 4.133 ),theouterintegralbecomes S … p;` ‹ u b “ dx " 1 2 ‹S S S p;` S “ O ‹ " 2 “ (4.136) Using( 4.134 )andtheJacobian,in( 2.63 ),theinnerintegraltakestheform S p;` ‹ u b “ dx " 2 S p S R 2 › ‹ U p b “ " ‹ 1 p; 2 d L 2 p W œ ‹ U p ““ O ‹ " 2 “ ” ‹ 1 "z Ñ “ dzds (4.137) " 2 2 ˇ S p S S 1 " 3 2 ˇ S p S 1 S ª 0 p; 2 RdR " 3 2 ˇ S p S d S ª 0 W œ ‹ U p “ RdR O ‹ " 4 S p S“ (4.138) Addingandsubtractingtheterm " 2 2 ˇ 1 2 S p S " 1 2 S p;` S to( 4.137 ),theinnerintegralbecomes S p;` ‹ u b “ dx " 2 2 ˇ S p S S 1 " 1 2 S p;` S " 3 2 ˇ S p S 1 S 3 O ‹ " 3 S p S“ (4.139) 81 whereweintroducetheconstant S 3 S 3 S ª 0 p; 2 2 RdR: (4.140) Combining( 4.139 )and( 4.136 )into( 4.135 )andassumingthat S p S O ‹ " 1 “ ,whichimpliesthat S p S " 1 1 O ‹ 1 “ ; (4.141) werewritethetotalmassinordersof " M " „ 1 2 S S 2 ˇS 1 1 ‚ O ‹ " 2 “ : (4.142) Takingthe ˝ "t timederivativeofthetotalmass,( 4.142 ),andsolvingfor 1 d˝ yields 1 d˝ S S 2 ˇ 2 S 1 1 d˝ : (4.143) Ontheotherhand,takingthetimederivativeof( 4.141 )yields d S p S d˝ " 1 1 d˝ O ‹ 1 “ : (4.144) Combiningequation( 2.62 ),whichrelatestheinterfacialsurfaceareagrowthwiththenormalvelocity,with equation( 4.144 )yields " 1 1 d˝ S p Ñ V ds 2 1 S 1 1 S 4 S 2 S S Ñ S 2 ds; (4.145) whereforthesecondequalityweusedtheexpressionof V obtainedinequation( 4.132 ).Plugging( 4.143 ) intoequation( 4.145 )andsolvingfor 1 d˝ weobtaintheleadingorderevolutionequation 1 d˝ " „ 4 ˇ 2 S 2 1 S S S 2 S p S Ñ S 2 ds ‚ 1 " 2 ˇ 2 S 4 S 1 1 S S S 2 S p S Ñ S 2 ds (4.146) Theseresultsshowthattheevolutionoftheinterfaceisgovernedbythecoupledsystem ‹ 4 : 132 “ V 2 1 S 1 1 S 4 S 2 Ñ (4.147) ‹ 4 : 146 “ 1 d˝ " 2 ˇ 2 S 1 S S S 2 ‹ 2 S 1 1 S 4 1 “ S p S Ñ S 2 ds (4.148) The H 1 gradientwdrivespureporeinterfacesbyamean-curvaturewandtheexternalchemical 82 potentialdecaysexponentiallytoitsequilibriavalue 1 Ð S 4 1 2 S 1 ; (4.149) Since 1 2 u 1 ,thedensityfunctiondecaysto u 1 Ð S 4 1 2 S 1 2 (4.150) andthebehaviourofthedensity, u ,takestheform u b " S 4 1 2 S 1 2 O ‹ " 2 “ : (4.151) 4.7CompetitiveGeometricEvolutionofBilayersandPores Afterobtainingaleadingorderexpressionfortheevolutionequationofthebilayermorphologyandthe poremorphology,wewouldliketoconsideracombinedsysteminwhichthetwomorphologiesco-exists. Let ` R 3 beagivendomainwithtwoadmissiblemanifolds b and p ,fortheco-dimensiononeandthe co-dimensiontwomorphology,respectively,whichsatisfy S b S O ‹ 1 “ and S p S O ‹ " 1 “ .Let b;l and p;l bethereachesof b and p ,respectively,onwhichthechangeofcoordinatestothewhiskeredcoordinates isunique. Awayfromtheinterface,attimescale ˝ "t ,theleadingorderexpressionforthemorphologiessolutions takestheform u p u b b " 1 2 O ‹ " 2 “ : (4.152) andthecompositesolutiontakestheform u b;p U b U p b " 1 2 O ‹ " 2 “ ; (4.153) andwealreadyseethatthetwostructureswillcompeteeachotherforsurfactantphasethroughthecommon, slowlyvarying,chemicalpotential 1 . For u ,acombinedbilayer-poresolutionoftheform( 4.153 ),thetotalmassconstraintofthecombinedsystem isgivenby M S ‹ u b “ dx S … b;l 8 p;l ‹ u b “ dx S b;l ‹ u b “ dx S p;l ‹ u b “ dx: (4.154) 83 Calculatingtheouterintegralyields S … b;l 8 p;l ‹ u b “ dx " 1 2 ‹S S S b;l 8 p;l S“ O ‹ " 2 “ ; (4.155) andplugging( 4.155 )backinto( 4.154 )yields M " 1 2 S S " 1 2 ‹S b;l S p;l S“ S b;l ‹ u b “ dx S p;l ‹ u b “ dx: (4.156) Using( 4.139 )andequation( 1.49 ),weobtainanexpressionfortheinnerintegrals S b;l ‹ u b “ dx "m b S b S " 1 2 S b;l S O ‹ " 2 “ ; (4.157) S p;l ‹ u b “ dx " 2 m p S p S " 1 2 S p;l S O ‹ " 3 “ ; (4.158) where m b R R ^ U b dz and m p 2 ˇS 1 and S 1 in( 4.70 ).Pluggingequations( 4.157 )and( 4.158 ) in( 4.156 )thetotalmasstakestheform M " 1 2 S S "m b S b S " 2 m p S p S : (4.159) Expanding M " ^ M O ‹ " 2 “ andusingourassumptionthat S b S O ‹ 1 “ and S p S O ‹ " 1 “ yields ^ M 1 2 S S m b S b S "m p S p S ; (4.160) Which,yieldstheconstraintonthechemicalpotential 1 , 1 2 S S › ^ M m b S b S "m p S p S ” : (4.161) Recallthattheinterfacialsurfaceareagrowthoftheporeisgiveninequation( 2.62 )andtheequivalent interfacialsurfaceareagrowthofthebilayerisgiveninequation( 3.110 ).Plugginginto( 2.62 )thenormal velocityofthepore, V p ,givenin( 4.132 ),andplugginginto( 3.110 )thenormalvelocityforthebilayer, V b , givenin( 3.102 ),yieldsleadingorderexpressionsforchangeinbilayersurfaceareaandporelength d S b S dt m b B 1 ‰ 1 ‹ 1 2 “ ˙ b 2 m b ’ S b H 2 0 ds; (4.162) d S p S dt 2 S 1 S 2 ‰ 1 S 4 2 S 1 1 ’ S p S Ñ S 2 ds; (4.163) 84 where B 1 isin( 3.93 ).Takingthetimederivativeof( 4.161 )yields 1 d˝ 2 S S „ m b d S b S d˝ "m p d S p S d˝ ‚ ; (4.164) andplugging( 4.162 )and( 4.163 )into( 4.164 )yields 1 d˝ 2 S S m b m b B 1 ‰ 1 ‹ 1 2 “ B 2 2 m b ’ S b H 2 0 ds "m p 2 S 1 S 2 ‰ 1 S 4 2 S 1 1 ’ S p S Ñ S 2 ds : (4.165) Theseresultsshowthatforinitialdatacorrespondingtospatiallyseparatedporeandbilayerstructuresyields acompetitiveevolutionthatcanbeunderstoodasatforsurfactant,mediatedthroughthecommonvalue ofthechemicalpotential 1 ,whoseevolutionisdeterminedtobytheconservationoftotalmass, ‹ 4 : 132 “ V p p ‹ 1 ⁄ p “ Ñ (4.166) ‹ 3 : 102 “ V b b ‹ 1 ⁄ b “ H 0 ; (4.167) ‹ 4 : 165 “ 1 d˝ 2 S S m b b ‹ 1 ⁄ b “ S b H 2 0 ds "m p p › 1 ⁄ p ” S p S Ñ S 2 ds ; (4.168) whereweintroducetheconstants b m b B 1 ; p 2 S 1 S 2 ; ⁄ b 1 2 ‹ 1 2 “ ˙ b m b ; ⁄ p 1 S 4 2 S 1 : (4.169) Thecompetitiveevolutionofthebilayersandporescouplesthroughcurvatureweightedsurfacearea.How- ever,thetwomorphologiesseekequilibriavalues,whichtypicallysatisfy ⁄ b A ⁄ p ,makingcoexistence ofbilayersandporesimpossibleunderthestrongfunctionalization,unlessoneofthestructuresissince zerocurvatureinterfacesareatequilibriumindependentofchemicalpotential.Forcurvedinterfaces,the range 1 > ⁄ p ; ⁄ b isinvariantunderthew,andonce 1 entersthisrangethebilayerswillshrink,while theporemorphologieswillgrow.Insection 7.2 wewillshownumericallytheequilibriaofeachsystemfora spchoiceofdouble-wellpotential,andthedynamicallyinvariantintervalisdescribedinFigure 7.2 . 85 Chapter5 ThePearlingEigenvalueProblem, Co-Dimension1 InthischapterweaddressthelinearstabilityofthebilayermorphologyinthestrongFCHandobtainan explicitexpressionforthepearlingstabilitycondition.Wepresentarigourousanalysisoftheeigenvalue problemcorrespondingtothestrongFCHfortheco-dimensiononestructure.Weshowthatinthe strong FCH scalingtheleadingorderbehaviorofthepearlingeigenvaluesisindependentoftheshapeofthe underlyingco-dimensiononemorphology.Underthe H 1 gradientwthepearlinginstabilitymanifests itselfonatimescalethatis O ‹ " 2 “ fasterthanthegeometricevolution,andhencecanbetakentobe instantaneousonthegeometricevolutiontimescale.Conversely,theinstabilityoccursonthe sametimescaleasthegeometricw,andmaynotnecessarilyimmediatelymanifestitselfonthegeometric evolutiontimescale. RecallthestrongFCHfreeenergywhichcorrespondstothechoice p 1in( 1.14 ), F ‹ u “ S 1 2 ‹ " 2 u W œ ‹ u ““ 2 " „ " 2 1 2 S © u S 2 2 W ‹ u “‚ dx; (5.1) where ` R d ;d C 2,isaboundeddomain, W ‹ u “ isatilteddouble-wellpotentialwithtwominimaat b , u R isthedensityofoneoftheamphiphilicspecies, " P 1controlsthewidthoftheboundarylayerand 1 and 2 arethefunctionalizationconstants.Thevariationof F ,introducedinequation( 1.18 ),isgiven by F u ‹ u “ ‹ " 2 W œœ ‹ u “ 1 “‹ " 2 u W œ ‹ u ““ d W œ ‹ u “ ; (5.2) 86 where d 1 2 .Thesecondvariationof F ,evaluatedatacriticalpointof E ,takestheform L b 2 F u 2 ‹ u “ › " 2 W œœ ‹ u “ 1 ”› " 2 W œœ ‹ u “ ” › " 2 u W œ ‹ u “ ” W œœœ ‹ u “ d W œœ ‹ u “ : (5.3) Weobtainapearlingstabilityconditionfortheco-dimensiononemorphologywhichissummarizedinthe followingtheorem- Theorem5.0.1. Foragivenadmissibleinterface, b ,theassociatedbilayersolutionconstructedin( 2.37 ), isstablewithrespecttothepearlingbifurcationifandonlyifthechemicalpotential 1 the pearlingstabilitycondition P ⁄ b b; 0 d Y b; 0 Y 2 L 2 ‹ “ 2 S b A 1 : (5.4) 5.1Overview Wewanttoinvestigatethepearlingeigenmodesoftheco-dimensiononebilayerstructure:givenanadmissible interface b > G K;` ,assumethatthesystemisatquasi-equilibrium,asin( 1.15 ),with u b U b ‹ z “ "u 1 ; (5.5) where U b isthehomoclinicbilayersolutionintroducedin( 2.37 ),and u 1 ,derivedinequation( 3.74 ),isgiven by L b; 0 u 1 1 b; 1 d ‰ z 2 U œ b ’ u 1 1 b; 2 d L 1 b; 0 ‰ z 2 U œ b ’ ; (5.6) where L b; 0 isthelinearoperatorintroducedin( 2.39 ),thechemicalpotential 1 isspatiallyconstantandthe functions b;j solves( 2.40 )for j 1 ; 2. Toshowthat u b ,inequation( 5.5 ),isaquasi-equilibrium,asin( 1.15 ),weplug( 5.5 )intothe variation,( 5.2 ),whichyields F u ‹ u b “ ‹ " 2 W œœ ‹ u b “ 1 “‹ " 2 u b W œ ‹ u b ““ d W œ ‹ u b “ : (5.7) ExpandingtheLaplacianinlocalcoordinates,( 2.9 ),andTaylorexpandingthepotentialterms W ‹ u b “ yields F u ‹ u b “ L b; 0 " ‹ H@ z W œœœ ‹ U b “ u 1 1 “ " 2 G X " ‹ H@ z U b L b; 0 u 1 “ " 2 H@ z u 1 d W œ ‹ U b “ (5.8) O ‹ " 2 “ " ‹ L 2 b; 0 u 1 d W œ ‹ U b ““ (5.9) 87 " 2 − H 2 U œœ b H@ z L b; 0 u 1 H ‹ W œœ ‹ U b ““ œ u 1 1 HU œ b W œœœ ‹ U b “ u 1 L b; 0 u 1 1 L b; 0 u 1 d W œœ ‹ U b “ u 1 ‘ O ‹ " 3 “ 1 O ‹ " 2 “ ; (5.10) wherethesecondequalityfollowsfromtheof u 1 in( 5.6 ).Theleadingordertermisspecially constant,whilethe O ‹ " 2 “ termsin( 5.8 )arelocalizedon b andconstantonthereach.Using 0 toproject awaytheconstantpartof( 5.8 ),yieldstermsthatare O ‹ " 2 “ in L ª andzeroofthereach,takingthe L 2 - normyields VV 0 F u ‹ u b “VV 2 L 2 ‹ “ O ‹ " 5 2 “ : (5.11) Weseethat u b theofquasi-equilibrium,givenin( 1.15 ). Weareinterestedinthepearlingeigenmodesofthesecondvariationof F , L b ,in( 5.3 ).Considerthe eigenvalueproblem L b : (5.12) BychangingcoordinatesoftheLaplacian,intheoperator L b ,tothewhiskeredcoordinates,using( 2.9 ),and plugging-intheexpansionof u b ,( 5.5 ),into u , L b canbewritteninordersof " as L b L 2 b " L 1 " 2 L 2 O ‹ " 3 “ ; (5.13) where L b introducedin( 2.45 ),andtheoperators L 1 ; L 2 taketheform L 1 L b X ‹ W œœœ ‹ U b “ u 1 “ ‹ W œœœ ‹ U b “ u 1 1 “ L b ‹ HU œ b L b u 1 “ W œœœ ‹ U b “ d W œœ ‹ U b “ ; (5.14) L 2 L b X ‰ W œœœ ‹ U b “ u 2 1 2 W ‹ 4 “ ‹ U b “ u 2 1 ’ ‰ W œœœ ‹ U b “ u 2 1 2 W ‹ 4 “ ‹ U b “ u 2 1 ’ L b (5.15) ‹ W œœœ ‹ U b “ u 1 1 “ W œœœ ‹ U b “ u 1 ‰ L b u 2 1 2 W œœœ ‹ U b “ u 2 1 ’ W œœœ ‹ U b “ ‹ L b u 1 HU œ b “ W ‹ 4 “ ‹ U b “ u 1 d W œœœ ‹ U b “ u 1 ; seeappendix D.1 fordetails.Notethatfor i C 1,theunboundedtermintheoperators L i is L b ,andwecan write L b inthefollowingform L b L 2 b " ~ L b ; (5.16) where L b isarelativelyboundedperturbationof L 2 b .Theeigenvaluesof L 2 b aredescribedinFigure 2.3 (right) wheretheboxedareacontainsthepearlingeigenvalues. 88 Recallthat b; 0 ,in( 2.48 ),isthesetofsmalleigenvaluesassociatedto L b ,and,accordingtoWeyl's asymptoticformula S b; 0 S O ‹ " 3 ~ 2 d “ .We P k " 1 ~ 2 ‹ b; 0 " 2 k “ ; (5.17) tobethedetuningconstantdependingonlyon k . 5.1. Thespace, X ,correspondingtothesmalleigenvaluesof L b isdas X Ÿ b; 0 k S k > b; 0 š 8 Ÿ b; 1 k S k > b; 1 š : (5.18) Themeandermodesareaccountedforinthegeometricmotion,however,weexpandonlythepearlingmodes, forbrevity.Lookingforsolutionsoftheeigenvalueproblem,( 5.12 ),weconsideraregularperturbation expansionoftheform j 0 ;j " 1 ;j O ‹ " 2 “ ; 0 ;j > X ; 0 ;j Q k > k b; 0 k ; 1 ;j > X Œ ; (5.19) j " 1 ;j O ‹ " 2 “ : (5.20) The L 2 -orthogonalprojection,onto X isgivenby f Q k > ‹ f; b; 0 k “ L 2 ‹ “ SS b; 0 k SS 2 L 2 ‹ “ b; 0 k Q k > ‹ f; b; 0 k “ L 2 ‹ “ b; 0 k ; (5.21) anditscomplementaryprojectionis ~ I Weconsideradecompositionoftheoperator L b intoa2 2blockform, < @ @ @ @ @ @ @ @ @ @ > MB B T C = A A A A A A A A A A ? ; (5.22) where M L b ;B L b ~ ;C ~ L b ~ : (5.23) Byabuseofnotationwedenote L b andits2 2decompositionwiththesamesymbol. [ HayrapetyanandPromislow,2014 ]haveshownthattherestrictedoperator C isuniformlycoerciveon X Œ 89 anditsspectrumisboundedfrombelowby A 0whichmaybechosenindependentoftlysmall " A 0.Insection 5.2 weinvestigatetheoperator M ,describingitsmatrixrepresentationand,developan expressionforitspearlingeigenvalues,presentedinTheorem 5.2.3 .Insection 5.3 ,wewillshowthat B;B T havean " -boundsasoperatorsfrom l 2 ‹ R N N “ to l 2 ‹ R N N “ .Insection 5.4 ,weconclude,usingbothsemi- groupestimatesandaperturbationargument,thatthespectrumof M determinesthepearlingeigenmodes of L b .Finally,insection 5.5 ,weconnectthepearlingeigenvaluesof L b tothoseof L b . 5.2Eigenvaluesof M L b Let v > X ,i.e., v canbewrittenas v Q k > b k b; 0 k ; (5.24) withoutlossofgenerality,assume SS v SS L 2 1.Theoperator L b actingon v ,takestheform L b Q j > b j b; 0 k Q k > ™ fl L b Q j > b j b; 0 j ; b; 0 k fi Ł L 2 ‹ “ b; 0 k Q k > Q j > b j ‹ L b b; 0 j ; b; 0 k “ L 2 ‹ “ b; 0 k : (5.25) Wetheoperatormatrixrepresentation M > R N d N d ,where N d " 3 ~ 2 d ,inthefollowingway M j;k ‹ L b b; 0 j ; b; 0 k “ L 2 ‹ “ : (5.26) Usingtheexpansionof L b ,( 5.13 ),wecanwriteeachentryof M inordersof " suchthat ‹ L b b; 0 j ; b; 0 k “ L 2 ‹ “ ‹ L 2 b b; 0 j ; b; 0 k “ L 2 ‹ “ " ‹ L 1 b; 0 j ; b; 0 k “ L 2 ‹ “ O ‹ " 2 “ ; (5.27) andcollectthematrixtermsintotwoclassessuchthat M M 0 " q ~ M; (5.28) where M 0 j;k ‹ L 2 b b; 0 j ; b; 0 k “ L 2 ‹ “ " ‹ L 1 b; 0 j ; b; 0 k “ L 2 ‹ “ q Q i 2 " i ‹ L i b; 0 j ; b; 0 k “ L 2 ‹ “ ; (5.29) ~ M j;k Q i C q " ‹ i q “ ‹ L i b; 0 j ; b; 0 k “ L 2 ‹ “ : (5.30) 90 Wewillshowthattheterm, M 0 ,canbesplitintoadiagonalandterms,thelatterofwhich canbeboundedindependentlyofthematrixsize N d ,ifthecurvaturesoftheinterface b aretly smooth.Theotherterm, ~ M ,canbebounded,independentofthedimension,viathe L ª norm,for q suitably large,dependinguponthedimension, d . 5.2.1Bounding ~ M Toestablishtheboundon ~ M ,westartwiththeofthe l 2 -normofamatrix,followedbyalemma whichestablishaboundonthe l 2 -normusingthe l ª norm: 5.2. Theinduced l 2 -normofamatrix A isgivenby, SS A SS l 2 sup SS v SS l 2 x 0 SS Av SS l 2 SS v SS l 2 : (5.31) Lemma5.1. Givenamatrix A > R N N ,thereexist C A 0 suchthat SS A SS l 2 @ C º N SS A SS l ª : (5.32) Proof. Let v > R N with SS v SS l 2 1.Then, SS A SS l 2 sup SS v SS l 2 x 0 SS Av SS l 2 SS v SS l 2 ¿ Á Á Á À N Q j 1 W N Q k 1 A j;k v k W 2 B ¿ Á Á Á À SS A SS 2 l ª N Q j 1 R R R R R R R R R R R N Q j 1 v k R R R R R R R R R R R 2 B SS A SS l ª ¿ Á Á À N Q j 1 SS v SS l 2 B SS A SS l ª º N: (5.33) Ì Corollary5.2.1. If q A 1 4 d 2 and SS ~ M SS l ª 1 then " p SS ~ M SS l 2 P C" ,where C isaconstantindependentof N d . Proof. Since SS ~ M SS l ª 1,applyingLemma( 5.1 )to ~ M yields SS ~ M SS l 2 B C º " 3 ~ 2 d C" 3 ~ 4 d ~ 2 ,forsomecon- stant C . Ì Corollary 5.2.1 impliesthatfor d 2 ; 3ittochoose q A 5 4 ; 7 4 respectively,torenderthe " q ~ M term O ‹ " “ intheinduced l 2 -norm. 91 5.2.2Bounding M 0 Next,wewanttoaboundin R d ;d C 2 ; forthematrix M 0 .Anexaminationofthetwoterms of M 0 ,giveninequation( 5.29 ),showsthattheyadmittheexpansions ‹ L 2 b b; 0 j ; b; 0 k “ L 2 ‹ “ ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ "P 2 k O ‹ " 2 “ if k j; " 2 R H 2 0 k j J 0 ds R l ~ " l ~ " ‹‹ 0 b; 0 “ œ “ 2 dz O ‹ " 2 º " “ if k x j; (5.34) ‹ L 1 b; 0 j ; b; 0 k “ L 2 ‹ “ ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ R l … " l … " W œœœ ‹ U b “‹ 0 b; 0 “ 2 L b; 0 u 1 dz d R l … " l … " W œœ ‹ U b “‹ 0 b; 0 “ 2 dz O ‹ º " “ if k j; " R H 1 k j ds R l … " l … " W œœœ ‹ U b “ U œ b ‹ 0 b; 0 “ 2 zdz O ‹ " 3 “ if k x j (5.35) (seeAppendix D.2 forcalculationdetails,sp,equations( D.5 )and( D.14 )).Wemaysplit M 0 into itsdiagonalmatrices M 0 M 0 diag M 0 (5.36) where M 0 diag ‹ j;k “ ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ M 0 k;k O ‹ " º " “ if j k; 0if j x k; (5.37) and M 0 ‹ j;k “ ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ 0if j k; M 0 j;k O ‹ " 2 º " “ if j x k: (5.38) withentriesgivenby M 0 k;k " „ P 2 k S l … " l … " W œœœ ‹ U b “ L b; 0 u 1 d W œœ ‹ U b 0 b; 0 “ 2 dz ‚ " › P 2 k 1 S d b; 0 SS b; 0 SS 2 2 ” ; (5.39) M 0 j;k " 2 S H 2 k j J 0 ds S l … " l … " ‹‹ 0 b; 0 “ œ “ 2 zdz " 2 S H 1 k j J 0 ds S l … " l … " W œœœ ‹ U b “ U œ b ‹ 0 b; 0 “ 2 zdz; " 2 ‰ S 1 S H 2 0 k j J 0 ds S 2 S H 1 k j J 0 ds ’ ; (5.40) withindices a" 3 ~ 2 d B j;k B ~ a" 3 ~ 2 d , a @ ~ a;a; ~ a > R , S b iscalledthe"shapefactor",and S b ;S 1 ;S 2 aregiven by S b S l ~ " l ~ " ' 1 W œœœ ‹ U “ 2 b; 0 dz; (5.41) S 1 S l ~ " l ~ " ‹‹ 0 b; 0 “ œ “ 2 dz; (5.42) 92 S 2 S l … " l … " W œœœ ‹ U b “ U œ b ‹ 0 b; 0 “ 2 zdz: (5.43) Seeappendix D.3 forthederivationofthesecondequalityinequation( 5.39 ). Theentriesof M 0 diag are O ‹ " “ .Ifwecanbound M 0 independentlyof " thentheeigenvaluesof M 0 are given,atleadingorder,bythediagonalentriesof M 0 diag .Tokeep M 0 orderof " 2 weneedtoboundthe twointegralsontheright-handsideofequation( 5.40 ).Since S 1 and S 2 arebounded,themainissueisto boundtheterms S H 1 k j J 0 ds; and S H 2 0 k j J 0 ds: (5.44) Wecanwritethetwotermsis( 5.44 )inamoregenericformas S f ‹ Ñ k b “ k j J 0 ds (5.45) where f ‹ Ñ k b “ isapolynomialofthecurvatures, Ñ k b inon 2.1 . Lemma5.2. Let b > R d beanadmissibleinterface,then,inparticular Ñ k b > W 2 ; ª .Let f R d 1 R bea boundedfunction,andthematrix M > R N N ;N > R withentries M i;j S f ‹ Ñ k b “ i j J 0 ds; (5.46) where k aretheeigenfunctionsofLaplace-Beltramioperator;then,thereexists C A 0 independentof " such that SS M SS l 2 l 2 B C: (5.47) Proof. Theoperatornormof M from l 2 to l 2 isby SS M SS l 2 l 2 inf Ÿ c A 0 US‹ Mv;w “S B c SS v SS l 2 SS w SS l 2 ; forall v;w > R N š : (5.48) Let v;w > R N ,usingtheof M ,( 5.46 ),wecanwrite S‹ Mv;w “S R R R R R R R R R R R Q i;j S f ‹ s “ i v i j w j J 0 ds R R R R R R R R R R R R R R R R R R R R R R S f ‹ s “‹ Q i i v i “‹ Q j j w j “ J 0 ds R R R R R R R R R R R ; (5.49) 93 andapplyingolder'sinequalitytothislastintegralyields S‹ Mv;w “S B SS f SS L ª ‹ “ WW Q i i v i WW L 2 ‹ “ R R R R R R R R R R R R R R R R R R R R R R Q j j w j R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ : (5.50) Calculatingthelastnormin( 5.50 )yields R R R R R R R R R R R R R R R R R R R R R R Q j j w j R R R R R R R R R R R R R R R R R R R R R R 2 L 2 ‹ “ S ™ fl Q j j w j fi Ł 2 J 0 ds Q j S 2 j w 2 j ds Q j w 2 j S 2 j dS SS w SS 2 l 2 (5.51) wherethesecondequalityfollowsformtheorthogonalityoftheLaplace-Beltramieigenfunctionsinthe innerproduct,seeequation( 2.71 ).Similarly,wehave R R R R R R R R R R R R R R R R R R R R R R Q j j v j R R R R R R R R R R R R R R R R R R R R R R 2 L 2 ‹ “ SS v SS 2 l 2 : (5.52) Plugging ‹ 5 : 51 “ and ‹ 5 : 52 “ backinto ‹ 5 : 50 “ yields S‹ Mv;w “S B SS f SS L ª ‹ “ SS v SS l 2 SS w SS l 2 ; (5.53) andbychoosing C SS f SS L ª ‹ “ andusingtheoperatornorm( 5.48 ),weobtainthedesiredbound, SS M SS l 2 l 2 B C: (5.54) Ì Corollary5.2.2. Thematrix M 0 ,din( 5.36 ),canbewrittenas M 0 M 0 diag M 0 ; (5.55) where M 0 isuniformlyboundedasanoperatorfrom l 2 to l 2 . Theorem5.2.3. Thepearlingeigenvaluesof L b ,( 5.20 ),taketheleadingorderform " 1 SS b; 0 SS 2 L 2 ‹ “ › 1 S b d b; 0 SS b; 0 SS 2 L 2 ‹ “ ” O ‹ " 2 “ ; (5.56) and,theassociatedco-dimensiononebilayernetworkispearlingstableifandonlyif 1 S b d b; 0 SS b; 0 SS 2 L 2 ‹ “ @ 0 : (5.57) 94 Proof. Corollary 5.2.2 impliesthattheeigenvaluesof M 0 , k ,are,atleadingorder,thediagonalentries of M 0 diag ,inequation( 5.37 ).Fromtheof M ,( 5.28 ),andCorollary 5.2.1 ,wededuce that k aretheeigenvaluesof M ,atleadingorder.Since M isthematrixrepresentationof L b the eigenvaluesof L b are,atleadingorder, k ,whichtakestheform k " ‹ P 2 k 1 S b d b; 0 SS b; 0 SS 2 L 2 ‹ “ “ ; (5.58) where S b istheshapefactorin( 5.41 )and P k isthedetuningconstantin( 5.17 ). Wewanttoalowerboundfortheeigenvaluesofthepearlingmodes: wehave o ‹ " 3 ~ 2 d “ possiblevalues for k > b; 0 ,forwhich ‹ b; 0 " 2 k “ O ‹ º " “ and,for d 2,theLaplace-Beltramieigenvaluestakesthe form k › 2 ˇk L ” 2 .Thedistancebetweentwosuccessivescaledeigenvaluesis " 2 k 1 " 2 k " 2 ‰ 2 ˇ L ’ 2 ‹ 2 k 1 “ : (5.59) Todeterminehowclose " 2 k cangetto b; 0 wechoose k 0 suchthat b; 0 " 2 › 2 ˇk 0 L ” 2 ,soif k 0 " 1 » b; 0 L 2 ˇ , theclosestwecanguaranteethat " 2 k approachesto b; 0 is " 2 ‰ 2 ˇ L ’ 2 ‹ 2 k 0 1 “ " 2 ‰ 2 ˇ L ’ 2 ™ fl " 1 » b; 0 L ˇ 1 fi Ł O ‹ " “ : (5.60) Recallthat P k " 1 ~ 2 ‹ b; 0 " 2 k “ then,thedistancebetweentwosequentialtermsis P k 1 P k " 1 ~ 2 ‹ b; 0 " 2 k 1 “ " 1 ~ 2 ‹ b; 0 " 2 k “ " 1 ~ 2 ‹ " 2 k " 2 k 1 “ " 1 ~ 2 " 2 ‰ 2 ˇ L ’ 2 ‹ 2 k 1 “ O ‹ " 1 ~ 2 “ : (5.61) Therefore,thedetuningparameter P k O ‹ " “ B P 2 k B O ‹ 1 “ for a" 1 ~ 2 B k B ~ a" 1 ~ 2 .Thisshowsthat P 2 k canbemadeassmallas O ‹ " “ andthereforeitislowerorderneartheturningpointofthepearlingspectrum. Weconcludethatthepearlingeigenvaluesof L b ( 5.20 ),takestheform " 1 SS b; 0 SS 2 L 2 ‹ “ › 1 S b d b; 0 SS b; 0 SS 2 L 2 ‹ “ ” O ‹ " 2 “ ; (5.62) Ì Notethatforagenericinterfacewerecoverthesamepearlingconditionsasforinterfaceswithconstant curvature,see[ Doelmanetal.,2014 ]formoredetails. Recallthatourmaingoalistoanexpressionforthepearlingeigenvaluesof L b usingour2 2represen- 95 tationof L b ,see( 5.22 ).Inthissectionwefoundanexpressionthepearlingeigenvaluesoftheoperator M . Thenextsectionestablishtheboundsontheterms B;B T . 5.3BoundingtheOperators Recallthe2 2blockformrepresentationof L b ,givenin( 5.22 ), < @ @ @ @ @ @ @ @ @ @ > MB B T C = A A A A A A A A A A ? : (5.63) Iftheblocks, B L b ~ and B T ~ L b aresmall(sameorderofthe L b blockorless) thenwecanrelatetheeigenvaluesof L b tothoseof M L b seesection 5.4 .Sincebothand ~ are self-adjointoperatorswehave ‹ L b ~ v;w “ L 2 ‹ L b ~ v; w “ L 2 ‹ ~ v; L b w “ L 2 ‹ v; ~ L b w “ L 2 : (5.64) So,itisenoughtoshowthatoneoftheblocksissmall,i.e.,wewanttoshowthatthereexista constant C ,independenton N d O ‹ " 3 ~ 2 d “ suchthat Y ~ L b v Y L 2 ‹ “ B "C Y v Y L 2 ‹ “ ; ¦ v > X : (5.65) withoutlossofgenerality,assume v > X ;v P j > b j b; 0 j and Y v Y L 2 ‹ “ 1.Notethat Y v Y 2 L 2 ‹ “ S Q j;k > b j b k j k 2 b; 0 dx Q j > b 2 j Y k b; 0 Y 2 L 2 ‹ “ 1 ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ S 2 j 2 b; 0 dx Q j;k > j x k b j b k 0 ; byorthogonalityof j ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ S j k 2 b; 0 dx Q j > b 2 j SS b SS 2 l 2 ; (5.66) where b ‹ b 1 ;b 2 ;:::;b N d “ . Since L b L 2 b " ~ L b ,and ~ L b isrelativelyboundedwithrespectto L 2 b ,wesplittheproofintothreeparts: weshowthatweappropriatelyboundtheoperator ~ L b v ,nextweappropriatelyboundtheoperator ~ L 2 b v andatallyweappropriatelybound ~ L b v . 96 5.3.1Bounding ~ L b v Recallthat v > X ;v P j > b j b; 0 j and Y v Y L 2 ‹ “ 1.Inparticular, v v; ~ v 0 : (5.67) Weneedtoshowthatthereexist C 1 suchthat Y ~ L b v Y L 2 ‹ “ B "C 1 Y v Y L 2 ‹ “ : (5.68) Usingtheexpressionfor L b ,( 2.45 ),yields L b v ‹ L b; 0 v "H@ z v " 2 G v “ Q j > b j ™ Œ Œ fl b; 0 b; 0 j ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ > X "H j œ b; 0 " 2 G b; 0 j fi Š Š Ł : (5.69) Theprojection ~ istheorthogonalprojectionto X ,thereiteliminatesthetermandtheoperator ~ L b v takestheform ~ L b v " ~ Q j > b j › H j œ b; 0 " G b; 0 j ” : (5.70) The L 2 -normofequation( 5.70 )isgivenby TT ~ L b v TT L 2 ‹ “ R R R R R R R R R R R R R R R R R R R R R R " ~ Q j > b j › H j œ b; 0 " G b; 0 j ” R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ B " R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R v ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ Q j > b j H j œ b; 0 R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ " 2 TT ~ G v TT L 2 ‹ “ ; (5.71) whereweusedtriangleinequalityandthefactthat Y ~ u Y L 2 ‹ “ B Y u Y L 2 ‹ “ foreach u . Wetheoperator R matrixrepresentation B > R N d N d ,with N d " 3 ~ 2 d suchthat B j;k a H j œ b; 0 ;H j œ b; 0 f L 2 ‹ “ : (5.72) Theentriesof B taketheform B k;j S f k j J 0 ds; (5.73) 97 andwecanapplyLemma 5.2 toconcludethat B isuniformlyboundedasanoperatorfrom l 2 ‹ R N “ to l 2 ‹ R N “ . Since B isthematrixrepresentationof R wehave SS R v SS 2 L 2 ‹ “ TT bBb T TT 2 L 2 ‹ “ B SS b SS 2 l 2 SS B SS 2 l 2 l 2 : (5.74) Theorem 5.2 and( 5.74 )implies SS R v SS L 2 ‹ “ B c SS v SS L 2 : (5.75) Goingbackto( 5.71 )andpluggingin( 5.75 )yields TT ~ L b v TT L 2 ‹ “ B "c SS v SS L 2 " 2 TT ~ G v TT L 2 ‹ “ (5.76) ThefollowingPropositionshowsthatthe L 2 ‹ “ -normof ~ G v canbeboundedasanoperatorinthe L 2 ‹ “ norm: Proposition5.3.1. Let f ‹ z “ beasmoothfunctionsuchthat S f ‹ z “S @ c 1 e c 2 S z S forsome c i > R ;c i A 0 ;i 1 ; 2 ;supp ‹ f “ ` l : (5.77) Theoperator ~ G ,where ~ istheprojectionofthespaceofsmalleigenvalues X ,edin( 5.18 ) and G isdin( 2.12 ),isboundedonthespace Y Ÿ f ‹ z “ k U k > ; š ; (5.78) i.e.,thereexists C A 0 suchthat TT ~ G v TT L 2 ‹ “ B C" 2 SS v SS L 2 ‹ “ ; (5.79) forevery v > Y .Particularly,for v > x weobtainthebound TT ~ G v TT L 2 ‹ “ B C" 1 SS v SS L 2 ‹ “ : (5.80) Proof. Fix ⁄ > ˆ ‹ s “ ,where ˆ ‹ s “ istheresolventsetoftheLaplace-Beltramioperator,thentheopera- tor G canbewrittenas G s "zD s; 2 ‹ s ⁄ “ "zD s; 2 ‹ s ⁄ “ 1 ‹ s ⁄ “ ⁄ : (5.81) 98 TheLaplace-Beltramioperatorisinvarianton Y ,i.e, s v > Y forevery v > Y andit SS s v SS L 2 ‹ “ B " 2 SS v SS L 2 ‹ “ (5.82) andsince D s; 2 isarelativelyboundedperturbationof s ,seeLemma 2.1 ,theoperator D s; 2 ‹ s ⁄ “ 1 is bounded,independentlyon " ,on Y . Let v > Y ~ X , v f ‹ z “ Q j > b j j ; (5.83) with Y v Y L 2 ‹ “ 1. Takingthe L 2 ‹ “ -normof ~ G actingon v yields TT ~ G v TT L 2 ‹ “ TT ~ ‹ s ⁄ “ v "z ~ D s; 2 ‹ s ⁄ “ 1 ‹ s ⁄ “ v TT L 2 ‹ “ (5.84) B TT ~ s v TT L 2 ‹ “ TT ~ ⁄ v TT L 2 ‹ “ " TT z ~ D s; 2 ‹ s ⁄ “ 1 ‹ s ⁄ “ v TT L 2 ‹ “ (5.85) B " 2 SS v SS L 2 ‹ “ S ⁄ SSS v SS L 2 ‹ “ " TT D s; 2 ‹ s ⁄ “ 1 TT L 2 ‹ “ SS‹ s ⁄ “ zv SS L 2 ‹ “ (5.86) B " 2 SS v SS S ⁄ SSS v SS c" 1 SS zv SS L 2 ‹ “ (5.87) Weconcludethat TT ~ G TT L 2 ‹ “ B C" 2 on Y: (5.88) Similarly,taking v > X ` Y , v b; 0 Q j > b j j ; (5.89) with Y v Y L 2 ‹ “ 1,the L 2 ‹ “ -normof ~ G actingon v yields TT ~ G v TT L 2 ‹ “ TT ~ ‹ s ⁄ “ v " ~ D s; 2 ‹ s ⁄ “ 1 ‹ s ⁄ “ zv TT L 2 ‹ “ (5.90) B R R R R R R R R R R R R R R R R R R R R R R R R * 0 ~ s v R R R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ R R R R R R R R R R R R R R R R R R R R R R * 0 ~ ⁄ v R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ " TT ~ D s; 2 ‹ s ⁄ “ 1 ‹ s ⁄ “ zv TT L 2 ‹ “ (5.91) B " TT D s; 2 ‹ s ⁄ “ 1 TT L 2 ‹ “ SS z s v SS L 2 ‹ “ (5.92) B c" 1 SS v SS L 2 ‹ “ (5.93) Weconcludethat TT ~ G TT L 2 ‹ “ B C" 1 on X : (5.94) 99 Ì CombiningProposition 5.3.1 withequation( 5.76 )weobtaintherequiredbound TT ~ L b v TT L 2 ‹ “ B "c SS v SS L 2 : (5.95) 5.3.2Bounding ~ L 2 b v Recallthat v > X ;v P j > b j b; 0 j and Y v Y L 2 ‹ “ 1.Inparticular, v v; ~ v 0 : (5.96) Wewanttoshowthereexist C 2 ,independenton " ,suchthat TT ~ L 2 b v TT L 2 ‹ “ B "C 2 SS v SS L 2 : (5.97) Writingthe L 2 b operatoractingon v explicitlywehave L 2 b v L b ‹ L b v “ ‹ L b; 0 "H@ z " 2 G “‹ L b; 0 v "H@ z v " 2 G v “ (5.98) ‹ L b; 0 "H@ z " 2 G “ Q j > b j › b; 0 b; 0 j "H j œ b; 0 " 2 G b; 0 j ” (5.99) Q j > b j > X ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ 2 b; 0 b; 0 j "L b; 0 ‹ H j œ b; 0 “ " 2 L b; 0 ‹ G b; 0 j “ b; 0 H œ b; 0 j " 2 H j @ z ‹ H œ b; 0 “ (5.100) " 3 H@ z ‹ G b; 0 j “ " 2 b; 0 b; 0 G j " 3 G œ b; 0 ‹ H j “ " 4 2 G b; 0 j Projectingawayfrom X using ~ andtakingthe L 2 -normyields TT ~ L 2 b v TT L 2 ‹ “ Y ~ Q j > b j > X ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ 2 b; 0 b; 0 j "L b; 0 ‹ H j œ b; 0 “ " 2 L b; 0 ‹ G b; 0 j “ b; 0 H œ b; 0 j " 2 H j @ z ‹ H œ b; 0 “ (5.101) " 3 H@ z ‹ G b; 0 j “ " 2 b; 0 b; 0 G j " 3 G œ b; 0 ‹ H j “ " 4 2 G b; 0 j L 2 ‹ “ B " Y R 1 v ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ Q j > b j L b; 0 ‹ H j œ b; 0 “ b; 0 H œ b; 0 j "H j @ z ‹ H œ b; 0 L 2 ‹ “ " 3 TT ~ H@ z ‹ G v “ TT L 2 ‹ “ (5.102) " 3 TT ~ G H@ z v TT L 2 ‹ “ " 2 TT ~ b; 0 G v TT L 2 ‹ “ " 4 TT ~ 2 G v TT L 2 ‹ “ " 2 TT ~ L b; 0 ‹ G v “ TT L 2 ‹ “ 100 whereweusedthetriangleinequalitytoderive( 5.102 )from( 5.101 ).Weintroducethematrix B > R N N , thematrixrepresentationof R 1 ,givenby B j;k a ‹ L b; 0 ‹ H œ b; 0 “ b; 0 H œ b; 0 "H@ z ‹ H œ b; 0 ““ j ; ‹ L b; 0 ‹ H œ b; 0 “ b; 0 H œ b; 0 "H@ z ‹ H œ b; 0 ““ k f L 2 : (5.103) Theentriesof B taketheform B j;k S S ` ~ " ` ~ " f ‹ z;s “ j k Jdzds; (5.104) andbyLemma 5.2 ,weconcludethatthereexists c A 0,independentof " ,suchthat TT B TT l 2 l 2 B c: (5.105) Using( 5.105 )weobtainaboundontheoperator R 1 SS R 1 v SS 2 L 2 ‹ “ B SS b SS 2 l 2 TT B TT 2 l 2 l 2 B c SS v SS 2 l 2 ; (5.106) andequation( 5.102 )reducesto TT ~ L 2 b v TT L 2 ‹ “ B c" SS v SS L 2 ‹ “ " 3 TT ~ H@ z ‹ G v “ TT L 2 ‹ “ " 3 TT ~ G H@ z v TT L 2 ‹ “ (5.107) " 2 TT ~ b; 0 G v TT L 2 ‹ “ " 4 TT ~ 2 G v TT L 2 ‹ “ " 2 TT ~ L b; 0 ‹ G v “ TT L 2 ‹ “ Considerthesecondtermintheright-handsideof( 5.107 ). " 3 TT ~ H@ z ‹ G v “ TT L 2 ‹ “ B " 3 TT ~ H G @ z v TT L 2 ‹ “ " 3 TT ~ H @ z G v TT L 2 ‹ “ (5.108) B "c 1 SS v SS L 2 ‹ “ " 3 TT ~ H @ z G v TT L 2 ‹ “ ; (5.109) wherethesecondinequalityfollowsapplyingProposition 5.3.1 tothetermin( 5.108 ).Inordertoshow thatthesecondtermin( 5.108 )isbounded,werecalltheof G ,givenin( 2.12 ).Takingthe z derivativeof( 2.12 )yields @ z G "D s; 2 "z @ z D s; 2 ; (5.110) where @ z D s; 2 denotesthemultiplieroperatorcomprisedofthe z derivativeofthecotsof D s; 2 ,which satisfythebounds( 2.15 ). 101 Proposition5.3.2. Let f ‹ z “ beasmoothfunctionsuchthat S f ‹ z “S @ c 1 e c 2 S z S forsome c i > R ;c i A 0 ;i 1 ; 2 ; and supp ‹ f “ ` l : (5.111) Theoperator @ z G ,where G isdin( 2.12 ),isboundedonthespace Y span Ÿ f ‹ z “ k U k > ; š ; (5.112) i.e.,thereexists C A 0 , C independentof " and f ,suchthat SS @ z G v SS L 2 ‹ “ B C" 1 SS v SS L 2 ‹ “ ; (5.113) forevery v > Y . Proof. Fix ⁄ > ˆ ‹ s “ ,where ˆ ‹ s “ istheresolventsetoftheLaplace-Beltramioperator,thentheopera- tor G canbewrittenas @ z G @ z ‹ s "zD s; 2 “ "z @ z D s; 2 s ⁄ “ 1 ‹ s ⁄ “ "D s; 2 ‹ s ⁄ “ 1 ‹ s ⁄ “ : (5.114) FromLemma 2.1 weknowthat D s; 2 isarelativelyboundedperturbationof s ,i.e.,thereexists C A 0, independentof " ,suchthat TT D s; 2 ‹ s ⁄ “ 1 TT l 2 l 2 B C: (5.115) Let v > Y , v f ‹ z “ Q j > b j j ; (5.116) with Y v Y L 2 ‹ “ 1. Takingthe L 2 ‹ “ -normof @ z G actingon v yields SS @ z G v SS L 2 ‹ “ TT "z @ z D s; 2 s ⁄ “ 1 ‹ s ⁄ “ "D s; 2 ‹ s ⁄ “ 1 ‹ s ⁄ “ v TT L 2 ‹ “ (5.117) B " TT D s; 2 ‹ s ⁄ “ 1 ‹ s ⁄ “ v TT L 2 ‹ “ " TT z @ z D s; 2 s ⁄ “ 1 ‹ s ⁄ “ v TT L 2 ‹ “ (5.118) B " TT D s; 2 ‹ s ⁄ “ 1 TT l 2 l 2 SS‹ s ⁄ “ v SS L 2 ‹ “ (5.119) " TT @ z D s; 2 s ⁄ “ 1 TT l 2 l 2 SS‹ s ⁄ “ zv SS L 2 ‹ “ 102 Notethat @ z D s; 2 d 1 Q i;j 1 ‹ @ z d i;j ‹ s;z ““ @ 2 @s i @s j d 1 Q j 1 ‹ @ z d j ‹ s;z ““ @ @s j (5.120) B SS @ z d i;j ‹ s;z “SS L ª ‹ l “ d 1 Q i;j 1 d i;j @ 2 @s i @s j SS @ z d j ‹ s;z “SS L ª ‹ l “ d 1 Q j 1 d j ‹ s;z “ @ @s j (5.121) B max @ z d i;j ‹ s;z “SS L ª ‹ l “ ; SS @ z d j SS L ª ‹ l “ D s; 2 B c"D s; 2 ; (5.122) forsome c A 0,whereforthelastinequalityweusedtheboundsonthe z -derivativesofthecots of D s; 2 ,giveninequation( 2.15 ).Combining( 5.120 )with( 5.119 )yields SS @ z G v SS L 2 ‹ “ B c" TT D s; 2 ‹ s ⁄ “ 1 TT L 2 ‹ “ SS‹ s ⁄ “ v SS L 2 ‹ “ ; (5.123) andweconcludethat SS @ z G SS L 2 ‹ “ B C" 1 on Y: (5.124) Ì Returningto( 5.108 )wehave " 3 TT ~ H@ z ‹ G v “ TT L 2 ‹ “ B "c 1 SS v SS L 2 ‹ “ " 3 TT ~ H ‹ @ z G “ v TT L 2 ‹ “ (5.125) B "c 1 SS v SS L 2 ‹ “ " 3 SS H ‹ @ z G “ v SS L 2 ‹ “ (5.126) B "c 1 SS v SS L 2 ‹ “ " 3 SS H SS L ª ‹ b;` “ SS @ z G v SS L 2 ‹ “ : (5.127) wherethethirdinequalityfollowsfromthegeneralizedolderinequality,see( D.36 ).CombiningProposi- tion 5.3.2 withequation( 5.127 )weobtainthebound " 3 TT ~ H@ z ‹ G v “ TT L 2 ‹ “ B "c 2 SS v SS L 2 ‹ “ ; (5.128) where c 2 isindependentof " ,butitdependupon SS H SS L ª ‹ b;` “ .Plugging( 5.128 )andintotheright-hand sideof( 5.107 )yields TT ~ L 2 b v TT L 2 ‹ “ B " ~ c SS v SS L 2 ‹ “ " 3 TT ~ G H@ z v TT L 2 ‹ “ " 2 TT ~ b; 0 G v TT L 2 ‹ “ " 4 TT ~ 2 G v TT L 2 ‹ “ (5.129) " 2 TT ~ L b; 0 ‹ G v “ TT L 2 ‹ “ : 103 Toshowthatthesecondtermin( 5.129 )isboundedasanoperatorinthe L 2 ‹ “ normwith " 3 TT ~ G H@ z v TT L 2 ‹ “ B c 1 " SS v SS L 2 ‹ “ (5.130) weusethefollowingLemma- Lemma5.3. Fix v > X , v Q j > b j j b; 0 ; (5.131) with SS v SS L 2 ‹ “ 1 .Then,if b isadmissible,and,inparticular,if Ñ k > W 2 ; ª ‹ “ ,thenthereexists C A 0 independentof " suchthat SS s ‹ Hv “SS L 2 ‹ “ B C" 1 SS v SS L 2 ‹ “ (5.132) Proof. Recallthat H ,givenin( 2.10 ),hastheexpansion H H 0 ‹ s “ "zH 1 ‹ s “ :::; (5.133) andthejacobiantakestheform J b ‹ s;z “ J 0 ‹ s “ ~ J b J 0 ‹ s “‹ " " 2 H 0 z ::: “ : (5.134) Theterm s ‹ Hv “ hastheexplicitform s ‹ Hv “ ‹ s H “ v ‹ s v “ H 2 © s H © s v: (5.135) Takingthe L 2 normofequation( 5.135 ) SS s ‹ Hv “SS L 2 ‹ “ B SS‹ s H “ v SS L 2 ‹ “ SS‹ s v “ H SS L 2 ‹ “ 2 SS © s H © s v SS L 2 ‹ “ : (5.136) Usingtheexpansionof H and J b wecanboundthetermontheright-handsideofequation( 5.136 ) SS‹ s H “ v SS 2 L 2 ‹ “ " S ‹ s H 0 “ 2 ‹ Q j > b j j “ 2 J 0 ds S ` ~ " ` ~ " 2 b; 0 dz (5.137) " 2 ™ fl S s ‹ H 0 H 1 “‹ Q j > b j j “ 2 J 0 ds S ` ~ " ` ~ " 2 b; 0 z 2 dz S H 0 s ‹ H 2 0 “‹ Q j > b j j “ 2 J 0 ds S ` ~ " ` ~ " 2 b; 0 z 2 dz fi Ł 104 O ‹ " 3 “ B c − " TT ‹ s H 0 “ 2 TT L ª ‹ “ SS v SS 2 L 2 ‹ “ (5.138) " 2 s ‹ H 0 H 1 “SS L ª ‹ “ SS v SS 2 L 2 ‹ “ TT H 0 s H 2 0 TT L ª ‹ “ SS v SS 2 L 2 ‹ “ ‘ O ‹ " 3 “ ; andaslongastheinterfaceisadmissible,i.e., Ñ k > W 2 ; ª ‹ “ ,wehave SS‹ s H “ v SS L 2 ‹ “ B c" 1 SS v SS L 2 ‹ “ : (5.139) Thesecondtermontheright-handsideofequation( 5.136 )canbebounded SS H s v SS 2 L 2 ‹ “ " S H 2 0 ‹ Q j > b j s j “ 2 J 0 ds S ` ~ " ` ~ " 2 b; 0 dz (5.140) " 2 ™ fl S H 0 H 1 ‹ Q j > b j s j “ 2 J 0 ds S ` ~ " ` ~ " 2 b; 0 z 2 dz S H 0 H 2 0 ‹ Q j > b j s j “ 2 J 0 ds S ` ~ " ` ~ " 2 b; 0 z 2 dz fi Ł O ‹ " 3 “ B c" TT f ‹ Ñ k “ TT L ª ‹ “ SS s v SS 2 L 2 ‹ “ : (5.141) Since SS s v SS L 2 ‹ “ B " 2 SS v SS L 2 ‹ “ ,andsincetheinterfaceisadmissibleand f isapolynomialofthecurvatures ofweknowthat TT f ‹ Ñ k “ TT L 2 ‹ “ isbounded,independentof " .Thisimpliesthat SS H s v SS L 2 ‹ “ B c" 1 SS v SS L 2 ‹ “ : (5.142) Weapplysimilarcalculationstothethirdtermontheright-handsideofequation( 5.136 )toobtain SS © s H © s v SS 2 L 2 ‹ “ B c" TT f ‹ © s Ñ k “ TT L ª ‹ “ SS © s v SS 2 L 2 ‹ “ ; (5.143) SS © s v SS 2 L 2 ‹ “ S ‹ Q j b j © j “ 2 J 0 ds S ` ~ " ` ~ " 2 b; 0 dz Q j b 2 j S ‹ j s j “ J 0 ds S ` ~ " ` ~ " 2 b; 0 dz (5.144) Q j b 2 j j S 2 j J 0 ds S ` ~ " ` ~ " 2 b; 0 dz B " 2 SS v SS 2 L 2 ‹ “ ; (5.145) andaslongastheinterfaceisadmissible,i.e., Ñ k > W 1 ; ª ‹ “ ,weobtainthebound SS © s H © s v SS L 2 ‹ “ B c" 1 SS v SS L 2 ‹ “ : (5.146) 105 Plugging( 5.139 ),( 5.142 )and( 5.146 )backinto( 5.136 )weconcludethat,aslongastheinterfaceisadmissible, thereexists C A 0independenton " suchthat SS s ‹ Hv “SS L 2 ‹ “ B C" 1 SS v SS L 2 ‹ “ : (5.147) Ì Proposition5.3.3. Fix v > X , v Q j > b j j b; 0 ; (5.148) with SS v SS L 2 ‹ “ 1 .Theoperator ~ G Hv ,where ~ istheprojectionofthespaceofsmalleigenvalues X , din( 5.18 )and G isdin( 2.12 ),isboundedonthespace Y Ÿ f ‹ z “ k U k > ; š ; (5.149) i.e.,thereexists C A 0 suchthat TT ~ G Hv TT L 2 ‹ “ B C" 2 SS v SS L 2 ‹ “ forall v > Y: (5.150) Proof. WerepeattheproofofProposition 5 : 3 : 1 butreplacingequation( 5.82 )withequation( 5.132 ),and takingthe L 2 ‹ “ normof ~ G actingon Hv yieldstherequiredresult. Ì Proposition 5.3.1 showsthatthethirdtermin( 5.129 )isboundedinthe L 2 ‹ “ operatornormwith " 2 TT ~ b; 0 G v TT L 2 ‹ “ B c 2 " SS v SS L 2 ‹ “ : (5.151) Inserting( 5.151 )and( 5.130 )into( 5.129 )yields TT ~ L 2 b v TT L 2 ‹ “ B " c SS v SS L 2 ‹ “ " 4 TT ~ 2 G v TT L 2 ‹ “ " 2 TT ~ L b; 0 ‹ G v “ TT L 2 ‹ “ : (5.152) Theboundonthesecondtermintheright-handsideof( 5.152 )followsfromthefollowinglemma Lemma5.4. Let v > X , v Q j > b j j b; 0 ; (5.153) 106 with Y v Y L 2 ‹ “ 1 .Thenthereexists C A 0 suchthat TT ~ 2 G v TT L 2 ‹ “ B C" 3 SS v SS L 2 ‹ “ : (5.154) Proof. Fix ⁄ > ˆ ‹ s “ ,where ˆ ‹ s “ istheresolventsetoftheLaplace-Beltramioperator,thentheopera- tor G canbewrittenas 2 G ‹ s "zD s; 2 “ 2 2 s 2 "zD s; 2 ‹ s ⁄ “ 1 ‹ s ⁄ “ s " 2 z 2 D 2 s; 2 ‹ s ⁄ “ 2 ‹ s ⁄ “ 2 : (5.155) TheLaplace-Beltramioperator TT 2 s v TT L 2 ‹ “ B " 4 SS v SS L 2 ‹ “ ; (5.156) andsince D s; 2 isarelativelyboundedperturbationof s ,weconcludethattheoperator D s; 2 ‹ s ⁄ “ 1 is boundedon Y . Takingthe L 2 ‹ “ -normof ~ 2 G v andusingequation( 5.155 )toexpress 2 G yields TT ~ ‹ 2 G “ v TT L 2 ‹ “ B UU ~ 2 s v UU L 2 ‹ “ 2 " TT zD s; 2 ‹ s ⁄ “ 1 ‹ s ⁄ “ s v TT L 2 ‹ “ (5.157) " 2 TT D 2 s; 2 ‹ s ⁄ “ 2 ‹ s ⁄ “ 2 z 2 v TT L 2 ‹ “ ; B 2 " TT D s; 2 ‹ s ⁄ “ 1 TT L 2 ‹ “ SS‹ s ⁄ “ s zv SS L 2 ‹ “ (5.158) " 2 TT D 2 s; 2 ‹ s ⁄ “ 2 TT L 2 ‹ “ TT ‹ s ⁄ “ 2 z 2 v TT L 2 ‹ “ : FromLemma 2.1 weknowthat D s; 2 isarelativelyboundedperturbationof s ,i.e.,thereexists C A 0, independentof " ,suchthat TT D s; 2 ‹ s ⁄ “ 1 TT L 2 ‹ “ B C: (5.159) Provingthat D 2 s; 2 isarelativelyboundedperturbationof 2 s issimilarand( 5.157 )reducesto TT ~ ‹ @ 2 z G “ v TT L 2 ‹ “ B C" 3 SS v SS L 2 ‹ “ : (5.160) Ì CombiningLemma 5.4 andequation( 5.129 )yields TT ~ L 2 b v TT L 2 ‹ “ B " c SS v SS L 2 ‹ “ " 2 TT ~ L b; 0 ‹ G v “ TT L 2 ‹ “ : (5.161) 107 Considerthesecondtermin( 5.161 ),andnotethat W œœ ‹ U b “ commuteswith G .Then " 2 TT ~ L b; 0 ‹ G v “ TT L 2 ‹ “ " 2 TT ~ ‹ @ 2 z W œœ ‹ U b ““‹ G v “ TT L 2 ‹ “ (5.162) B " 2 TT ~ ‹ @ 2 z G “ v TT L 2 ‹ “ " 2 TT 2 ~ ‹ @ z G “‹ @ z v “ TT L 2 ‹ “ " 2 TT ~ G L b; 0 v TT L 2 ‹ “ (5.163) " 2 TT ~ ‹ @ 2 z G “ v TT L 2 ‹ “ " 2 TT 2 ~ ‹ @ z G “‹ @ z v “ TT L 2 ‹ “ " 2 TT b; 0 ~ G v TT L 2 ‹ “ : (5.164) Forthetermin( 5.164 )wehavethefollowinglemma Lemma5.5. Let v > X , v Q j > b j j b; 0 ; (5.165) with Y v Y L 2 ‹ “ 1 .Thenthereexists C A 0 suchthat TT ~ ‹ @ 2 z G “ v TT L 2 ‹ “ B C SS v SS L 2 ‹ “ : (5.166) Proof. Fix ⁄ > ˆ ‹ s “ ,where ˆ ‹ s “ istheresolventsetoftheLaplace-Beltramioperator,thentheopera- tor G canbewrittenas @ 2 z G @ 2 z ‹ s "zD s; 2 “ "z@ 2 z ‹ D s; 2 “‹ s ⁄ “ 1 ‹ s ⁄ “ " 2 @ z D s; 2 ‹ s ⁄ “ 1 ‹ s ⁄ “ : (5.167) FromLemma 2.1 weknowthat D s; 2 isarelativelyboundedperturbationof s ,i.e.,thereexists C A 0, independentof " ,suchthat TT D s; 2 ‹ s ⁄ “ 1 TT L 2 ‹ “ B C: (5.168) Takingthe L 2 ‹ “ -normof ~ ‹ @ 2 z G “ v combinedwithequation( 5.167 )yields TT ~ ‹ @ 2 z G “ v TT L 2 ‹ “ TT ~ "z ‹ @ 2 z D s; 2 “‹ s ⁄ “ 1 ‹ s ⁄ “ 2 " ‹ @ z D s; 2 “‹ s ⁄ “ 1 ‹ s ⁄ “ v TT L 2 ‹ “ ; (5.169) B 2 " TT ‹ @ z D s; 2 “‹ s ⁄ “ 1 ‹ s ⁄ “ v TT L 2 ‹ “ " TT z ‹ @ 2 z D s; 2 “‹ s ⁄ “ 1 ‹ s ⁄ “ v TT L 2 ‹ “ ; B 2 " TT ‹ @ z D s; 2 “‹ s ⁄ “ 1 TT L 2 ‹ “ SS‹ s ⁄ “ v SS L 2 ‹ “ (5.170) " TT ‹ @ 2 z D s; 2 “‹ s ⁄ “ 1 TT l 2 l 2 SS‹ s ⁄ “ zv SS L 2 ‹ “ : From( 5.120 )weknowthat @ z D s; 2 B c"D s; 2 ; (5.171) 108 andsimilarcalculationsshowsthatthereexistsaconstant~ c A 0suchthat @ 2 z D s; 2 B ~ c" 2 D s; 2 : (5.172) Plugging( 5.172 )and( 5.120 )into( 5.169 )yields TT ~ ‹ @ 2 z G “ v TT L 2 ‹ “ B 2 " 2 TT ‹ D s; 2 “‹ s ⁄ “ 1 TT l 2 l 2 SS‹ s ⁄ “ v SS L 2 ‹ “ (5.173) " 3 TT ‹ D s; 2 “‹ s ⁄ “ 1 TT l 2 l 2 SS‹ s ⁄ “ zv SS L 2 ‹ “ ; B C SS v SS L 2 ‹ “ : (5.174) Ì Returningto( 5.164 ),andusingLemma 5.5 wehave " 2 TT ~ L b; 0 ‹ G v “ TT L 2 ‹ “ B C" 2 SS v SS L 2 ‹ “ " 2 TT 2 ~ ‹ @ z G “‹ @ z v “ TT L 2 ‹ “ " 2 TT b; 0 ~ G v TT L 2 ‹ “ : (5.175) ApplyingProposition 5.3.2 tothesecondtermandProposition 5.3.1 tothethirdterm,equation( 5.175 ) reducesto " 2 TT ~ L b; 0 ‹ G v “ TT L 2 ‹ “ B ~ C" SS v SS L 2 ‹ “ : (5.176) Pluggingequation( 5.176 )into( 5.161 )weobtaintherequiredbound TT ~ L 2 b v TT L 2 ‹ “ B "C SS v SS L 2 ‹ “ ; for v > X ; (5.177) where C isindependentof " ,butitdependupon TT Ñ k b TT L ª ‹ “ . 5.3.3Bounding ~ L b Finally,thefollowingPropositionshowsthattheoperator ~ L b isbounded. Proposition5.3.4. Theoperator ~ L b l 2 ‹ R N N “ ( l 2 ‹ R N N “ hasan O ‹ " “ operatornorm. Proof. Let v > X ;v P j > b j b; 0 j with Y v Y L 2 ‹ “ 1.Inparticular, v v; ~ v 0 ; (5.178) 109 whereistheprojectiononto X and ~ itscomplementary. Fix ⁄ > ˆ ‹ L 2 b “ ,where ˆ ‹ L 2 b “ istheresolventsetof L 2 b ,andrewrite L b inthefollowingway L b L 2 b " ~ L b ‹ L 2 b ⁄ “ " ~ L b ‹ L 2 b ⁄ “ 1 ‹ L 2 b ⁄ “ ⁄ : (5.179) Takingthe L 2 -normof ~ L b actingon v yields TT ~ L b v TT L 2 ‹ “ TT ~ ‹ L 2 b ⁄ “ v " ~ ~ L b ‹ L 2 b ⁄ “ 1 ‹ L 2 b ⁄ “ v TT L 2 ‹ “ ; (5.180) B TT ~ L 2 b v TT L 2 ‹ “ " TT ~ ~ L b ‹ L 2 b ⁄ “ 1 ‹ ~ “‹ L 2 b ⁄ “ v TT L 2 ‹ “ ; (5.181) B TT ~ L 2 b v TT L 2 ‹ “ (5.182) " − TT ~ L b ‹ L 2 b ⁄ “ 1 ‹ L 2 b ⁄ “ v TT L 2 ‹ “ TT ~ L b ‹ L 2 b ⁄ “ 1 ~ ‹ L 2 b ⁄ “ v TT L 2 ‹ “ ‘ ; B TT ~ L 2 b v TT L 2 ‹ “ (5.183) " − TT ~ L b ‹ L 2 b ⁄ “ 1 TT l 2 l 2 TT L 2 b v TT L 2 ‹ “ TT ~ L b ‹ L 2 b ⁄ “ 1 TT l 2 l 2 TT ~ L 2 b v TT L 2 ‹ “ ‘ : Since ~ L b isrelativelyboundedwithrespectto L 2 b ,theoperator ~ L b ‹ L 2 b ⁄ “ 1 hasan O ‹ " “ boundasan operatorfrom l 2 ‹ R N N “ to l 2 ‹ R N N “ .Insection 5.3.2 wehaveshownthat TT ~ L 2 b TT L 2 B c": (5.184) Therefore,combiningbound( 5.184 )withtheboundednessof TT ~ L b ‹ L 2 b ⁄ “ 1 TT l 2 l 2 ,aninspectionofequa- tion( 5.93 )yields TT ~ L b v TT L 2 B ~ c" › ‹ 1 " “SS v SS L 2 TT L 2 b v TT L 2 ” (5.185) Tocompletetheboundon TT ~ L b TT L 2 weneedshowthat TT L 2 b TT L 2 isbounded.Usingthe of( 5.21 ),wecanrewritetheoperator L 2 b v as L 2 b Q j > b j b; 0 k Q k > Q j > b j ‹ L 2 b b; 0 j ; b; 0 k “ L 2 b; 0 k ; (5.186) anditsmatrixrepresentation M > R N d N d ,where N d O ‹ " 3 ~ 2 d “ ,tobe M j;k › L 2 b b; 0 j ; b; 0 k ” L 2 : (5.187) 110 Usingequation( 5.34 )wecanwrite M M diag M where M diag ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ ‹ b; 0 " 2 k “ 2 O ‹ " 2 “ if k j; 0if k x j; (5.188) M ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ 0if k j; " 2 R H 2 0 k j J 0 ds R l ~ " l ~ " ‹‹ 0 b; 0 “ œ “ 2 dz O ‹ " 2 º " “ if k x j; (5.189) ByTheorem 5.2 weknowthat M hasan O ‹ " “ boundasanoperatorfrom l 2 ‹ R N N “ to l 2 ‹ R N N “ . Moreover,weconsider k > forwhich b; 0 " 2 k O ‹ " “ .Hence, M and M diag haveasimilarboundand TT L 2 b v TT L 2 B c" SS v SS L 2 : (5.190) Pluggingthisboundbackto( 5.185 )yields TT ~ L b v TT L 2 B "c SS v SS L 2 ; (5.191) whichimpliesthat ~ L b has O ‹ " “ boundasanoperator l 2 ‹ R N N “ l 2 ‹ R N N “ . Ì Recallthe2 2blockformof L b ,( 5.22 ).Inthissectionwehaveshownthatthediagonalblocksare O ‹ " “ boundedasoperatorsfrom l 2 ‹ R N N “ l 2 ‹ R N N “ .Thefollowingsectionfocusesonthepearlingspectrum of L b andweusetheboundson B;B T toshowthatthepearlingeigenvaluesof L b are,atleadingorder,the pearlingeigenvaluesof M . 5.4RelatingtheEigenvaluesof L b and L b Recallthe2 2blockformrepresentationof L b ,givenin( 5.22 ), < @ @ @ @ @ @ @ @ @ @ > MB B T C = A A A A A A A A A A ? ; (5.192) wherethesubmatricesaregivenby M L b ;B L b ~ ;B T ~ L b and C ~ L b ~ : (5.193) 111 Weusetheestimatesfromtheprevioussectiontorelatethesmalleigenvaluesof L b tothoseof M .If wereaspectralprojectionassociatedto L b thenthetwooperatorswouldcommute,andsince ~ 0,the termswouldbezero.However, X onlyapproximatesaspectralsubsetof L b ,andtheestimates Y B Y L 2 ‹ “ Y B T Y L 2 ‹ “ B c" ,foundinSection 5.3 ,aresharp.However,therestrictedoperator C isuniformly coerciveon X Œ withitsspectrumisboundedfrombelowby A 0whichmaybechosenindependent oftlysmall " A 0,(see[ HayrapetyanandPromislow,2014 ]formoredetails).Consider v 1 > X and v 2 > X Œ .Then,forany @ wereducethe2 2representationoftheedimensionaleigenvalue problem < @ @ @ @ @ @ @ @ @ @ > MB B T C = A A A A A A A A A A ? < @ @ @ @ @ @ @ @ @ > v 1 v 2 = A A A A A A A A A ? < @ @ @ @ @ @ @ @ @ > v 1 v 2 = A A A A A A A A A ? (5.194) toadimensionalsystemforthecomponent v 1 ,whichsolves ‹ M “ v 1 B ‹ C “ 1 B T v 1 : (5.195) Wewillusetwomethodstoshowthatthepearlingconditionsestablishedfor M inCorollary 5.2.3 doin factcharacterizethesmallspectrumof L b :First,weshowthattheeigenvaluesof M areinfactasmall perturbationofthesmalleigenvaluesof L b ,andweobtainaperturbationestimate.Second,welookatthe solutionofthelinearwgeneratedby L b .Assumingtheeigenvaluesof M arestableunderpearling,in termsofCorollary 5.2.3 ,wewillshowthatthesemi-groupsgeneratedby L b decayexponentiallyfastand describetheresultingexponentialdichotomy. 5.4.1Perturbationestimate Consider > ˙ ‹ L b “ 9 ‹ ; “ ,takingthe l 2 -normofbothsidesof( 5.195 )andestimatingtheright-handside yields Y‹ M “ v 1 Y l 2 B SS B SS L 2 ‹ “ TT ‹ C “ 1 TT L 2 ‹ “ TT B T TT L 2 ‹ “ Y v 1 Y l 2 : (5.196) Usingtheestimatesonthenormsof B and B T ,giveninequation( 5.191 ),weknowthereexists c > R independentof " suchthat Y‹ M “ v 1 Y l 2 B c" 2 SS R ‹ C “SS L 2 ‹ “ Y v 1 Y l 2 : (5.197) where R ‹ C “ istheresolventoperatorof C ,as R ‹ ; C “ ‹ C “ 1 ; (5.198) 112 for ¶ ˙ ‹ C “ .Since C isself-adjoint,standardestimatesbaseduponspectraldecompositionoftheresolvent allowustoboundthe L 2 ‹ “ -normoftheresolventoperator SS R ‹ ; C “SS L 2 ‹ “ B ‹ dist ‹ ˙ ‹ C “““ 1 B 1 S S : (5.199) Pluggingthebound( 5.199 )in( 5.197 )yields Y‹ M “ v 1 Y l 2 B c" 2 S S Y v 1 Y l 2 : (5.200) Let Ÿ w i š bethesettheeigenvectorsof M withthecorrespondingeigenvalues Ÿ i š .Thespectraldecompo- sitionof v isgivenby v Q i w i ; (5.201) andtheright-handsideofequation( 5.200 )canbewrittenas Y‹ M “ v 1 Y l 2 Y Q i ‹ i “ w i Y l 2 C dist ‹ ˙ ‹ M “ ; “Y Q i w i Y l 2 dist ‹ ˙ ‹ M “ ; “Y v 1 Y l 2 (5.202) Combiningequations( 5.202 )and( 5.200 )yields dist ‹ ˙ ‹ M “ ; “ B c" 2 S S ; for @ : (5.203) Therefore,for > R anorderofonedistancebelow thisestimateimpliesthatdist ‹ ˙ ‹ M ““ O ‹ " 2 “ .We inferthatthespectrumof L b below lieswithin O ‹ " 2 “ ofthespectrumof M .Inparticular,ifthespectrum of M isboundedfrombelowbyapositive O ‹ " “ quantity,thensoisthespectrumof L b . Thespectrumof M istoleadingordergivenbyitsdiagonalterms M k;k ,whichareoftheform M k;k ‹ L b b; 0 k ; b; 0 k “ L 2 ‹ “ ,see( 5.26 ).Sincethebasiselements b; 0 k havenormone,weinferfromthe RayleighRitzvariationalcharacterizationofeigenvaluesthatthesmallesteigenvalueof L b issmallerthan thesmallesteigenvalueof M . Wededucefromthesecalculationsthatthepearlingcondition( 5.2.3 )appliesto L b : 5.4.2Semi-groupestimates Let V v 1 ;v 2 T where v 1 > X and v 2 > X Œ .Wederivedecayestimatesonthelinearevolutionequation V t L b V: (5.204) 113 Usingthe2 2blockformrepresentationfor L b weobtain v 1 ;t Mv 1 Bv 2 ; (5.205) v 2 ;t B T v 1 Cv 2 : (5.206) Equations( 5.191 )and( 5.64 )boundsonthematrices SS B SS L 2 ‹ “ ; TT B T TT L 2 ‹ “ B "c: (5.207) WeassumetheeigenvaluesofMarepositive,Corollary 5.2.3 impliesthattheyareorderof .ByRayleigh- Ritzformulaweknowthat SS M SS L 2 ‹ “ B "c .Since M isaself-adjointmatrixwecanapplytheSpectral MappingTheoremto M andobtainthedecayestimate TT e Mt v TT L 2 ‹ “ B ce "˙t SS v SS L 2 ‹ “ ; (5.208) where "˙ A 0isalowerboundonthespectrumof M .From[ HayrapetyanandPromislow,2014 ,Thm2.5] weknowthat C isuniformlycoerciveon X Œ ,itsspectrumisboundedbelowbysomeconstant whichmay bechosenindependentoftlysmall " A 0.Since C isself-adjointitissectorialanditgeneratesan analyticsemi-groupforwhichwehavethesemi-groupestimates TT e Ct v TT L 2 ‹ “ B ce t SS v SS L 2 ‹ “ : (5.209) We > ‹ 0 ;˙ “ andintroducethequantities M 1 ‹ t “ sup 0 B s B t − e "s SS v 1 ‹ s “SS L 2 ‹ “ ‘ ; (5.210) M 2 ‹ t “ sup 0 B s B t − e "s SS v 2 ‹ s “SS L 2 ‹ “ ‘ : (5.211) Thequantity M 1 ‹ t “ theestimate SS v 1 ‹ s “SS L 2 ‹ “ B e "s M 1 ‹ t “ ; 0 B s B t; (5.212) soif M 1 isuniformlybounded,then SS v 1 SS L 2 decayswithexponentialrate as t Ð ª ,asdo v 2 ‹ t “ and M 2 ‹ t “ . 114 ApplyingVariationofConstantsformulatoequation( 5.206 )yields v 2 ‹ t “ e Ct v 2 ‹ 0 “ S t 0 e C ‹ t s “ B T v 1 ‹ s “ ds; (5.213) whichinlightoftheboundfor B ,see( 5.207 ),andthesemi-groupestimate( 5.209 )reducesto SS v 2 ‹ t “SS L 2 ‹ “ B e t SS v 2 ‹ 0 “SS L 2 ‹ “ "c S t 0 e ‹ t s “ SS v 1 ‹ s “SS L 2 ‹ “ ds; (5.214) e t SS v 2 ‹ 0 “SS L 2 ‹ “ "ce t S t 0 e s SS v 1 ‹ s “SS L 2 ‹ “ ds; (5.215) B e t SS v 2 ‹ 0 “SS L 2 ‹ “ "ce t S t 0 e ‹ " “ s M 1 ‹ t “ ds; (5.216) andthesecondinequalityfollowsfromestimate( 5.212 ).Integrationofthelastlineyields SS v 2 ‹ t “SS L 2 ‹ “ B e t SS v 2 ‹ 0 “SS L 2 ‹ “ "ce t M 1 ‹ t “ e ‹ " “ t 1 " : (5.217) Fixing > ‹ 0 ; “ impliesthat e t › e ‹ " “ t 1 ” e "t e t @ e "t isdecaying,theequationabovereduces to SS v 2 ‹ t “SS L 2 ‹ “ B ~ c − e t SS v 2 ‹ 0 “SS L 2 ‹ “ "e "t M 1 ‹ t “‘ ; (5.218) where~ c ~ c ‹ “ .Since t > 0 ;T isarbitrary,wecan0 @ t œ @ t ,replace t with t œ andmultiplyby e "t œ , obtaining e "t œ SS v 2 ‹ t œ “SS L 2 ‹ “ B ~ ce "t œ − e t œ SS v 2 ‹ 0 “SS L 2 ‹ “ "e "t œ M 1 ‹ t œ “‘ ; (5.219) since M 1 ‹ t œ “ B M 1 ‹ t “ ,takingthesupremumover0 @ t œ @ t yields M 2 ‹ t “ B ~ c −SS v 2 ‹ 0 “SS L 2 ‹ “ "M 1 ‹ t “‘ : (5.220) Toobtainaboundon M 1 weapplythevariationofconstantformulatotheODEof v 1 ,eq( 5.205 ),which yields v 1 ‹ t “ e Mt v 1 ‹ 0 “ S t 0 e M ‹ t s “ Bv 2 ‹ s “ ds (5.221) applyingtheboundon B ,( 5.64 ),andthesemi-groupestimateon M ,( 5.208 ),yields SS v 1 ‹ t “SS L 2 ‹ “ B e ˙"t SS v 1 ‹ 0 “SS L 2 ‹ “ "c S t 0 e ˙" ‹ t s “ SS v 2 ‹ s “SS L 2 ‹ “ ds; (5.222) 115 e ˙"t SS v 1 ‹ 0 “SS L 2 ‹ “ "ce ˙"t S t 0 e "˙s SS v 2 ‹ s “SS L 2 ‹ “ ds; (5.223) andrecallthat SS v 2 ‹ s “SS L 2 ‹ “ B e s M 2 ‹ t “ whichyields SS v 1 ‹ t “SS L 2 ‹ “ B e ˙"t SS v 1 ‹ 0 “SS L 2 ‹ “ "ce ˙"t S t 0 e " ‹ ˙ “ s M 2 ‹ t “ ds; (5.224) B e ˙"t SS v 1 ‹ 0 “SS L 2 ‹ “ "ce ˙"t e " ‹ ˙ “ t 1 " ‹ ˙ “ M 2 ‹ t “ ; (5.225) B e ˙"t SS v 1 ‹ 0 “SS L 2 ‹ “ ~ c ‹ e t e ˙"t “ M 2 ‹ t “ : (5.226) where~ c ~ c ‹ “ .Since > ‹ 0 ;˙ “ ,wehave ‹ e t e ˙"t “ B e t andtheinequalityreducesto SS v 1 ‹ t “SS L 2 ‹ “ B e ˙"t SS v 1 ‹ 0 “SS L 2 ‹ “ ~ ce t M 2 ‹ t “ : (5.227) Applyingthebound( 5.220 )to M 2 ‹ t “ yields SS v 1 ‹ t “SS L 2 ‹ “ B c e ˙"t SS v 1 ‹ 0 “SS L 2 ‹ “ e t −SS v 2 ‹ 0 “SS L 2 ‹ “ "M 1 ‹ t : (5.228) Since t > 0 ;T arbitrary,wecan0 @ t œ @ t ,replace t with t œ andmultiplyby e "t œ ,obtaining e "t œ SS v 1 ‹ t œ “SS L 2 ‹ “ B c e ‹ ˙ “ "t œ SS v 1 ‹ 0 “SS L 2 ‹ “ −SS v 2 ‹ 0 “SS L 2 ‹ “ "M 1 ‹ t œ : (5.229) notethat M 1 ‹ t œ “ B M 1 ‹ t “ andtakingthesupremumover0 @ t œ @ t yields M 1 ‹ t “ B c v 1 ‹ 0 “SS L 2 ‹ “ SS v 2 ‹ 0 “SS L 2 ‹ “ "M 1 ‹ t : (5.230) For " tlysmallweobtainauniformboundon M 1 ‹ t “ M 1 ‹ t “ B c 1 " −SS v 1 ‹ 0 “SS L 2 ‹ “ SS v 2 ‹ 0 “SS L 2 ‹ “ ‘ ; (5.231) validforall t A 0.Combiningbound( 5.231 )on M 1 andestimate( 5.212 )on v 1 ‹ t “ ,yields SS v 1 ‹ t “SS L 2 ‹ “ B ce t −SS v 1 ‹ 0 “SS L 2 ‹ “ SS v 2 ‹ 0 “SS L 2 ‹ “ ‘ ; (5.232) andweseethat SS v 1 SS L 2 ‹ “ decayswithexponentialrate as t Ð ª .Toobtainaboundon SS v 2 SS L 2 ‹ “ ,we 116 combinebound( 5.231 )on M 1 ‹ t “ withequation( 5.218 )andthisyields SS v 2 ‹ t “SS L 2 ‹ “ B ^ c − "e "t SS v 1 ‹ 0 “SS L 2 ‹ “ ‹ e t "e "t “SS v 2 ‹ 0 “SS L 2 ‹ “ ‘ ; (5.233) B ^ ce "t − " SS v 1 ‹ 0 “SS L 2 ‹ “ ‹ 1 " “SS v 2 ‹ 0 “SS L 2 ‹ “ ‘ ; (5.234) Returningtotheoriginalequation( 5.204 ),andrecallthat V v 1 ;v 2 T wecanboundthenormof V using( 5.232 )and( 5.233 )whichyields SS V SS L 2 ‹ “ B SS v 1 ‹ t “SS L 2 ‹ “ SS v 2 ‹ t “SS L 2 ‹ “ ; (5.235) B ce t 1 " “SS v 1 ‹ 0 “SS L 2 ‹ “ ‹ 2 " “SS v 2 ‹ 0 “SS L 2 ‹ “ : (5.236) forsomeconstant c dependupon andforevery t A 0.Therefore,thesemi-groupsgeneratedby L b manifest decaywithanexponentialrate . 5.5Connectingthepearlingeigenvaluesof L b andthoseof L b Inthepartofthissectionwederivedconditionsunderwhichthebilayerdressingofanadmissible interfaceispearlingstable.Thisreducestoanunderstandingofthethespectrumof L b ,thesecondvariation of F atthebilayer u b .However,tounderstandthedynamicstabilityofabilayerunderthe H 1 gradient w,requirestheanalysisofthepearlingeigenvaluesofthelinearization, L b ofthegradientw.This analysisiscompletedinthetheorembelow. Proposition5.5.1. Fixanadmissibleinterface b andlet u b denotetheassociatedbilayersolution.Let L b bethesecondvariationof F evaluatedat u b .Then,thespectrumof L b isreal,andthereexist U A 0 such thatforeach > ˙ ‹ L b “ 9 ‹ ;U “ thereexists > ˙ ‹ L b “ 9 ‹ ;" 2 Ua “ ,suchthat " 2 a O ‹ " 3 ~ 4 d ~ 2 “ ; (5.237) wheretheconstant a isdvia a S R b; 0 ‹ @ 2 z b; 0 “ 1 b; 0 dz A 0 : (5.238) Inparticular,forspacedimension d 2 or d 3 thetermgivestheleadingorderformof . 117 ProofofTheorem 5.5.1 . Considertheeigenvalueproblem L ; (5.239) andlet Ÿ v i š N i 1 betheorthonormalbasisofthespace X givenin 5.1 .Weconsiderthedecompo- sitionoftheeigenfunction v v Œ ; (5.240) where v > X , v P N i 1 i v i , SS v SS L 2 ‹ “ SS Ñ SS l 2 ; (5.241) and v Œ > V Œ .Inserting( 5.240 )into( 5.239 )yields L b ‹ v v Œ “ 1 ‹ v v Œ “ : (5.242) Theprojectionof( 5.242 )onto v j > X yields ‹ L b v;v j “ L 2 ‹ “ ‹ L b v Œ ;v j “ L 2 ‹ “ ‹ 1 v;v j “ L 2 ‹ “ ‹ 1 v Œ ;v j “ L 2 ‹ “ : (5.243) Weintroducethefollowingmatrices D i;j ‹ L b v i ;v j “ L 2 ‹ “ ; (5.244) E i;j ‹ 1 v i ;v j “ L 2 ‹ “ : (5.245) Usingthematrices D and E ,in( 5.244 )and( 5.245 ),respectively,werewriteequation( 5.243 )as ‹ D E “ Ñ Ñ ; (5.246) whereweintroducethevector Ñ Ñ j ‹ 1 v Œ ;v j “ L 2 ‹ “ ‹ L b v Œ ;v j “ L 2 ‹ “ : (5.247) Toboundtheright-handsideof( 5.247 )weneedtobound SS v Œ SS L 2 ‹ “ .Considerthecomplementaryprojection 118 of( 5.242 )onto v Œ > V Œ whichyields ‹ L b v Œ ;v Œ “ L 2 ‹ “ ‹ 1 v;v Œ “ L 2 ‹ “ ‹ 1 v Œ ;v Œ “ L 2 ‹ “ ‹ L b v;v Œ “ L 2 ‹ “ : (5.248) Theoperator L b iscoerciveon X Œ andso,thereexists A 0sothat TT v Œ TT 2 L 2 ‹ “ B ‹ L b v Œ ;v Œ “ L 2 ‹ “ : (5.249) Proposition( 5.3.4 )impliesthatthereexists c A 0sothatthebilinearform ‹ L b v;v Œ “ L 2 ‹ “ ‹ v; L b v Œ “ L 2 ‹ “ hasisbounded ‹ L b v;v Œ “ L 2 ‹ “ B c" SS v SS L 2 ‹ “ TT v Œ TT L 2 ‹ “ : (5.250) Sincetheterm ‹ 1 v Œ ;v Œ “ L 2 ‹ “ ispositive,weneedtoconsiderthesignofIf @ 0,thelasttermon theright-handsideof( 5.248 )isnegative,andwecandropitwhenweareboundingfromabove.It A 0, thenthereexists c A 0sothat ‹ 1 v Œ ;v Œ “ L 2 ‹ “ B c TT v Œ TT 2 L 2 ‹ “ : (5.251) Moreover,fromequation( 5.266 )ofLemma 5.6 ,thereexists c A 0sothat ‹ 1 v;v Œ “ L 2 ‹ “ B c 1 S S " 2 SS v SS L 2 ‹ “ TT v Œ TT L 2 ‹ “ : (5.252) Usingthebounds( 5.249 ),( 5.250 ),( 5.251 )and( 5.252 )inequation( 5.248 )weobtain TT v Œ TT 2 L 2 ‹ “ B c 1 S S " 2 SS v SS L 2 ‹ “ TT v Œ TT L 2 ‹ “ c 2 TT v Œ TT 2 L 2 ‹ “ c 3 " SS v SS L 2 ‹ “ TT v Œ TT L 2 ‹ “ ; (5.253) where ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ 0if @ 0 ; if A 0 : (5.254) Solving( 5.253 )for SS v Œ SS L 2 ‹ “ ,weobtaintheupperbound, TT v Œ TT L 2 ‹ “ B c" SS v SS L 2 ‹ “ ; (5.255) validsolongas @ U ~ c 2 .Bounding S j S ,in( 5.247 ),via( 5.250 )and( 5.252 )yields S j S B ‹ c 1 " c 2 " 2 S S“SS v j SS L 2 ‹ “ TT v Œ TT L 2 ‹ “ : (5.256) 119 Combining( 5.256 )and( 5.255 )weobtainaboundonthe l 2 normof Ñ SS Ñ SS l 2 B c ‹ " 2 " 3 S S“ º N SS SS l 2 ; (5.257) where N O ‹ " 3 ~ 2 d “ . Goingbacktoequation( 5.246 ),wenotethatthematrix D ,in( 5.244 ),ispreciselythematrix M introducedin( 5.26 ),andfromCorollary 5.2.2 weknowthat,atleadingorder, D takestheform D D 0 " 2 D 1 ; (5.258) where D 0 isadiagonalmatrixwhoseentriesaretheeigenvaluesof L b ,and D 1 is O ‹ 1 “ intheoperatornorm. WecanexpresstheLaplaceinverseoperatorusingthewhiskeredcoordinatessystem,( 2.9 ), 1 U x ‹ " 2 @ 2 z " 1 H@ z G “ 1 ‹‹ 1 T “ L 0 “ 1 L 1 0 L 1 0 T :::; (5.259) whereweintroducetheoperators L 0 " 2 @ 2 z s ; (5.260) T ‹ " 1 H@ z "zD s; 2 “ L 1 0 : (5.261) Plugging( 5.259 )into( 5.245 )weseethat E takestheform E ~ E 0 ~ E 1 ; (5.262) wheretheentriesofthematrices ~ E 0 and ~ E 1 taketheform ~ E 0 i;j ‹ L 1 0 v i ;v j “ L 2 ‹ “ ; (5.263) ~ E 1 i;j ‹ L 1 0 Tv i ;v j “ L 2 ‹ “ : (5.264) Toestimatetheentriesof ~ E 0 weprovethefollowingLemma Lemma5.6. Theinverseoperator L 1 0 actingon v > X , v P k > b; 0 k b; 0 k ,takestheform L 1 0 b; 0 k " 2 k ‹ @ 2 z " 2 k “ 1 b; 0 ; ¦ k > b; 0 : (5.265) 120 Inparticular, L 1 0 hasan O ‹ " 2 “ boundon X ,i.e.,thereexists c A 0 sothat TT L 1 0 v TT L 2 ‹ “ B c" 2 SS v SS L 2 ‹ “ ; ¦ v > X : (5.266) Proof. Toobtainanexpressionfor L 1 0 v ,considerthefollowingequation L 0 f b; 0 k ; (5.267) andrecallthat k isaneigenfunctionof s ,whichimplicatesthatthefunction f isoftheform f g ‹ z “ k , andequation( 5.267 )reducesto ‹ " 2 @ 2 z k “ g ‹ z “ k b; 0 k : (5.268) Factoring " 2 fromtheleft-handsideof( 5.268 )yields " 2 ‹ @ 2 z " 2 k “ g ‹ z “ k b; 0 k : (5.269) Invertingtheoperatorsinequations( 5.269 )and( 5.267 )weconcludethat L 1 0 ‹ b; 0 k “ " 2 k ‹ @ 2 z " 2 k “ 1 b; 0 : (5.270) Forageneral v > X , v P k > b; 0 n k b; 0 ,equation( 5.270 )takestheform L 1 0 v " 2 Q k > b; 0 k k ‹ @ 2 z " 2 k “ 1 b; 0 : (5.271) Since j areorthonormalinthe b weightedinnerproduct,see( 2.29 ),the L 2 ‹ “ -normoftheinverse operatoractingon v > X yields TT L 1 0 v TT L 2 ‹ “ " 2 SS A Ñ SS l 2 ; (5.272) where A isthediagonalmatrixwithentries A i Y‹ @ z " 2 i “ 1 0 Y L 2 ‹ R “ : (5.273) Since " 2 i b; 0 O ‹ º " “ wededucethat A isuniformlyboundedandhence TT L 1 0 v TT L 2 ‹ “ c" 2 SS v SS L 2 ‹ “ : (5.274) 121 Ì Lemma 5.6 impliesthattheentriesof ~ E 0 ,in( 5.263 ),admitthefollowingexpansion ~ E 0 i;j S b S ` ~ " ` ~ " ‹ " 2 @ 2 z s “ 1 ‹ b; 0 i “ j b; 0 J b dzds (5.275) " 2 S b S ` ~ " ` ~ " i j b; 0 ‹ @ 2 z " 2 i “ 1 b; 0 J b dzds (5.276) ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ " 2 R ` ~ " ` ~ " b; 0 ‹ @ 2 z b; 0 “ 1 b; 0 dz O ‹ " 2 º " “ if i j; " 3 R R ` ~ " ` ~ " b; 0 ‹ @ 2 z " 2 i “ 1 b; 0 i j H 0 dzds O ‹ " 4 “ if i x j; (5.277) wherefor i j ,since k > b; 0 ,weobtained( 5.277 )byexpanding " 2 i b; 0 O ‹ º " “ .Consequentlywemay write ~ E 0 " 2 › E 0 º "B ” ; (5.278) where E 0 aI d isadiagonalmatrix, aI d istheidentitymatrixmultipliesbytheconstant a in( 5.238 ), andthematrix B hasentriesoftheform R b f ‹ Ñ k “ i j J 0 ds .ByLemma 5.2 ,thematrix B has O ‹ 1 “ bound intheoperatornorm.ThenextLemmashowsthatthematrix ~ E 1 ,deedin( 5.264 ),isboundedinthe operatornorm. Lemma5.7. Thematrix ~ E 1 ,din( 5.264 )hasan O ‹ " 3 “ boundedintheoperatornorm,i.e.,there exists c A 0 sothat TT ~ E 1 TT l 2 l 2 B c" 3 : (5.279) Proof. Theoperator L 1 0 isself-adjointinthe J 0 innerproduct,asin( 2.35 ),andsotheentriesof ~ E 1 , givenin( 5.264 )canbewrittenas E 1 i;j ‹ L 1 0 Tv i ;v j “ J 0 ‹ Tv i ; L 1 0 v j “ J 0 : (5.280) For v k > X , v k ' b; 0 k ,theoperator T actingon v k takestheform Tv k "H k g œ k " 3 zg k D s; 2 k ; (5.281) where g k ‹ @ 2 z " 2 k “ 1 b; 0 : (5.282) 122 Usingequation( 5.281 )andLemma 5.6 wewrite ‹ Tv i ; L 1 0 v j “ J 0 S b S ` ~ " ` ~ " ‹ "H i g œ i " 3 zg i D s; 2 i “ " 2 j g j J 0 dzds (5.283) " 3 E 1 i;j " 5 E 2 i;j (5.284) wherethematrices E 1 and E 2 aregivenby E 1 i;j S b S ` ~ " ` ~ " Hg œ i g j i j J 0 dzds; (5.285) E 2 i;j S b S ` ~ " ` ~ " zg i g j j D s; 2 i J 0 dzds: (5.286) Lemma 5.2 impliesthat E 1 hasan O ‹ 1 “ boundintheoperatornorm,andaccordingtoCorollary 5.2.1 ,the " 5 termsarenegligible.Goingbackto( 5.280 )weconcludethat TT ~ E 1 TT l 2 l 2 B c" 3 : (5.287) Ì Equation( 5.278 )andLemma 5.7 impliesthat E ,in( 5.262 ),canbewrittenas E " 2 aI d " 5 ~ 2 E 1 ; (5.288) wherethematrix E 1 hasan O ‹ 1 “ boundintheoperatornorm. Weusetheexpansionsofthematrices D and E ,in( 5.258 )and( 5.288 ),respectively,toexpand equation( 5.246 )sothat ‹ D 0 " 2 a “ Ñ Ñ " 2 ‹ D 1 º " E 1 “ Ñ ; (5.289) which,dividingby " 2 a ,takestheform „ " 2 D 0 ‚ Ñ " 2 › " 2 ‹ D 1 º " E 1 “ Ñ ” ; (5.290) Takingthe L 2 normofequation( 5.290 ),andusinginequality( 5.257 )wededucethat WW„ " 2 a D 0 ‚ Ñ WW l 2 B C ‹ 1 " S S“ º N SS SS l 2 ; (5.291) 123 howeveras D 0 isself-adjoint,itfollowsthat isclosetoaneigenvectorof D 0 and dist ‹ ; " 2 a ˙ ‹ D 0 ““ B C ‹ 1 " S S“ " 3 ~ 4 d ~ 2 : (5.292) Sincethespectrumof D 0 constitutesthepearlingeigenvaluesof L b ,and ˙ ‹ D 0 “ O ‹ " “ ,weknowthat " 2 a ˙ ‹ D 0 “ O ‹ " 1 “ ,andhence S " S O ‹ 1 “ .Aslongastheright-handsideofequation( 5.292 )is O ‹ " r “ with r A 1, i.e.,aslongasthedimensionsizeis d @ 3 : 5,wemayconcludethatfor > ˙ ‹ L b “ 9 ‹ ;U “ thereexists > ˙ ‹ L b “ 9 ‹ ;" 2 Ua “ sothat " 2 a O ‹ " 3 ~ 4 d ~ 2 “ : (5.293) Ì 124 Chapter6 ThePearlingEigenvalueProblem, Co-Dimension2 InthisChapterweaddressthestabilityofthebilayermorphologyinthestrongFCHandobtainanexplicit expressionforthepearlingstabilitycondition.Wepresentarigourousanalysisoftheeigenvalueproblem correspondingtothestrongFCHfortheco-dimensiontwostructure.Weshowthatinthe strongFCH scalingtheleadingorderbehaviorofthepearlingeigenvaluesisindependentoftheshapeoftheunderlying co-dimensiontwomorphology.Underthe H 1 gradientwthepearlinginstabilitymanifestsitselfona timescalethatis O ‹ " 2 “ fasterthanthegeometricevolution,andhencecanbetakentobeinstantaneous onthegeometricevolutiontimescale.Conversely,theinstabilityoccursonthesametimescale asthegeometricw,andmaynotnecessarilyimmediatelymanifestitselfonthegeometricevolutiontime scale. RecallthestrongFCHfreeenergywhichcorrespondstothechoice p 1in( 1.14 ), F ‹ u “ S 1 2 ‹ " 2 u W œ ‹ u ““ 2 " „ " 2 1 2 S © u S 2 2 W ‹ u “‚ dx; (6.1) where ` R d ;d C 2,isaboundeddomain, W ‹ u “ isatilteddouble-wellpotentialwithtwominimaat b , u R isthedensityoftheamphiphilicspecies, " P 1controlsthewidthoftheboundarylayerand 1 and 2 arethefunctionalizationconstants.Thevariationof F ,introducedinequation( 1.18 ),isgivenby F u ‹ u “ ‹ " 2 W œœ ‹ u “ 1 “‹ " 2 u W œ ‹ u ““ d W œ ‹ u “ ; (6.2) 125 where d 1 2 .Thesecondvariationsof F takestheform L p 2 F u 2 ‹ u “ › " 2 W œœ ‹ u “ 1 ”› " 2 W œœ ‹ u “ ” › " 2 u W œ ‹ u “ ” W œœœ ‹ u “ d W œœ ‹ u “ : (6.3) Weobtainapearlingstabilityconditionfortheco-dimensiontwomorphologywhichissummarizedinthe followingtheorem- Theorem6.0.2. Foragivenadmissibleinterface, p ,theassociatedporesolutionconstructedin( 2.75 ), isstablewithrespecttothepearlingbifurcationifandonlyifthechemicalpotential 1 the pearlingstabilitycondition P ⁄ p d − TT œ p; 0 TT 2 L R p; 0 SS p; 0 SS 2 L R ‘ S p A 1 ; (6.4) 6.1Overview Wewanttoinvestigatethepearlingeigenmodesoftheco-dimensiontwoporestructures:Givenanadmissible interfaceassumethatthesystemisatquasi-equilibriumwith u p U p ‹ R “ "u 1 ; (6.5) where U p istheradialsymmetricporesolutionof( 2.75 )and u 1 ,derivedinequation( 4.103 ),isgivenby u 1 1 p; 2 d L 2 p W œ ‹ U p “ ; (6.6) where L p isthelinearoperatorintroducedin( 2.78 ),Thechemicalpotential 1 isspatiallyconstantandthe functions p;j solves( 2.86 )for j 1 ; 2. Weareinterestedinthepearlingeigenmodesofthesecondvariationof F , L p ,in( 6.3 ).Considerthe eigenvalueproblem L p : (6.7) BychangingcoordinatesoftheLaplacian,intheoperator L p ,tothewhiskeredcoordinates,using( 2.65 ), andplugging-intheexpansionof u p ,( 6.5 ),into u ,wecanrewrite L p inordersof " suchthat L p L 2 p " L 1 O ‹ " 2 “ ; (6.8) 126 where L 1 ‹ W œœœ ‹ U p “ u 1 1 “ X L p L p X ‹ W œœœ ‹ U p “ u 1 “ ‹ L p u 1 D z U p “ W œœœ ‹ U p “ d W œœ ‹ U p “ ; (6.9) and L p isin( 2.77 ).Seeappendix( E.2 )fordetailedcalculationsoftheexpansionof L p . Recallthat p; 0 ,in( 2.92 ),isthesetofsmalleigenvaluesassociatedto L p ,and,accordingtoWeyl's asymptoticformula S p; 0 S O ‹ " 3 ~ 2 d “ .We P k " 1 ~ 2 ‹ p; 0 " 2 k “ ; (6.10) tobethedetuningconstantdependingonlyon k . 6.1. Thespace, X ,correspondingtothesmalleigenvaluesof L p isdas X Ÿ p; 0 k S k > š ; (6.11) Lookingforsolutionsoftheeigenvalueproblem,( 6.7 ),weconsideraregularperturbationexpansionofthe form j 0 ;j " 1 ;j O ‹ " 2 “ ; 0 ;j > X ; 0 ;j Q k > k 0 k ; 1 ;j > X Œ ; (6.12) j " 1 ;j O ‹ " 2 “ : (6.13) The L 2 -orthogonalprojection,onto X isgivenby f Q k > ‹ f; 0 k “ L 2 ‹ “ SS 0 k SS 2 L 2 ‹ “ 0 k Q k > ‹ f; 0 k “ L 2 ‹ “ 0 k ; (6.14) anditscomplementaryprojectionis ~ I Weconsideradecompositionoftheoperator L intoa2 2blockform, < @ @ @ @ @ @ @ @ @ @ > MB B T C = A A A A A A A A A A ? ; (6.15) 127 where M L p ;B L p ~ ;C ~ L p ~ : (6.16) Byabuseofnotationwedenote L p andits2 2decompositionwiththesamesymbol. ByAssumption 2.2.3 ,therestrictedoperator C isuniformlycoerciveon X Œ anditsspectrumisbounded frombelowby A 0whichmaybechosenindependentoftlysmall " A 0. 6.2Eigenvaluesof M L p Let v > X , v canbewrittenas v Q k > b k 0 k ; (6.17) withoutlossofgenerality,assume SS v SS L 2 ‹ “ 1.Theoperator L actingon v ,takestheform L Q j > b j 0 k Q k > ™ fl L p Q j > b j 0 j ; 0 k fi Ł L 2 ‹ “ 0 k Q k > Q j > b j ‹ L p 0 j ; 0 k “ L 2 ‹ “ 0 k : (6.18) Wetheoperatormatrixrepresentation M > R N N ,where N " 3 ~ 2 ,inthefollowingway M j;k ‹ L p 0 j ; 0 k “ L 2 ‹ “ : (6.19) Usingtheexpansionof L p ,( 6.8 ),wecanwriteeachentryof M inordersof " suchthat ‹ L p 0 j ; 0 k “ L 2 ‹ “ ‹ L 2 p 0 j ; 0 k “ L 2 ‹ “ " ‹ L 1 0 j ; 0 k “ L 2 ‹ “ O ‹ " 2 “ ; (6.20) anddecomposethematrixintotermsoforder B " r andtermsoforder A " r ,suchthat M M 0 " r ~ M; (6.21) where M 0 j;k ‹ L 2 p 0 j ; 0 k “ L 2 ‹ “ " ‹ L 1 0 j ; 0 k “ L 2 ‹ “ r Q i 2 " i ‹ L i 0 j ; 0 k “ L 2 ‹ “ ; (6.22) ~ M j;k Q i C r " ‹ i r “ ‹ L i 0 j ; 0 k “ L 2 ‹ “ : (6.23) 128 Wewillshowthattheterm, M 0 ,canbesplitintoadiagonalandterms,thelatterofwhich canbeboundedindependentlyofthematrixsize N ,assumingthatthecurvaturesoftheinterfaceare tlysmooth.Theotherterm, ~ M ,canbebounded,independentofthedimension,viathe L ª norm. Inparticular,wehaveshowninSection 5.2.1 ,thatfora3-dimensionalspace,thematrix ~ M isnegligiblefor r 2. 6.2.1Bounding M 0 Next,wewanttoaboundforthematrix M 0 in R 3 .Anexaminationofthetwotermsof M 0 ,given inequation( 6.22 ),showsthattheyadmittheexpansions ‹ L 2 p 0 j ; 0 k “ L 2 ‹ “ ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ "P 2 k O ‹ " 2 º " “ if k j; O ‹ " 2 º " “ if k x j; (6.24) ‹ L 1 0 j ; 0 k “ L 2 ‹ “ ‹‹‹ Lu 1 “ W œœœ ‹ U p “ d W œœ ‹ U p ““ 0 j ; 0 k “ L 2 ‹ “ O ‹ º " “ if k j; (6.25) ‹ L 1 0 j ; 0 k “ L 2 ‹ “ " ‹‹‹ 2 W œœœ ‹ U p “ u 1 1 “ Ñ © z 0 Ñ ‹ © z u 1 U p “ W œœœ ‹ U p “ 0 “ j ; 0 k “ L 2 ‹ “ (6.26) O ‹ " 2 “ if k x j see( E.14 )and( E.33 )formoredetails.Wemaysplit M 0 intoitsdiagonalmatrices M 0 M 0 diag M 0 (6.27) where M 0 diag ‹ j;k “ ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ M 0 k;k O ‹ " º " “ if j k; 0if j x k; (6.28) and M 0 ‹ j;k “ ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ 0if j k; M 0 j;k O ‹ " 2 º " “ if j x k: (6.29) withentriesgivenby M 0 k;k " „ P 2 k S l … " 0 W œœœ ‹ U p “ Lu 1 d W œœ ‹ U p 0 0 “ 2 dz ‚ ; (6.30) " P 2 k 1 S p d − TT ‹ 0 0 “ œ TT 2 L R p; 0 TT ‹ 0 0 “ TT 2 L R ; (6.31) M 0 j;k " 2 ‹‹‹ 2 W œœœ ‹ U p “ u 1 1 “ Ñ © z 0 Ñ ‹ © z u 1 U p “ W œœœ ‹ U p “ 0 “ j ; 0 k “ L 2 ‹ “ ; (6.32) 129 withindices a" 1 ~ 2 B j;k B ~ a" 1 ~ 2 , a @ ~ a;a; ~ a > R ,and S p istheshapefactoroftheporestructure,givenby, S p 2 ˇ S ª 0 W œœœ ‹ U p “ 1 ‹ 0 0 “ 2 RdR: (6.33) UsingTheorem 5.2 wededucethat M 0 isuniformlyboundedasanoperatorfrom l 2 to l 2 . Corollary6.2.1. Thematrix M 0 ,din( 6.27 ),canbewrittenas M 0 M 0 diag M 0 ; (6.34) where M 0 isuniformlyboundedasanoperatorfrom l 2 to l 2 . Atthispointweconcludethattheeigenvaluesof M 0 , k ,are,atleadingorder,thediagonalentriesof M 0 diag , inequation( 6.28 ).Bytheof M ,( 6.21 ),wededucethat k aretheeigenvaluesof M ,at leadingorder.Since M isthematrixrepresentationof L p theeigenvaluesof L p are,atleading order, k ,whichtakestheform k " P 2 k 1 S p d − TT œ p; 0 TT 2 L R p; 0 SS p; 0 SS 2 L R ; (6.35) where S p istheshapefactorin( 6.33 )and P k isthedetuningconstantin( 6.10 ). Since P 2 k canbemadeassmallas O ‹ " “ (seeequation( 5.61 )),itfollowsthattheterminvolving P 2 k islower orderneartheturningpointofthepearlingspectrum.Thisleadsustothefollowingcorollary- Corollary6.2.2. Thepearlingeigenvaluesof L p ,( 6.13 ),takestheform " 1 SS p; 0 SS 2 L R 1 S p d − TT œ p; 0 TT 2 L R p; 0 SS p; 0 SS 2 L R O ‹ " 2 “ ; (6.36) and,inordertohave,atleadingorder, pearlingstability weneed 1 S p d − TT œ p; 0 TT 2 L R p; 0 SS p; 0 SS 2 L R ‘ @ 0 : (6.37) Recallthatourmaingoalistoanexpressionforthepearlingeigenvaluesof L usingour2 2representation of L p ,see( 6.15 ).Inthissectionwefoundanexpressionthepearlingeigenvaluesoftheoperator M .The nextsectionestablishtheboundsontheterms B;B T . 130 6.3BoundingtheOperators Recallthe2 2blockformrepresentationof L p ,givenin( 6.15 ), < @ @ @ @ @ @ @ @ @ @ > MB B T C = A A A A A A A A A A ? : (6.38) Iftheblocks, B L p ~ and B T ~ L p aresmall(sameorderofthe M L p blockor less)thenwecanrelatetheeigenvaluesof L p tothoseof M L p seesection 6.2 .Sincebothand ~ areself-adjointoperatorswehave ‹ L p ~ v;w “ L 2 ‹ L p ~ v; w “ L 2 ‹ ~ v; L p w “ L 2 ‹ v; ~ L p w “ L 2 : (6.39) So,itisenoughtoshowthatoneoftheblocksissmall,i.e.,wewanttoshowthatthereexista constant C ,independenton N O ‹ " 3 ~ 2 “ suchthat Y ~ L p v Y L 2 ‹ “ B "C Y v Y L 2 ‹ “ ; ¦ v > X : (6.40) withoutlossofgenerality,assume v > X ;v P j > b j 0 j and Y v Y L 2 ‹ “ 1.Notethat Y v Y 2 L 2 ‹ “ S Q j;k > b j b k j k 2 0 dx Q j > b 2 j Y k 0 Y 2 L 2 ‹ “ 1 ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ S 2 j 2 0 dx Q j;k > j x k b j b k 0 ; byorthogonalityof j ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ S j k 2 0 dx; (6.41) Q j > b 2 j SS b SS 2 l 2 ; (6.42) where b ‹ b 1 ;b 2 ;:::;b N d “ . Notethatwecanwrite L p inthefollowingform L p L 2 p " ~ L p ; (6.43) where L p isarelativelyboundedperturbationof L 2 p .Wesplittheproofintothreeparts:weshowthat wecanboundtheoperator ~ L p v ,nextweboundtheoperator ~ L 2 p v andthenwebound ~ L p v . 131 6.3.1Bounding ~ L p v Recallthat v > X ;v P j > b j 0 j and Y v Y L 2 ‹ “ 1.Inparticular, v v; ~ v 0 : (6.44) Weneedtoshowthatthereexist C ,independentof " ,suchthat Y ~ L p v Y L 2 ‹ “ B "C Y v Y L 2 ‹ “ : (6.45) Usingtheexpressionfor L p ,( 2.77 ), L p v takestheform L p v ‹ Lv "D z v " 2 @ 2 G v “ Q j > b j ™ Œ Œ fl p; 0 0 j ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ > X "D z 0 j " 2 @ 2 G 0 j fi Š Š Ł : (6.46) Notethat ~ projectof X ,thereforeiteliminatesthetermandtheterm ~ L p v becomes ~ L p v " ~ Q j > b j › "@ 2 G j 0 D z 0 j ” : (6.47) The L 2 ‹ “ -normof( 6.47 )hasthebound TT ~ L p v TT L 2 ‹ “ B " SS D z v SS L 2 ‹ “ " 2 TT @ 2 G v TT L 2 ‹ “ ; (6.48) andwewillshowthateachofthetermsontheright-handsideofequation( 6.48 )isbounded.Toshowthat thenormontheright-handsideofequation( 6.48 )isbounded,weintroducethematrix B > R N N , with N O ‹ " 3 ~ 2 “ suchthat B j;k ` D z 0 j ;D z 0 k e L 2 ‹ “ ; (6.49) which,usingtheof D z ,givenin( 2.66 ),takestheform B j;k S f ‹ Ñ “ j k ds; where f S ª 0 „ Ñ ~ J p © z 0 ‚ 2 Jdz: (6.50) ApplyingTheorem 5.2 weconcludethat B isuniformlyboundedoperatorfrom l 2 l 2 .Since B isthematrix representationoftheoperator D z weobtainthebound SS D z v SS L 2 ‹ “ B c SS v SS L 2 ‹ “ for v > X ; (6.51) 132 where c isindependenton " .Equation( 6.48 )reducesto TT ~ L p v TT L 2 ‹ “ B "c SS v SS L 2 ‹ “ " 2 TT @ 2 G v TT L 2 ‹ “ ; (6.52) ThefollowingPropositionshowsthat,overtheappropriatespace,theoperator ~ @ 2 G isboundedinthe L 2 ‹ “ norm: Proposition6.3.1. Let f ‹ z “ beasmoothfunctionsuchthat S f ‹ z “S @ c 1 e c 2 S z S forsome c i > R ;c i A 0 ;i 1 ; 2 ;supp ‹ f “ ` l : (6.53) Theoperator ~ @ 2 G ,where ~ istheprojectionofthespaceofsmalleigenvalues X ,din( 6.11 ) and @ 2 G isdin( 2.67 ),isboundedonthespace Y span Ÿ f ‹ z “ k U k > ; š ; (6.54) i.e.,thereexists C A 0 ,independentof " ,suchthat TT ~ @ 2 G v TT L 2 ‹ “ B C" 2 SS v SS L 2 ‹ “ ; (6.55) forevery v > Y .Particularly,for v > X ,i.e.,when f ‹ z “ 0 ,weobtainthebound TT ~ @ 2 G v TT L 2 ‹ “ B C" 1 SS v SS L 2 ‹ “ : (6.56) Proof. Fix ⁄ > ˆ ‹ @ 2 s “ ,where ˆ ‹ @ 2 s “ istheresolventsetoftheco-dimeaniontwoLaplacianoperator,thenthe operator @ 2 G canbewrittenas @ 2 G 1 ~ J 2 p @ 2 s " z @ s Ñ ~ J p @ s 1 ~ J 2 p @ 2 s ⁄ “ " z @ s Ñ ~ J p @ s ‹ @ 2 s ⁄ “ 1 ‹ @ 2 s ⁄ “ ⁄ : (6.57) where ~ J p isin( 2.64 )and 1 ~ J 2 p havetheexpansion 1 ~ J 2 p 1 " R ; where R ª Q i 0 ‹ ~ J 2 p 1 “ i 1 : (6.58) Withoutlossofgenerality,weassumethat ⁄ 0.Wenotethatevery v > Y thefollowinginequality 133 TT @ 2 s v TT L 2 ‹ “ B " 2 SS v SS L 2 ‹ “ ; (6.59) andsince @ s isarelativelyboundedperturbationof @ 2 s ,theoperator @ s ‹ @ 2 s ⁄ “ 1 isbounded,independent of " ,on Y . Consideringthecasewhen f x 0 ,then v takestheform, v f ‹ z “ ‹ s “ ; Q j > b j j ; Y v Y L 2 ‹ “ 1(6.60) Takingthe L 2 ‹ “ -normof ~ G actingon v yields TT ~ @ 2 G v TT L 2 ‹ “ R R R R R R R R R R R R R R R R R R R R R R ~ 1 ~ J 2 p @ 2 s " z @ s Ñ ~ J p @ s ‹ @ 2 s “ 1 @ 2 s v R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ ; (6.61) B R R R R R R R R R R R R R R R R R R R R R R ~ 1 ~ J 2 p @ 2 s v R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ " R R R R R R R R R R R R R R R R R R R R R R ~ 1 ~ J 2 p z @ s Ñ ~ J p @ s ‹ @ 2 s “ 1 @ 2 s v R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ : (6.62) Usingtheexpansionof ~ J p ,givenin( 6.58 ),wecanboundthetermontheright-handsideofequa- tion( 6.61 ) R R R R R R R R R R R R R R R R R R R R R R ~ 1 ~ J 2 p @ 2 s v R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ B TT ~ @ 2 s v TT L 2 ‹ “ " TT ~ R @ 2 s v TT L 2 ‹ “ B TT @ 2 s v TT L 2 ‹ “ " TT R @ 2 s v TT L 2 ‹ “ ; (6.63) B " 2 SS v SS L 2 ‹ “ " SS R f ‹ z “SS L ª ‹ “ TT @ 2 s TT L 2 ‹ “ ; B › " 2 " 1 ” c SS v SS L 2 ‹ “ ; (6.64) where c isindependentof " andthethirdinequalityfollowsfromthefactthat v f ‹ z “ ‹ s “ ,and f decays at O ‹ 1 “ ratein z .Asforthesecondtermontheright-handsideofequation( 6.61 ) " 2 R R R R R R R R R R R R R R R R R R R R R R ~ 1 ~ J 2 p z @ s Ñ ~ J p @ s ‹ @ 2 s “ 1 @ 2 s v R R R R R R R R R R R R R R R R R R R R R R 2 L 2 ‹ “ B " 2 R R R R R R R R R R R R R R R R R R R R R R 1 ~ J 2 p z @ s Ñ ~ J p @ s ‹ @ 2 s “ 1 @ 2 s v R R R R R R R R R R R R R R R R R R R R R R 2 L 2 ‹ “ ; (6.65) B " 2 R R R R R R R R R R R R R R R R R R R R R R 1 ~ J 3 p @ s ‹ @ 2 s “ 1 ‹ @ s Ñ “ @ 2 s zv R R R R R R R R R R R R R R R R R R R R R R 2 L 2 ‹ “ ; (6.66) B " 2 R R R R R R R R R R R R R R R R R R R R R R 1 ~ J 3 p R R R R R R R R R R R R R R R R R R R R R R 2 L ª ‹ “ TT @ s ‹ @ 2 s “ 1 TT 2 l 2 l 2 TT ‹ @ s Ñ “ @ 2 s zv TT 2 L 2 ‹ “ ; (6.67) B " 2 c 1 SS‹ @ s Ñ “ zf ‹ z “SS 2 L ª ‹ “ TT @ 2 s TT 2 L 2 ‹ “ B " 2 c 2 SS v SS 2 L 2 ‹ “ ; (6.68) whereforthethirdinequalityweusedthefactthat f ‹ z “ decayat O ‹ 1 “ in z .Plugging( 6.70 )and( 6.68 ) 134 into( 6.61 )yields TT ~ @ 2 G v TT L 2 ‹ “ B " 2 c 3 SS v SS L 2 ‹ “ : (6.69) For v > X ,theonlyinisequation( 6.70 )whichthebecomes R R R R R R R R R R R R R R R R R R R R R R ~ 1 ~ J 2 p @ 2 s v R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ B TT ~ @ 2 s v TT L 2 ‹ “ " TT ~ R @ 2 s v TT L 2 ‹ “ B " TT R @ 2 s v TT L 2 ‹ “ ; (6.70) B " SS R f ‹ z “SS L ª ‹ “ TT @ 2 s TT L 2 ‹ “ B " 1 c 4 SS v SS L 2 ‹ “ ; (6.71) andforthiscasewehave TT ~ @ 2 G v TT L 2 ‹ “ B " 1 c SS v SS L 2 ‹ “ : (6.72) Ì CombiningProposition 6.3.1 withequation( 6.52 )weobtaintherequiredbound TT ~ L p v TT L 2 B "C SS v SS l 2 : (6.73) 6.3.2Bounding ~ L 2 p v Recallthat v > X ;v P j > b j 0 j and Y v Y L 2 ‹ “ 1,inparticular, v v; ~ v 0 : (6.74) Wewanttoshowthereexist C 2 ,independenton " ,suchthat TT ~ L 2 p v TT L 2 ‹ “ B "C 2 SS v SS L 2 : (6.75) Writingthe L 2 p operatoractingon v explicitlywehave L 2 p v L p ‹ L p v “ ‹ L "D z " 2 @ 2 G “‹ Lv "D z v " 2 @ 2 G v “ ; (6.76) ‹ L 0 "D z " 2 @ 2 G “ Q j > b j › p; 0 0 j "D z 0 j " 2 @ G 0 j ” ; Q j > b j > X ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ 2 p; 0 0 j "L ‹ D z 0 j “ " 2 L ‹ @ 2 G 0 j “ p; 0 D z 0 j " 2 D z ‹ D z 0 j “ 135 " 3 D z ‹ @ 2 G 0 j “ " 2 p; 0 @ 2 G 0 j " 3 @ G ‹ D z 0 j “ " 4 @ 2 G ‹ @ 2 G 0 j : Projectingawayfrom X using ~ andtakingthe L 2 -normyields TT ~ L 2 p v TT L 2 ‹ “ Y ~ Q j > b j > X ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ 2 p; 0 0 j "L ‹ D z 0 j “ " 2 L ‹ @ 2 G 0 j “ p; 0 D z 0 j " 2 D z ‹ D z 0 j “ (6.77) " 3 D z ‹ @ 2 G 0 j “ " 2 p; 0 @ 2 G 0 j " 3 @ G ‹ D z 0 j “ " 4 @ 2 G ‹ @ 2 G 0 j L 2 ‹ “ ; B " Y R 1 v ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ Q j > b j L ‹ D z 0 j “ p; 0 D z 0 j "D z ‹ D z 0 j “Y L 2 ‹ “ " 6 TT ~ D z ‹ @ 2 G v “ TT L 2 ‹ “ (6.78) " 6 TT ~ @ 2 G D z v TT L 2 ‹ “ " 4 TT ~ p; 0 @ 2 G v TT L 2 ‹ “ " 8 TT ~ @ 2 G ‹ @ 2 G v “ TT L 2 ‹ “ " 4 TT ~ L ‹ @ 2 G v “ TT L 2 ‹ “ ; whereweusedthetriangleinequality.Thematrixrepresentation B > R N N of R 1 isgivenby B j;k `‹ L ‹ D z 0 “ p; 0 D z 0 "D z ‹ D z 0 ““ j ; L ‹ D z 0 “ p; 0 D z 0 "D z ‹ D z 0 ““ k e L 2 : (6.79) Theentriesof B taketheform B j;k S S l ~ " 0 f ‹ z;s “ j k Jdzds; (6.80) and,forasmoothfunctionf,Theorem 5.2 impliesthatthereexists c A 0,independentof " ,suchthat TT B TT l 2 l 2 B c: (6.81) Using( 6.81 )weobtainaboundontheoperator R 1 SS R 1 v SS 2 L 2 ‹ “ B SS b SS 2 l 2 TT B TT 2 l 2 l 2 B c SS v SS 2 l 2 ; (6.82) andequation( 6.78 )reducesto TT ~ L 2 p v TT L 2 ‹ “ B c" SS v SS L 2 ‹ “ " 3 TT ~ D z ‹ @ 2 G v “ TT L 2 ‹ “ " 3 TT ~ @ 2 G D z v TT L 2 ‹ “ (6.83) " 2 TT ~ p; 0 @ 2 G v TT L 2 ‹ “ " 4 TT ~ @ 2 G ‹ @ 2 G v “ TT L 2 ‹ “ " 2 TT ~ L ‹ @ 2 G v “ TT L 2 ‹ “ : Considerthesecondtermintheright-handsideof( 6.83 ). " 3 TT ~ D z ‹ @ 2 G v “ TT L 2 ‹ “ B " 3 WW ~ J p © z › @ 2 G v ” WW L 2 ‹ “ ; (6.84) 136 B " 3 WW ~ J p © z @ 2 G v WW L 2 ‹ “ " 3 WW ~ J p @ 2 G ‹ © z v “WW L 2 ‹ “ ; (6.85) B " 3 WW ~ J p WW L ª ‹ “ TT © z @ 2 G v TT L 2 ‹ “ " 3 WW ~ J p WW L ª ‹ “ TT @ 2 G ‹ © z v “ TT L 2 ‹ “ ; (6.86) B " 3 c 1 TT © z @ 2 G v TT L 2 ‹ “ c 2 " SS v SS L 2 ‹ “ ; (6.87) wherethelastinequalityfollowsapplyingProposition 6.3.1 .Inordertoshowthatthetermin( 6.87 )is bounded,wenotethat © z @ 2 G ™ fl © z 1 ~ J 2 p fi Ł @ 2 s " ™ fl © z z @ s Ñ ~ J 3 p fi Ł @ s ; (6.88) andweconsiderthefollowingProposition. Proposition6.3.2. Let f ‹ z “ beasmoothfunctionsuchthat S f ‹ z “S @ c 1 e c 2 S z S forsome c i > R ;c i A 0 ;i 1 ; 2 ; and supp ‹ f “ ` l : (6.89) Theoperator © z @ 2 G ,where @ 2 G isdin( 2.67 ),isboundedonthespace Y span Ÿ f ‹ z “ k U k > ; š ; (6.90) i.e.,thereexists C A 0 , C independentof " and f ,suchthat TT © z @ 2 G v TT L 2 ‹ “ B C" 1 SS v SS L 2 ‹ “ ; (6.91) forevery v > Y . Proof. Fix ⁄ > ˆ ‹ @ 2 s “ ,where ˆ ‹ @ 2 s “ istheresolventsetoftheco-dimeaniontwoLaplacianoperator,and, withoutlossofgenerality,assumethat ⁄ 0.Theoperator @ 2 G canbewrittenas @ 2 G 1 ~ J 2 p @ 2 s " z @ s Ñ ~ J 3 p @ s 1 ~ J 2 p @ 2 s " z @ s Ñ ~ J 3 p @ s ‹ @ 2 s “ 1 ‹ @ 2 s “ : (6.92) Let v > Y suchthat f x 0 ,then v takestheform, v f ‹ z “ ‹ s “ ; Q j > b j j ; (6.93) and,withoutlossofgenerality, Y v Y L 2 ‹ “ 1. 137 Takingthe L 2 ‹ “ -normof @ z G actingon v yields TT ‹ © z @ 2 G “ v TT L 2 ‹ “ R R R R R R R R R R R R R R R R R R R R R R ‹ © z 1 ~ J 2 p “ @ 2 s v " © z ‹ z @ s Ñ ~ J 3 p “ @ s ‹ @ 2 s “ 1 ‹ @ 2 s “ v R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ (6.94) B R R R R R R R R R R R R R R R R R R R R R R ‹ © z 1 ~ J 2 p “ f ‹ z “ R R R R R R R R R R R R R R R R R R R R R R L ª ‹ “ TT @ 2 s TT L 2 ‹ “ (6.95) " R R R R R R R R R R R R R R R R R R R R R R © z ‹ z @ s Ñ ~ J 3 p “ f ‹ z “ R R R R R R R R R R R R R R R R R R R R R R L ª ‹ “ TT @ s ‹ @ 2 s “ 1 TT l 2 l 2 TT ‹ @ 2 s “ TT L 2 ‹ “ B " 2 c 1 R R R R R R R R R R R R R R R R R R R R R R ‹ © z 1 ~ J 2 p “ f ‹ z “ R R R R R R R R R R R R R R R R R R R R R R L ª ‹ “ TT TT L 2 ‹ “ " 1 c 2 SS v SS L 2 ‹ “ ; (6.96) wheretheinequalityfollowsfromthetriangleinequalitycombinedwiththegeneralizedolderinequality. Notethat © z 1 ~ J 2 p " ‹ 1 ; 1 “ Ñ 2 ~ J 3 p ; (6.97) whichimpliesthatthereexists c independentof " suchthat R R R R R R R R R R R R R R R R R R R R R R ‹ © z 1 ~ J 2 p “ f ‹ z “ R R R R R R R R R R R R R R R R R R R R R R L ª ‹ “ B "c 3 : (6.98) Plugging( 6.98 )into( 6.96 )yields TT ‹ © z @ 2 G “ v TT L 2 ‹ “ B " 1 c 4 SS v SS L 2 ‹ “ ; for v > X ; (6.99) andweconcludethat TT © z @ 2 G TT L 2 ‹ “ B C" 1 on Y: (6.100) Ì Returningto( 6.87 ),andusingProposition 6.3.2 ,wehave " 3 TT ~ D z ‹ @ 2 G v “ TT L 2 ‹ “ B "c 1 SS v SS L 2 ‹ “ : (6.101) Plugging( 6.101 )andintotheright-handsideof( 6.83 )yields TT ~ L 2 p v TT L 2 ‹ “ B c" SS v SS L 2 ‹ “ " 3 TT ~ @ 2 G D z v TT L 2 ‹ “ " 2 TT ~ p; 0 @ 2 G v TT L 2 ‹ “ (6.102) " 4 TT ~ @ 2 G ‹ @ 2 G v “ TT L 2 ‹ “ " 2 TT ~ L ‹ @ 2 G v “ TT L 2 ‹ “ : 138 Thesecondtermin( 6.102 )involvestheoperator @ 2 G D z .Usingtheof @ 2 G and D z ,givenin( 2.67 ) and( 2.66 ),respectively,wecanwrite @ 2 G D z v @ 2 G „ 1 ~ J p © z v ‚ Ñ ~ J p › @ 2 G ‹ © z v “ ” „ @ 2 G Ñ ~ J p ‚ © z v 2 @ s „ Ñ ~ J p ‚ @ s © z v: (6.103) Takingthe L 2 ‹ “ -normyields TT ~ @ 2 G D z v TT L 2 ‹ “ WW Ñ ~ J p › @ 2 G ‹ © z v “ ” „ @ 2 G Ñ ~ J p ‚ © z v 2 @ s „ Ñ ~ J p ‚ @ s © z v WW L 2 ‹ “ ; (6.104) B WW Ñ ~ J p › @ 2 G ‹ © z v “ ” WW L 2 ‹ “ WW„ @ 2 G Ñ ~ J p ‚ © z v WW L 2 ‹ “ 2 WW @ s „ Ñ ~ J p ‚ @ s © z v WW L 2 ‹ “ ; (6.105) B WW Ñ ~ J p WW L ª TT› @ 2 G ‹ © z v “ ”TT L 2 ‹ “ WW„ @ 2 G Ñ ~ J p ‚ © z ‹ f ‹ z ““WW L ª ‹ “ TT TT L 2 ‹ “ (6.106) 2 WW @ s „ Ñ ~ J p ‚WW L ª ‹ “ SS @ s © z v SS L 2 ‹ “ ; B " 2 c 1 SS v SS L 2 ‹ “ c 2 SS v SS L 2 ‹ “ (6.107) 2 WW @ s „ Ñ ~ J p ‚WW L ª ‹ “ TT @ s ‹ @ 2 s “ TT l 2 l 2 TT @ 2 s © z v TT L 2 ‹ “ ; B " 2 C SS v SS L 2 ‹ “ ; (6.108) wheretheinequalityisthetriangleinequality,forthesecondinequalityweuseolderandthethirdin- equalityfollowsfromLemma 6.3.1 ,combinedwiththeassumptionthat > W 2 ; ª .Plugging( 6.108 )into( 6.102 ) yields TT ~ L 2 p v TT L 2 ‹ “ B c" SS v SS L 2 ‹ “ " 2 TT ~ p; 0 @ 2 G v TT L 2 ‹ “ " 4 TT ~ @ 2 G ‹ @ 2 G v “ TT L 2 ‹ “ " 2 TT ~ L ‹ @ 2 G v “ TT L 2 ‹ “ (6.109) Proposition 6.3.1 showsthatthesecondtermin( 6.109 )isboundedasanoperatorinthe L 2 ‹ “ normwith " 2 TT ~ p; 0 @ 2 G v TT L 2 ‹ “ B c 2 " SS v SS L 2 ‹ “ ; (6.110) and( 6.109 )reducesto TT ~ L 2 p v TT L 2 ‹ “ B c" SS v SS L 2 ‹ “ " 4 TT ~ @ 2 G ‹ @ 2 G v “ TT L 2 ‹ “ " 2 TT ~ L ‹ @ 2 G v “ TT L 2 ‹ “ : (6.111) Theboundonthesecondtermintheright-handsideof( 6.111 )followsfromthefollowinglemma 139 Lemma6.1. Let v > X , v Q j > b j j 0 ; (6.112) with Y v Y L 2 ‹ “ 1 .Thenthereexists C A 0 suchthat TT ~ @ 2 G ‹ @ 2 G v “ TT L 2 ‹ “ B C" 3 SS v SS L 2 ‹ “ : (6.113) Proof. Thereexists c ,independentof " ,suchthat TT @ 4 s v TT L 2 ‹ “ B c" 4 SS v SS L 2 ‹ “ : (6.114) Fix ⁄ > ˆ ‹ @ 2 s “ ,where ˆ ‹ @ 2 s “ istheresolventsetofthesecondorderoperator.withoutlossofgenerality, assume ⁄ 0.Then,wewritetheoperator @ 2 G ‹ @ 2 G v “ explicitly- @ 2 G ‹ @ 2 G v “ 1 ~ J 2 p < @ @ @ @ > ‹ @ 2 s 1 ~ J 2 p “ @ 2 s 2 ‹ @ s 1 ~ J 2 p “ @ 3 s 1 ~ J 2 p @ 4 s " ™ fl @ 2 s z ~ J 3 p @ s z ~ J 3 p @ 3 s ‹ @ s z ~ J 3 p “ @ 2 s fi Ł = A A A A ? v (6.115) " Ñ z ~ J 3 p ™ fl ™ fl @ s 1 ~ J 2 p fi Ł @ 2 s " 1 ~ J 2 p @ 3 s fi Ł v " 2 ™ fl ‹ z ~ J 3 p “ 2 @ 2 s z ~ J 3 p ‹ @ s z ~ J 3 p “ @ s fi Ł v; 1 ~ J 4 p @ 4 s ™ fl 2 1 ~ J 2 p ‹ @ s 1 ~ J 2 p “ " 1 ~ J 2 p Ñ z ~ J 3 p " 2 Ñ z ~ J 3 p 1 ~ J 2 p fi Ł @ 3 s (6.116) ™ fl 1 ~ J 2 p ‹ @ 2 s 1 ~ J 2 p “ " 1 ~ J 2 p ‹ @ s Ñ z ~ J 3 p “ " Ñ z ~ J 3 p ‹ @ s 1 ~ J 2 p “ " 2 ‹ Ñ z ~ J 3 p “ 2 fi Ł @ 2 s ™ fl " 1 ~ J 2 p ‹ @ 2 s Ñ z ~ J 3 p “ " 2 Ñ z ~ J 3 p ‹ @ s Ñ z ~ J 3 p “ fi Ł @ s : Toboundthetermontheright-handsideofequation( 6.116 ),notethattheTaylorexpansion 1 ~ J 4 p can bewrittenas 1 ~ J 4 p 1 " R ; R R ‹ z; Ñ “ ª Q i 0 ‹ ~ J 4 p 1 “ i 1 : (6.117) andweobtainthebound R R R R R R R R R R R R R R R R R R R R R R ~ 1 ~ J 4 p @ 4 s v R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ B ˘ ˘ ˘ ˘ ˘ ˘ ˘: 0 TT ~ @ 4 s v TT L 2 ‹ “ " SS R f ‹ z “SS L ª ‹ “ TT @ 4 s TT L 2 ‹ “ B c 1 " 3 SS v SS L 2 ‹ “ : (6.118) Similarly,aslongas Ñ > W 1 ; ª ,thenwehave R R R R R R R R R R R R R R R R R R R R R R ~ ™ fl 2 1 ~ J 2 p ‹ @ s 1 ~ J 2 p “ " 1 ~ J 2 p Ñ z ~ J 3 p " 2 Ñ z ~ J 3 p 1 ~ J 2 p fi Ł @ 3 s v R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ B c 1 " 3 SS v SS L 2 ‹ “ : (6.119) 140 Toboundthesecondoperatorontheright-handsideofequation( 6.116 )werequirethat Ñ > W 2 ; ª ,then R R R R R R R R R R R R R R R R R R R R R R ~ ™ fl 1 ~ J 2 p ‹ @ 2 s 1 ~ J 2 p “ " 1 ~ J 2 p ‹ @ s Ñ z ~ J 3 p “ " Ñ z ~ J 3 p ‹ @ s 1 ~ J 2 p “ " 2 ‹ Ñ z ~ J 3 p “ 2 fi Ł @ 2 s v R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ B c 3 " 1 SS v SS L 2 ‹ “ ; (6.120) andsimilarly, R R R R R R R R R R R R R R R R R R R R R R ™ fl " 1 ~ J 2 p ‹ @ 2 s Ñ z ~ J 3 p “ " 2 Ñ z ~ J 3 p ‹ @ s Ñ z ~ J 3 p “ fi Ł @ s v R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ B c 4 " 1 SS v SS L 2 ‹ “ : (6.121) Plugging( 6.118 ),( 6.119 ),( 6.120 )and( 6.121 )backinto( 6.116 )weobtainthebound TT ~ @ 2 G ‹ @ 2 G v “ TT L 2 ‹ “ B C" 3 SS v SS L 2 ‹ “ : (6.122) Ì CombiningLemma 6.1 andequation( 6.111 )yields TT ~ L 2 p v TT L 2 ‹ “ B " c SS v SS L 2 ‹ “ " 2 TT ~ L ‹ G v “ TT L 2 ‹ “ : (6.123) Considerthesecondtermin( 6.123 ),andnotethat W œœ ‹ U p “ commuteswith @ 2 G .Then " 2 TT ~ L ‹ @ 2 G v “ TT L 2 ‹ “ " 2 TT ~ ‹ z W œœ ‹ U p ““‹ @ 2 G v “ TT L 2 ‹ “ ; (6.124) B " 2 TT ~ z @ 2 G v TT L 2 ‹ “ " 2 TT 2 ~ © z @ 2 G © z v “ TT L 2 ‹ “ " 2 TT ~ @ 2 G Lv TT L 2 ‹ “ ; (6.125) B " 2 TT ~ z @ 2 G v TT L 2 ‹ “ " 4 TT 2 ~ © z @ 2 G © z v “ TT L 2 ‹ “ " 2 TT p; 0 ~ @ 2 G v TT L 2 ‹ “ ; (6.126) B " 2 TT ~ z @ 2 G v TT L 2 ‹ “ c 2 " 3 SS v SS L 2 ‹ “ c 3 " SS v SS L 2 ‹ “ ; (6.127) wherethelastinequalityfollowsfromPropositions 6.3.1 and 6.3.2 .Thetermin( 6.127 )hasthefollowing bound " 2 TT ~ z @ 2 G v TT L 2 ‹ “ B " 2 R R R R R R R R R R R R R R R R R R R R R R R R " < @ @ @ @ > ‹‹ 1 ; 1 “ Ñ “ 2 6 ~ J 4 p @ 2 s ™ fl ‹‹ 1 ; 1 “ Ñ “ 2 3 ~ J 4 p " ‹‹ 1 ; 1 “ Ñ “ 2 z Ñ 12 ~ J 5 p fi Ł @ s = A A A A ? v R R R R R R R R R R R R R R R R R R R R R R R R L 2 ‹ “ ; (6.128) B c 1 " SS v SS L 2 ‹ “ ; (6.129) forsome c 1 independentof " (TheexplicitcalculationissimilartothatofPropositions 6.3.1 and 6.3.2 ). Plugging( 6.128 )into( 6.127 )yields " 2 TT ~ L ‹ @ 2 G v “ TT L 2 ‹ “ B c 5 " SS v SS L 2 ‹ “ : (6.130) 141 Pluggingequation( 6.130 )into( 6.123 )weobtaintherequiredbound TT ~ L 2 p v TT L 2 ‹ “ B "C SS v SS L 2 ‹ “ ; for v > X (6.131) where C isindependentof " ,butitdependupon SS SS L ª ‹ “ ,whichisuniformlyboundedfor p admissible. 6.3.3Bounding ~ L p Finally,thefollowingPropositionshowsthattheoperator ~ L p isbounded. Proposition6.3.3. Theoperator ~ L p l 2 ‹ R N N “ ( l 2 ‹ R N N “ hasan O ‹ " “ operatornorm. Proof. Let v > X ;v P j > b j 0 j with Y v Y L 2 ‹ “ 1.Inparticular, v v; ~ v 0 ; (6.132) whereistheprojectiononto X and ~ isthecomplementaryprojection. Fix ⁄ > ˆ ‹ L 2 p “ ,where ˆ ‹ L 2 p “ istheresolventsetof L 2 p ,andrewrite L p inthefollowingway L p L 2 p " ~ L p ‹ L 2 p ⁄ “ " ~ L p ‹ L 2 p ⁄ “ 1 ‹ L 2 p ⁄ “ ⁄ : (6.133) Takingthe L 2 -normof ~ L actingon v yields TT ~ L p v TT L 2 ‹ “ TT ~ ‹ L 2 p ⁄ “ v " ~ ~ L ‹ L 2 p ⁄ “ 1 ‹ L 2 p ⁄ “ v TT L 2 ‹ “ ; (6.134) B TT ~ L 2 p v TT L 2 ‹ “ " TT ~ ~ L ‹ L 2 p ⁄ “ 1 ‹ ~ “‹ L 2 p ⁄ “ v TT L 2 ‹ “ ; (6.135) B TT ~ L 2 p v TT L 2 ‹ “ (6.136) " − TT ~ L ‹ L 2 p ⁄ “ 1 ‹ L 2 p ⁄ “ v TT L 2 ‹ “ TT ~ L ‹ L 2 p ⁄ “ 1 ~ ‹ L 2 p ⁄ “ v TT L 2 ‹ “ ‘ ; B TT ~ L 2 p v TT L 2 ‹ “ (6.137) " − TT ~ L ‹ L 2 p ⁄ “ 1 TT L 2 ‹ “ TT L 2 p v TT L 2 ‹ “ TT ~ L ‹ L 2 p ⁄ “ 1 TT L 2 ‹ “ TT ~ L 2 p v TT L 2 ‹ “ ‘ : Since ~ L isrelativelyboundedwithrespectto L 2 p ,theoperator ~ L ‹ L 2 p ⁄ “ 1 hasan O ‹ " “ boundasanoperator from l 2 ‹ R N N “ to l 2 ‹ R N N “ .Insection 6.3.2 wehaveshownthat TT ~ L 2 p TT L 2 B c": (6.138) Therefore,combiningbound( 6.138 )withtheboundednessof TT ~ L ‹ L 2 p ⁄ “ 1 TT L 2 ,aninspectionofequa- 142 tion( 6.137 )yields TT ~ L p v TT L 2 B ~ c" › ‹ 1 " “SS v SS L 2 TT L 2 p v TT L 2 ” (6.139) Tocompletetheboundon TT ~ L p TT L 2 weneedshowthat TT L 2 p TT L 2 isbounded.Usingthe of( 6.14 ),wecanrewritetheoperator L 2 p v as L 2 p Q j > b j 0 k Q k > Q j > b j ‹ L 2 p 0 j ; 0 k “ L 2 0 k ; (6.140) anditsmatrixrepresentation M > R N N ,where N O ‹ " 3 ~ 2 “ ,withentries M j;k › L 2 p 0 j ; 0 k ” L 2 : (6.141) Calculating L 2 p v explicitly,wecanwrite M M diag M where M diag ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ ‹ p; 0 " 2 k “ 2 O ‹ " 2 “ if k j; 0if k x j; (6.142) M ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ 0if k j; " 2 º " ‹ P k P j “ R R l ~ " 0 ‹ z Ñ “ Ñ ‹ © 0 “ 0 k j dzds O ‹ " 3 “ if k x j; (6.143) seeappendix( E.3 )fordetailedcalculations.ByTheorem 5.2 weknowthat M hasan O ‹ " “ bound asanoperatorfrom l 2 ‹ R N N “ to l 2 ‹ R N N “ .Moreover,weconsider k > forwhich ‹ p; 0 " 2 k “ 2 O ‹ " “ . Hence, M and M diag hasasimilarboundand TT L 2 p v TT L 2 B c" SS v SS L 2 : (6.144) Insertingthisboundbackto( 6.139 )yields TT ~ L v TT L 2 B "c SS v SS L 2 ; (6.145) whichimpliesthat ~ L p hasan O ‹ " “ boundasanoperator l 2 ‹ R N N “ l 2 ‹ R N N “ . Ì Recallthe2 2blockformof L ,( 6.15 ).Inthissectionwehaveshownthatthediagonalblockshas O ‹ " “ boundasoperatorsfrom l 2 ‹ R N N “ l 2 ‹ R N N “ .Thenextstepintheanalysiswillbetoshowthat,at 143 leadingorder,thespectrumof L isdeterminedbythespectrumof M .However,wealreadyprovedthis fortheBilayer,andsincetheanalysisdoesnotdependupontheinterfaceco-dimension,thesameresult holdshere.Weconcludethatthepearlingstabilityconditionof L ,fortheco-dimension2structureis,at leadingorder,thepearlingconditionof M ,giveninCorollary 6.2.2 .Thatis,wehaveshownthatforagiven admissibleinterface, p ,theassociatedporesolutionconstructedin( 2.75 ),isstablewithrespecttothe pearlingbifurcationifandonlyiftheeldchemicalpotential 1 the pearlingstabilitycondition statedinTheorem 6.0.2 . 144 Chapter7 AnalysisofNetworkBifurcations 7.1Introduction Inthechapters 3 and 4 wedevelopasymptoticexpressionsforthegeometricevolutionofadmissiblebilayer andporemorphologies.Thesearequenchedcurvaturedrivenws,whichyieldastablemotionbymean curvatureforvaluesofthespatiallyconstantchemicalpotential, ,thatarelessthan ⁄ b forbilayersand lessthan ⁄ p forpores.Ifthechemicalpotentialexceedseitherofthesecriticalvalues,thentheevolution becomesmotionagainstmeancurvature,whichisunstabletogrowths.Insection 4.7 ,thecombined evolutionofwellseparatedbilayersandporesisgivenbyequations( 4.166 - 4.169 ),whichcoupletheevolution ofthetwomorphologiesthroughthespatiallyconstantvalueofthechemicalpotential.Thestability ofbilayersandporestothepearlingbifurcationischaracterizedinchapters 5 and 6 ,respectively.Again thestabilityconditioncanbeexpressedintermsofthemagnitudeofthefareldchemicalpotentialwith respecttocriticalvaluesthatdependonlyuponthefunctionalizationparameters, 2 ; 2 andthepotential well W .Asthestabilityoftheunderlyingporeandbilayermorphologiesisindependentoftheirshape,for apotentialwell W ,wemayanalyzethestabilityregionsofbilayersandporeswithrespectto 1 , 1 , and 2 .Forsimplicitywe 1 1andpresentthestabilityregionsintermsof 1 and d 1 2 : Underthe H 1 gradientwthechemicalpotential 1 isdynamiconthe ˝ O ‹ " 1 “ time-scale.Thisisthe sametimescaleasthegeometricw,andhenceoftheinstability.Howeverthetimescaleofthe pearlinginstabilityisgovernedbythepearlingeigenvaluesof L whicharetwoordersof " largerthanthe pearlingeigenvaluesof L ,thatistheyscalewith O ‹ " 1 “ andthetime-scaleoftheoftheonsetofthepearling instabilityis t O ‹ " “ .Thusthepearlinginstabilitymanifestsitselfonatimescalethatisinstantaneous withrespecttotheunderlyinggeometricevolution. 145 7.2MeanderingEquilibria Weinvestigatedthegeometricevolutionofco-dimensionsoneandtwomorphologiesinChapters 3 and 4 .It wasshownthat,inthecombinedsystem,solongastheunderlyingnetworkmorphologiesremainadmissible andhavenon-zerocurvature,thentheleadingorderchemicalpotential 1 willdecayexponentiallytoa constant,seeequation( 4.165 ).Theexplicitexpressionfortheequilibriapointsofthechemicalpotentialfor asystemwithbilayers, ⁄ b ,andforasystemwithpores, ⁄ p ,takestheform ‹ 3 : 116 “ ⁄ b ‹ d “ 1 2 ‹ 2 1 d “ TT ^ U œ b TT 2 L 2 R R ^ U b dz ; (7.1) ‹ 4 : 149 “ ⁄ p ‹ d “ 1 R ª 0 ‹ ^ U œ p “ 2 RdR 2 R ª 0 ^ U p RdR : (7.2) AccordingtotheresultsinSection 4.7 ,forinterfaces,therange 1 > ⁄ p ; ⁄ b isinvariantunderthe w,andonce 1 entersthisrangeonestructurewillshrink,whiletheothermorphologywillgrow. Onceadouble-wellpotential W ˘ hasbeenchosen,seeequation( 7.10 ),wecancalculatethebilayerandpore U b and U p fromequations 2.37 and 2.75 respectively,seeFigure 7.1 . Figure7.1: Thebilayer(left)andthepore(right)correspondingtothetilteddouble-well potential W Thefunctionalizationparameter 2 playsanimportantrule:sinceitcanbeeitherpositiveornegativeitwill determinetherelativesizeof ⁄ b and ⁄ p ,andsoweconsider 2 asafreeparameterthrough d .Oncethe well, W ,and 1 aretheonlytwovaryingparametersinthesystemare 1 and 2 .Figure 7.2 depicts theequilibriapoints,ofthetwostructures,asafunctionof d .Thetwoequilibrialinesintersectanddivide theplaneintofourregions:abovethetwolines-bothstructuresgrow,andthechemicalpotentialdecays, 146 Figure7.2: Themeanderingequilibrialines:Theblue(solid)lineisforthebilayersystem, ⁄ b ,andthe red(dashed)lineisfortheporesystem ⁄ p ,asafunctionof d ,where 1 1andadouble-well potential. belowthetwolines-bothstructuresshrink,andthechemicalpotentialgrows,andtworegionsbetweenthe twolines,whereonestructuregrowsandtheseconddecays.Theintersectionpointofthetwoequilibria linesisgivenby ⁄ d 1 ™ fl 1 ™ fl SS Up SS L R R ª 0 ^ U p RdR SS U b SS 2 L 2 2 R R ^ U b dz fi Ł R R ^ U b dz SS U b SS 2 L 2 fi Ł ; (7.3) and,afterchoosing W ˘ and 1 ,theintersectionpoint, ⁄ d ,isForthisspvalueof d thestrong FCHmay,atleadingorder,supportacoexistenceofthetwomorphologies.However,foranyvalueof d A ⁄ d thesystemgivesprioritytopores,and,similarly,when d @ ⁄ d thesystemprefersbilayers. 7.3PearlingStability InChapters 5 and 6 wederivedanexplicitleadingorderexpressionforthepearlingeigenvalues.Thepearling stabilityconditionistheconditionon 1 forwhichthepearlingeigenvaluesremainpositive.Thepearling stabilityconditions,forthebilayersandthepores,respectively,taketheform ‹ 5 : 2 : 3 “ 1 S b d b; 0 SS b; 0 SS 2 2 @ 0 ; forbilayers(7.4) ‹ 6 : 2 : 2 “ 1 S p d − TT œ p; 0 TT 2 L R p; 0 SS p; 0 SS 2 L R ‘ @ 0 ; forpores ; (7.5) 147 where b; 0 istheground-stateeigenvalueofthelinearoperator L b; 0 ,in( 1.29 ),withthecorresponding eigenfunction b; 0 , p; 0 isthegroundstateeigenvalueofthelinearoperator L p; 0 ,in( 2.81 ),withthe correspondingeigenfunction p; 0 ,and S b ;S p arethe shapefactors ofthebilayersandthepores,respectively, by ‹ 5 : 41 “ S b S R b; 1 W œœœ ‹ U b “ 2 b; 0 dz; (7.6) ‹ 6 : 33 “ S p 2 ˇ S ª 0 p; 1 W œœœ ‹ U p “ 2 p; 0 RdR: (7.7) Withinthe 1 d planethepearlingbifurcationoccursalongthetwo"pearlingbifurcationlines" ‹ 5 : 2 : 3 “ P ⁄ b ¢ ¨ ¨ ¦ ¨ ¨ ¤ 1 d b; 0 SS b; 0 SS 2 2 S b R R R R R R R R R R R d > R £ ¨ ¨ § ¨ ¨ ¥ ; (7.8) ‹ 6 : 2 : 2 “ P ⁄ p ¢ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¤ 1 d − TT œ p; 0 TT 2 L R p; 0 SS p; 0 SS 2 L R ‘ S p R R R R R R R R R R R d > R £ ¨ ¨ ¨ § ¨ ¨ ¨ ¥ : (7.9) Thesignof shapefactors S b ;S p determinesifthemorphologyispearlingstablewhen 1 isabovethe pearlingbifurcationlines P ⁄ b ;P ⁄ p ,orifthemorphologypearls. 7.4NumericalEvaluationofBifurcationRegions Inthissectionwenumericallydeterminethepearlinglinesandthemeanderstability/meanderngeringlines andpresenttheirpartitioningofthe 1 d plane.Wethebackgroundstate, b ,tobe 1,andchoosea tilteddouble-wellpotentialoftheform W ˘ ‹ u “ ‹ u 2 b 2 “ 2 4 ˘ 3 ‹ u 2 b “ ; (7.10) wheretheparameter ˘ determinesthedepthoftherightwell.Weconsider3twelltilts,corresponding to ˘ 0 : 9 ; 0 : 7 ; 0 : 5,seeFigure 7.3 .Westartby ˘ 0 : 9. Tocalculateeachofthestabilitylineswemustevaluatethegroundstateeigenvalue,thegroundstate eigenfunctions,andthevalueoftheshapefactor.WeusetheEvansfunctionstocalculatethegroundstate eigenvalues,andforthepotentialin( 7.10 )wendthat b; 0 0 : 7421and p; 0 0 : 4648.Wenormalize theassociatedeigenfunctionssothat SS b; 0 SS 2 2 SS p; 0 SS 2 L R 1,seeFigure 7.4 .Next,toevaluatetheshape factors S b and S p weneedanexpressionforthe" L 1 1 functions " b; 1 L 1 b; 0 1and p; 1 L 1 p; 0 1, in( 2.40 )and( 2.86 ),respectively,anddepictedinFigure 7.5 ,andtheresultingvaluesfortheshapefactors 148 Figure7.3: Thetilteddouble-wellpotential W ‹ u “ . Figure7.4: Thegroundstateeigenfunctions: b; 0 (left)and p; 0 (right). aregiveninTable 7.1 .Thepearlingbifurcationlinesdividethe 1 d planeintofourregions,whenboth shapefactorsarenegative.Theregion-abovethetwolines,bothmorphologiesarepearlingstable, secondregion-belowthetwolines,bothmorphologiesarepearlingunstable,thethirdregionwherethe bilayerispearlingunstablewhiletheporeisstableandthefourthregionitisviseversa,seeFigure 7.6 (left).Theequilibrialines, ⁄ b ‹ d “ and ⁄ p ‹ d “ ,inequations( 7.1 )and( 7.2 )arealsofunctionsof d , and 7.6 (right)showsthepartitioningoftheplanefromallfourlines.Theinvariantintervalfor 1 isbetweenthetwohorizontallines,wheretheblue(solid)linerepresentsthebilayerequilibriaandthered (solid)linerepresentstheporeequilibrialine. Recallingthattheparameter 1 isdynamicontheslow ˝ "t timescale,weconsiderinitialdataforwhich thechemicalpotential 1 lieswithintheregionwherebothmorphologiesarepearlingstableandaregrowing, 149 Figure7.5: The L 1 1functions: b; 1 (left)and p; 1 (right). ˘ S b S p -0.9 -2.1958 -0.0431 -0.7 -0.616 -0.033 -0.5 0.8736 -2.1756 Table7.1: Numericalevaluationsoftheshapefactorofthebilayers S b andtheshapefactorofthepores S p asafunctionofthetiltofthedoublewellpotential W ˘ ‹ u “ . seearrowonFigure 7.6 (right).Thenfor d @ 0thechemicalpotentialwillshrinkwhileboththebilayerand poremorphologies, b and p ,willgrow,whiletheymaythisisaslowprocesswhichcanbedominated bytheevolutionof 1 ifthecurvatureweightedintegralsin( 4.168 )aretlylarge.Assumingthat bothmorphologiesremainadmissible,thenatsometime t O ‹ " 1 “ thechemicalpotentialwillcrossthe P ⁄ b line,andthebilayerswillpearlonafast O ‹ " “ time-scale. Genericallythecoupledbilayerandporeevolutioniscompetitive,withthetwomorphologiesseekingin- compatiblevaluesofthechemicalpotentialatequilibrium.However,bytuningthevalueof d , theequilibriumvaluescanbalance, ⁄ b ⁄ p ,andthecodimensiononeandtwomorphologiescanpotentially co-exist.ThegreencircleinFigure 7.6 markedthelocationofacommonequilibria,andforthesp double-wellpotential,with ˘ 0 : 9,weseethattheequilibriapointislocatedinthepearlingstableregion. Forthisvalueof d ,andaninitialvalueof 1 belowtheequilibriapoint,thetwomorphologieswithshrink, untilreachingtheequilibria,withoutfrompearlinginstabilityormeanderinginstability. Foradoubletiltedwellpotential W ˘ with ˘ 0 : 7 theresultsarequalitativelysameasthecase ˘ 0 : 9. ThevaluesoftheshapefactorsareindicatedinTable 7.1 andFigure 7.7 (above)depictsthedivisionof the 1 d planebythefourmeanderingequilibria/pearlingbifurcationlines. 150 Figure7.6: Pearlingbifurcationlinesasafunctionof d (topleft),PearlingandEquilibrialines(topright), theco-existenceequilibriaismarkedbyagreencircle.Zoomingontotheblackcircleintheonthe right(bottomcenter). Weconsideranevenmoretilteddoublewellpotential W ˘ ,with ˘ 0 : 5. Inthiscase,thevalue theshapefactorofthebilayerispositive,seeTable 7.1 .Thechangeinthesignof S b impliesthatthe bilayermorphologyisstableaslongas 1 liesbelow P ⁄ b .Furthermore,theshapefactoroftheporeremains negativeandtheporestructureisstablefor 1 above P ⁄ p .For d closetozero,theareawherebothstructures arepearlingstableislocatedinsidethedynamicallyinvariantinterval.Thechemicalpotentialwilldecayto itsequilibria,andwillstayinsidetheinterval,whiletheporeswillshrink,andthebilayerswillgrow.As 1 decays,itwillcrosstheporebifurcationline, P ⁄ p ,whichwillcausetheporestructuretopearlbeforethe systemreacheditsequilibria,seeFigure 7.7 (bottom). Note8. Thechoiceofthepotential W ˘ ethesignof S b .Bychangingthepotentialtilt,weareable tochangethesignof S b .Iftheshapefactorisidenticallyzero,thecorrespondingpearlingbifurcationline 151 Figure7.7: Thetwoequilibrialinesandthetwobifurcationlinesforadoubletiltedwellwith ˘ 0 : 7 (top)andwith ˘ 0 : 5(bottom). willbeverticalinthe 1 d plane. 152 7.5ComparisontoExperimentandFullFCHsimulations TheFunctionalizedCahn-Hilliardfreeenergyprovidesacompactdescriptionoftheenergylandscapedriving morphologicalselectioninamphiphilicmixtures,suchaslipidbilayers.Wehaveshownthatthestrengthof theinteractionsofthehydrophilicunitswiththesolventphase,parameterizedby 1 A 0,thepackingentropy ofthehydrophobictails,parameterizedby 2 ,andthepressurejumpbetweenamphiphilicandhydrophobic phases,characterizedbytheinselfenergies, W ‹ b “ oftheamphiphilicandbulkphases,can triggerarangeofbifurcations.Sptheandpearlinginstabilitiesobservedexperimentally in[ BudinandSzostak,2011 ]and[ Zhuetal.,2012 ]byadjustingthebulkvaluesoflipidsandthecharge densityofthelipids,respectively,canbeinducedintheFCHframeworkbyvaryingthecorresponding controlparameters.Theinstabilityistriggeredbyajumpinthevalueofthechemicalpotential 1 . Assumingwestartwiththecombinedsystematitsequilibriapoint,andinstantaneouslyincrease 1 .Then, atleastonemorphologywillstartgrowing,as 1 decaysbacktoitsequilibria,andthismorphologymay startseeFigure 7.8 (left).Ontheotherhand,thepearlinginstabilitycanbetriggeredbyan Figure7.8: (left)Increasingthebackgroundstate 1 movestheblackpointfromitsequilibria,which resultsingrowthofbothmorphologies.(right)Szostak'sexperiment:raisingthebackgroundconcentration oflipidsinducesthevesicletogrowworm-like(co-dimensiontwo)protrusionsovera74nano-secondtime period[ BudinandSzostak,2011 ] increasein d whichmovesthesystemfromitsequilibriainapearlingstableregionintoapearlingunstable region,see 7.9 (left). Anotherwaytochangethestabilityofasystemisbychangingthetiltofthedouble-wellpotential.Fig- 153 Figure7.9: (left)Increasing d movestheblackpointtoatpearlingstabilityarea,whichquickly leadstopearlinginthebilayers.(right)Changingthedensityofchargedgroupsonthesurfaceviaa photochemicallyinducedredoxreactionincitestheporetopearlandbreakintomicelles[ Zhuetal.,2012 ]. ure 7.10 describesthenumericalresultsfor ˘ 0 : 7(left)for ˘ 0 : 9(right),wherethegreendotmarksthe initialdataforwhichthefunctionalizationtermssatisfy d 1andthechemicalpotential,attheinitial state,is 1 ‹ 0 “ 0 : 4.For ˘ 0 : 7thegreendotislocatedataregionwhichisbothpearlingstableandme- anderingstable.Accordingtotheanalysis,startingwithacombinedsystem,withinitialdatacorresponding tothegreendot,bothmorphologieswillshrinkwhile 1 growsuntil 1 ⁄ b ,andbothstructuresshould remainpearlingstable.However,for ˘ 0 : 9,thesamegreendotwouldbelocatedinaregioncorresponding tomeanderingstability,i.e.,thetwostructureswillshrink,however,theregionispearlingunstableforthe bilayer.TheseresultsareinconcordwithFigure 1.10 whichdescribesthecompetitionfortheamphiphilic phasebetweenabilayerandporesasafunctionofthetilt. Therearehowevermanyavenuestoexplore,forexamplethepearlingbifurcationinducesaperiodicdimpling ofabilayersurfacewhichcanleadtoperforation.Withinthebiologicalcontextofcellmembranes,itis ofparticularinteresttounderstandtheenergyrequiredtoopenasinglehole.Canalocaladjustmentof parametervalues,suchasaspatialvariationin 1 ,inducetheopeningofisolatedholesinthemembrane? 154 Figure7.10: Pearlingandmeanderingstabilityregionsforttiltsofthepotential W ˘ .Forthe tilt ˘ 0 : 7,thegreendotisbothmeanderingandprealingstable(left).For ˘ 0 : 9,thegreendotis locatedinthebilayerunstablepearlingregion. 7.6VerifyingtheNumericalResults Toverifythenumericalresultspresentedhere,wecomparethenumericalvalueoftheshapefactor, S b , aswellasthenumericalvaluesofotherkeyparameters,tothevalueoftheiralgebraicexpressions,given in[ Doelmanetal.,2014 ].Itwasshownby[ Doelmanetal.,2014 ]thattheshapefactor S b ,ofthebilayeris negativeforafamilyoftilteddouble-wellpotentialsoftheform W p ‹ u “ ~ W p ‹ u 1 “ 20 ‹ u ~ m p 1 “ p 1 H ‹ u ~ m p 1 “ ; (7.11) where ~ W p ‹ u “ 1 p 2 ‹ pu 2 2 u p “ ;m p p 2 1 p 2 A 1 ; (7.12) H istheHeavisidefunctionand p A 2.Figure 7.11 depicts W 3 .Moreover,in[ Doelmanetal.,2014 ]algebraic expressionsarederivedforkeyquantities,whichwerepeatinthefollowingLemma. Lemma7.1 ([ Doelmanetal.,2014 ]) . Fix p A 2 andlet U b bethehomoclinicsolutionof @ 2 z U b W œ ‹ U b “ for W W p .Thenthegroundstateeigenvalueofthelinearizedoperator L b; 0 ,din( 1.29 ), b; 0 1 2 p ‹ p 2 “ A 0 : (7.13) Moreover,thefollowingequalitieshold SS U œ b SS L 2 ‹ “ SS b; 0 SS L 2 ‹ “ 2 º p 2 ~ m p 2 2 p I ‰ 2 p 2 ’ ; (7.14) 155 Figure7.11: Atilteddouble-wellpotentialoftheform( 7.11 )for p 3(blue)andoftheform( 7.10 ) for ˘ 0 : 9(red). where I ‹ q “ 4 q R 1 0 z q ‹ 1 z “ q dz ,whiletheshapefactor S b satisfy S b 2 ‹ p 1 “ º p 2 ~ m 1 2 ‹ 3 p 4 “ p I ‰ 1 p 2 ’ : (7.15) For p 3,weuse W 3 toobtainthevaluesof b; 0 ; SS U œ b SS L 2 ‹ “ and S b numericallyandcomparethemtothe algebraicvalues,giveninLemma 7.1 .Theresultsareshownintable 7.2 . Parameter NumericalValue Algebraicvalue b; 0 7.4985 7.5 SS U œ b SS L 2 ‹ “ 2.9394 2.9394 S b -7.3242 -7.3485 Table7.2: Numericalevaluationsofthekeyparameterscomparedtotheiralgebraicvalues,forthedouble wellpotential W 3 ‹ u “ . Notethatthetilteddouble-wellpotential W ˘ ,in( 7.10 ),doesnotbelongtothisclassofpotentials, in( 7.11 ).Thedoublewellpotentialsgivenby(refNV-eq:ArjenW)havestronglyunequaldepthsof thetwolocalminimaandalargervalueforthelocalmaximabetweenthem.Thevalueof S b turnspositive for W ˘ asthevalueofthelocalminimabecomeproximalandtheheightofthelocalmaximadecreases. 156 APPENDICES blankspace 157 AppendixA CoordinatesSystem A.1PolarCoordinates Theco-dimension2morphologiescanbeformedfromcylindrically,symmetriccriticalpointsoftheCahn- Hilliardfreeenergy.Inpolarcoordinates,thescaleddestancevector z isgivenby z ‹ R cos ;R sin “ ; (A.1) where R isthe " -scaledradialdistanceto p .Werewritethe z -gradientandthe z -Laplaceoperatorsinpolar coordinates © z ‹ cos @ R 1 R sin @ ; sin @ R 1 R cos @ “ ; (A.2) z @ 2 R 1 R @ R 1 R 2 @ 2 : (A.3) Forradialsymmetricfunctionsthegradient( A.2 )andtheLaplacian( A.3 )reduceto © z ‹ cos @ R ; sin @ R “ ; (A.4) z @ 2 R 1 R @ R : (A.5) Plugging( A.5 )into( 2.70 ),weobtainaradiallysymmetricrepresentationoftheLaplacianintheinner coordinates x " 2 ‰ @ 2 R 1 R @ R ’ " 1 Ñ ‹ cos @ R ; sin @ R “ @ 2 s ‹ z Ñ “ Ñ © z O ‹ " “ ; (A.6) 158 A.2Detailedinvestigationofthespectrumof L b; 0 . Considerthelinear,closed,limitoperator L b; ª @ 2 z W œœ ‹ b “ : (A.7) AccordingtoStrum-LiouvilleTheoryforoperatorsontherealline,thepointspectrumof L b; ª consistsof numberofsimpleeigenvalueswhichcanbeenumeratedinastrictlydescendingorder 0 A 1 A ::: A N A W œœ ‹ b “ : (A.8) Wethematrix A ™ Œ Œ Œ Œ Œ fl 01 W œœ ‹ b “ 0 fi Š Š Š Š Š Ł (A.9) Thematrixeigenvaluesaregivenby » W œœ ‹ b “ (A.10) Theessentialspectrumof L b; ª ˙ ess ‹ L b; ª “ Ÿ > R dim E c ‹ A ‹ ““ x 0 š W œœ ‹ b “ : (A.11) Notethat L b; ª isthelimitoperatorof L b; 0 ,edin( 2.39 ),and L b; 0 isaclose,linearoperator.Moreover, sincetheoperator ‹ L b; 0 L b; ª “‹ L b; 0 ⁄ “ 1 ‹ W œœ ‹ U b “ W œœ ‹ b ““‹ L b; 0 ⁄ “ 1 (A.12) isacompactforevery ⁄ > ˙ ‹ L b; 0 ““ ,weknowthat L b; 0 isarelativelycompactperturbationof L b; ª .We applyWeyl'sTheoremtoconclude ˙ ess ‹ L b; ª “ ˙ ess ‹ L b; 0 “ : (A.13) A.3Self-adjointoperators Considerthe L 2 ‹ “ innerproductin( 2.73 )andthetwooperators ~ L b @ 2 z W œœ ‹ u “ " 2 s andthe fulloperator L b @ 2 z z W œœ ‹ u “ " 2 G .TheLaplace-beltramioperatoris not self-adjointinthisinner 159 product, ‹ s f;g “ L 2 ‹ “ S S l … " l … " gJ 1 0 © s ‹ g 1 J 0 © s “ fJ 0 ~ Jdzds S S l … " l … " g 1 J 0 © s ‹ g ~ J “ © s fdzds (A.14) S S l … " l … " © s ‹ g 1 J 0 © s ‹ g ~ J ““ fdzds x ‹ f; s g “ L 2 ‹ “ (A.15) althoughitisself-adjointinthe b -innerproduct.However, G isself-adjointinthe L 2 ‹ “ innerproduct. ‹ G f;g “ L 2 ‹ “ S S l … " l … " gJ 1 © s ‹ g 1 J © s “ fJdzds S S l … " l … " g 1 J © s ‹ g “ © s fdzds (A.16) S S l … " l … " © s ‹ g 1 J © s ‹ g ““ fdzds ‹ f; G g “ L 2 ‹ “ : (A.17) Calculatingthe L 2 ‹ “ -innerproducttotherestofthetermsintheoperator L b yields, ‹ @ 2 z f;g “ L 2 ‹ “ S S l … " l … " f œœ gJdzds S S l … " l … " f ‹ gJ “ œœ dzds S S l … " l … " fg œœ J 2 fg œ J œ fgJ œœ dzds (A.18) ‹ f;@ 2 z g “ L 2 ‹ “ 2 " S S l … " l … " fg œ dzds " S S l … " l … " fg œ Jdzds " 2 S S l … " l … " fg 2 Jdzds; (A.19) " ‹ z f;g “ L 2 ‹ “ " S S l … " l … " œ gJdzds " S S l … " l … " f ‹ gJ “ œ dzds S S l … " l … " f œ J "fgJ œ fgJ œ dzds (A.20) " ‹ f; z g “ L 2 ‹ “ " S S l … " l … " fg œ Jdzds " 2 S S l … " l … " fg 2 Jdzds: (A.21) Eachofthetermsseparatelyisnotself-adjointinthe L 2 ‹ “ -innerproduct,howevertheirsum ‹ @ 2 z f z f;g “ L 2 ‹ “ ‹ f;@ 2 z g z g “ L 2 ‹ “ : Therefore,ourfulloperator L b isself-adjointinthe L 2 ‹ “ -innerproductwherethe ~ L b operatorisnot. 160 AppendixB GeometricEvolutionCo-dimension1 B.1Outerexpansionofthe 1 st variationof F Recallthatthe1 st variationof F isgivenby F u ‹ u “ " 2 W œœ ‹ u “ 1 ”› " 2 u W œ ‹ u “ ” d W œ ‹ u “ : (B.1) Pluggingaformalexpansionof u ‹ x “ u 0 ‹ x “ "u 1 ‹ x “ ::: andexpandingyields › " 2 W œœ ‹ u “ 1 ”› " 2 u W œ ‹ u “ ” d W œ ‹ u “ (B.2) ‰ " 2 W œœ ‹ u 0 “ " ‹ W œœœ ‹ u 0 “ u 1 1 “ " 2 ‹ W œœœ ‹ u 0 “ u 2 1 2 W œœœœ ‹ u 0 “ u 2 1 “ :: ’ ‰ " 2 ‹ u 0 " u 1 " 2 u 2 “ W œ ‹ u 0 “ "W œœ ‹ u 0 “ u 1 " 2 ‹ W œœ ‹ u 0 “ u 2 1 2 W œœœ ‹ u 0 “ u 2 1 “ ::: ’ d ‰ W œ ‹ u 0 “ "W œœ ‹ u 0 “ u 1 " 2 ‹ W œœ ‹ u 0 “ u 2 1 2 W œœœ ‹ u 0 “ u 2 1 “ ::: ’ : Since ‹ x;t “ F u ‹ u “ wecanrewriteitinorderof " suchthat F u ‹ u “ 0 ‹ x;t “ 1 ‹ x;t “ " 2 2 ‹ x;t “ O ‹ " 3 “ ; (B.3) where 0 W œœ ‹ u 0 “ W œ ‹ u 0 “ ; (B.4) 1 ‹ W œœœ ‹ u 0 “ u 1 1 “ W œ ‹ u 0 “ ‹ W œœ ‹ u 0 ““ 2 u 1 d W œ ‹ u 0 “ ; (B.5) 161 2 ‰ W œœœ ‹ u 0 “ u 2 1 2 W ‹ 4 “ ‹ u 0 “ u 1 ’ W œ ‹ u 0 “ ‹ W œœœ ‹ u 0 “ u 1 1 “ W œœ ‹ u 0 “ u 1 (B.6) W œœ ‹ u 0 “‰ u 0 W œœ ‹ u 0 “ u 2 1 2 W œœœ ‹ u 0 “ u 2 1 ’ d W œœ ‹ u 0 “ u 1 B.2Innerexpansionofthe t -derivativeof u Togetanexpressionfortheleft-handsideusingthewhiskeredcoordinateswetakethe t -derivativeof u and consider u ~ u ‹ s;z;˝ “ .Treatingboth s and z asfunctionsof t ,theuseofthechainruleisrequired,and resultsin u t © S © s ~ u @ ~ u @z @z @t @ ~ u @˝ @˝ @t : (B.7) Assumingthat u donotchangewhen s variesnormalto b with z helded,seeequation( ?? ),andusing thenormalvelocity,denedin( 3.9 ),equation( B.7 )reducesto @u @t " 1 V ˝ ‹ s “ @ ~ u @z @ ~ u @˝ @˝ @t : (B.8) Expanding~ u inordersof " suchthat~ u ‹ s;z;˝ “ ~ u 0 ‹ s;z;˝ “ " ~ u 1 ‹ s;z;˝ “ O ‹ " 2 “ andpluggingitbackinto equation( B.8 )yields @u @t " 1 V ˝ ‹ s “ @ ~ u 0 @z V ˝ ‹ s “ @ ~ u 1 @z @ ~ u 0 @˝ @˝ @t V ˝ ‹ s “ @ ~ u 2 @z @ ~ u 1 @˝ @˝ @t :::: (B.9) B.3Innerexpansionofthe 1 st variationof F Recallthatthe1 st variationof F isgivenby F u ‹ u “ › " 2 W œœ ‹ u “ 1 ”› " 2 u W œ ‹ u “ ” d W œ ‹ u “ ; (B.10) andthechemicalpotential isby F u ‹ u “ : (B.11) Atagiventimescale ˝ ,theinnerspatialexpansionforthedensityfunction u ‹ t;x “ isgivenby u ‹ x;t “ ~ u 0 ‹ s;z;˝ “ " ~ u 1 ‹ s;z;˝ “ " 2 ~ u 2 ‹ s;z;˝ “ :::; (B.12) 162 andinlocalcoordinates,recallthattheLaplacianoperator,see( 2.9 ),takestheform " 2 x @ 2 z z " 2 G : (B.13) First,consideranexpansionofeachofthetermsontheright-handsideof( B.10 ): " 2 @ 2 z "H 0 @ z " 2 ‹ zH 1 @ z s “ " 3 1 :::; (B.14) " 2 ~ u @ 2 z ~ u 0 " ‹ @ 2 z ~ u 1 H 0 @ z ~ u 0 “ " 2 ‹ @ 2 z ~ u 2 H 0 @ z ~ u 1 zH 1 @ z ~ u 0 s ~ u 0 “ " 3 ‹ @ 2 z ~ u 3 H 0 @ z ~ u 2 zH 1 @ z ~ u 1 s ~ u 1 1 ~ u 0 “ :::; (B.15) W œœ ‹ ~ u “ W œœ ‹ ~ u 0 “ "W œœœ ‹ ~ u 0 “ ~ u 1 " 2 W œœœ ‹ ~ u 0 “ ~ u 2 1 2 W ‹ 4 “ ‹ ~ u 0 “ ~ u 2 1 " 3 W œœœ ‹ ~ u 0 “ u 3 W ‹ 4 “ ‹ ~ u 0 “ ~ u 1 ~ u 2 1 6 W ‹ 5 “ ‹ ~ u 0 “ ~ u 3 1 O ‹ " 4 “ ; (B.16) W œ ‹ ~ u “ W œ ‹ ~ u 0 “ "W œœ ‹ ~ u 0 “ ~ u 1 " 2 W œœ ‹ ~ u 0 “ ~ u 2 1 2 W œœœ ‹ ~ u 0 “ ~ u 2 1 " 3 W œœ ‹ ~ u 0 “ ~ u 3 W œœœ ‹ ~ u 0 “ ~ u 1 ~ u 2 1 6 W ‹ 4 “ ‹ ~ u 0 “ ~ u 3 1 O ‹ " 4 “ : (B.17) Next,wecollectingthetermsandwritetheminorderof " ‹ " 2 W œœ ‹ u “ 1 “ ‹ @ 2 z W œœ ‹ ~ u 0 ““ (B.18) " ‹ H 0 @ z W œœœ ‹ ~ u 0 “ ~ u 1 1 “ " 2 ‰ zH 1 @ z s W œœœ ‹ ~ u 0 “ ~ u 2 1 2 W ‹ 4 “ ‹ ~ u 0 “ ~ u 2 1 ’ " 3 ‰ 1 W œœœ ‹ ~ u 0 “ ~ u 3 W ‹ 4 “ ‹ ~ u 0 “ ~ u 1 ~ u 2 1 6 W ‹ 5 “ ‹ ~ u 0 “ ~ u 3 1 ’ O ‹ " 4 “ ; ‹ " 2 u W œ ‹ u ““ ‹ @ 2 z ~ u 0 W œ ‹ ~ u 0 ““ (B.19) " › @ 2 z ~ u 1 H 0 @ z ~ u 0 W œœ ‹ ~ u 0 “ ~ u 1 ” " 2 ‰ @ 2 z ~ u 2 H 0 @ z ~ u 1 H 1 z@ z ~ u 0 s ~ u 0 W œœ ‹ ~ u 0 “ ~ u 2 1 2 W œœœ ‹ ~ u 0 “ ~ u 2 1 ’ " 3 ™ fl @ 2 z ~ u 3 H 0 @ z ~ u 2 H 1 z@ z ~ u 1 s ~ u 1 1 ~ u 0 W œœ ‹ ~ u 0 “ ~ u 3 W œœœ ‹ ~ u 0 “ ~ u 1 ~ u 2 1 6 W ‹ 4 “ ‹ ~ u 0 “ ~ u 3 1 fi Ł O ‹ " 4 “ : 163 Usingexpansion( B.18 )and( B.19 )wecanrewritethe1 st variationof F inthefollowingform F u ‹ u “ ~ 0 ‹ s;z;˝ “ " ~ 1 ‹ s;z;˝ “ " 2 ~ 2 ‹ s;z;˝ “ O ‹ " 3 “ ; (B.20) where~ i for i C 0aretheinnerexpansionofthechemicalpotential ‹ x;t “ ~ 0 ‹ s;z;˝ “ " ~ 1 ‹ s;z;˝ “ " 2 ~ 2 ‹ s;z;˝ “ :::; (B.21) givenby ~ 0 ‹ @ 2 z W œœ ‹ ~ u 0 ““‹ @ 2 z ~ u 0 W œ ‹ ~ u 0 ““ ; (B.22) ~ 1 ‹ @ 2 z W œœ ‹ ~ u 0 ““‹ H 0 @ z ~ u 0 @ 2 z ~ u 1 W œœ ‹ ~ u 0 “ ~ u 1 “ (B.23) ‹ H 0 @ z W œœœ ‹ ~ u 0 “ ~ u 1 1 “‹ @ 2 z ~ u 0 W œ ‹ ~ u 0 ““ d W œ ‹ ~ u 0 “ ; ~ 2 ‹ @ 2 z W œœ ‹ ~ u 0 ““‹ @ 2 z ~ u 2 zH 1 @ z ~ u 0 H 0 @ z ~ u 1 s ~ u 0 W œœ ‹ ~ u 0 “ ~ u 2 1 2 W œœœ ‹ ~ u 0 “ ~ u 2 1 “ (B.24) ‹ H 0 @ z W œœœ ‹ ~ u 0 “ ~ u 1 1 “‹ @ 2 z ~ u 1 H 0 @ z ~ u 0 W œœ ‹ ~ u 0 “ ~ u 1 “ ‹ zH 1 @ z s W œœœ ‹ ~ u 0 “ ~ u 2 1 2 W ‹ 4 “ ‹ ~ u 0 “ ~ u 2 1 “‹ @ 2 z ~ u 0 W œ ‹ ~ u 0 ““ d W œœ ‹ ~ u 0 “ ~ u 1 ; and,merelyfrompedanticreasons(Idon'tknowifIusethistermlateron)wealsohave ~ 3 ‹ @ 2 z W œœ ‹ ~ u 0 ““‰ L ~ u 3 H 0 @ z ~ u 2 zH 1 @ z ~ u 1 s ~ u 1 1 ~ u 0 W œœœ ‹ ~ u 0 “ ~ u 1 ~ u 2 1 6 W ‹ 4 “ ‹ ~ u 0 “ ~ u 3 1 ’ (B.25) ‹ H 0 @ z W œœœ ‹ ~ u 0 “ ~ u 1 1 “‰ L ~ u 2 H 0 @ z ~ u 1 zH 1 @ z ~ u 0 s ~ u 0 1 2 W œœœ ‹ ~ u 0 “ ~ u 2 1 ’ ‰ zH 1 @ z s W œœœ ‹ ~ u 0 “ ~ u 2 1 2 W ‹ 4 “ ‹ ~ u 0 “ ~ u 2 1 ’‹ L ~ u 1 H 0 @ z ~ u 0 “ ‰ 1 W œœœ ‹ ~ u 0 “ ~ u 3 W ‹ 4 “ ‹ ~ u 0 “ ~ u 1 ~ u 2 1 6 W ‹ 5 “ ‹ ~ u 0 “ ~ u 3 1 ’‹ @ 2 z ~ u 0 W œ ‹ ~ u 0 ““ d ‰ W œœ ‹ ~ u 0 “ ~ u 2 1 2 W œœœ ‹ ~ u 0 “ ~ u 2 1 ’ : B.4NormalVelocityCalculationsfor ˝ "t Recallequation( 3.91 )givenby ‰ H 0 @ z L b; 0 ~ u 1 H 0 W œœœ ‹ U b “ ~ u 1 U œ b 1 H 0 U œ b z@ n 1 V ˝ ‹ s “ S z 0 ^ U b ‹ w “ dw;U œ b ’ L 2 ‹ R “ : (B.26) 164 Tocalculateeachoftheinnerproductswenotethattheinnerproductof L b; 0 ~ u 1 ,givenin( 3.76 ),with U œœ b yields ‹ L b; 0 ~ u 1 ;U œœ b “ L 2 ‹ R “ ‹ ~ 1 ' 1 d z 2 U œ b ;U œœ b “ L 2 ‹ R “ 1 S R ' 1 U œœ b dz d S R z 4 ‹‹ U œ b “ 2 “ œ dz (B.27) 1 S R ' 1 L b; 0 ‹ z 2 U œ b “ dz d S R z 4 ‹‹ U œ b “ 2 “ œ dz (B.28) 1 S R z 2 U œ b dz d S R z 4 ‹‹ U œ b “ 2 “ œ dz (B.29) 1 2 S R U b dz d 4 S R ‹ U œ b “ 2 dz (B.30) 1 2 m d 4 ˙ b (B.31) (B.32) whereweuseidentity( B.43 )toget( B.28 ),( B.29 )followsfromthefactthat L b; 0 isself-adjointand( B.30 ) followsfromintegratingbypartseachoftheintegrals.Thelastequalitywerecall m b and ˙ b are in( 1.50 )and( 3.94 ),respectively. Thetermintheinnerproductinequation( B.26 )canbewrittenas ‹ H 0 @ z L b; 0 ~ u 1 ;U œ b “ H 0 ‹ L b; 0 ~ u 1 ;U œœ b “ ; (B.33) andthesecondtermintheinnerproductinequation( B.26 )canbewrittenas ‹ H 0 W œœœ ‹ U b “ ~ u 1 U œ b ;U œ b “ H 0 ‹ ~ u 1 ;W œœœ ‹ U b “‹ U œ b “ 2 “ H 0 ‹ ~ u 1 ;L b; 0 U œœ b “ H 0 ‹ L b; 0 ~ u 1 ;U œœ b “ (B.34) Summingthesetwoinnerproducttogetheryields ‹ H 0 @ z L b; 0 ~ u 1 H 0 W œœœ ‹ U b “ ~ u 1 U œ b ;U œ b “ 2 H 0 ‹ L b; 0 ~ u 1 ;U œœ b “ H 0 1 m b H 0 d 2 ˙ b ; (B.35) wherethelastequalityweusedequation( B.31 ).Calculatingtheintegralsinthenexttwoterminequa- tion( B.26 )yields ‹ 1 H 0 U œ b ;U œ b “ 1 H 0 S R ‹ ^ U œ b “ 2 dz 1 H 0 ˙ b ; (B.36) ‹ z@ n 1 ;U œ b “ @ n 1 S R z ^ U œ b dz @ n 1 m b ; (B.37) whereforthelastequalityweintegratedbyparts.Thelastterminequation( B.26 )involvethenormal 165 velocity V ˝ ‹ s “ ‰ V ˝ ‹ s “ S z 0 ^ U b ‹ w “ dw;U œ b ’ V ˝ ‹ s “ S R S z 0 ^ U b ‹ w “ dwU œ b dz V ˝ ‹ s “ S R ^ U 2 b dz V ˝ ‹ s “ B 1 (B.38) Settingequation( B.26 )equaltozero,summarizingthecalculationofeachinnerproductandsolvingfor V ˝ ‹ s “ yields V ˝ ‹ s “ ‹ H 0 1 @ n 1 “ m b 1 2 H 0 ‹ 1 2 “ ˙ b B 1 : (B.39) B.5UsefulIdentities Recallthat U b solves W œ ‹ U b “ @ 2 z U b : (B.40) Takingthe z derivativeof( B.40 )yields L b; 0 U œ b 0 ; (B.41) andtakingthe z derivativeagainyields L b; 0 U œœ b W œœœ ‹ U b “‹ U œ b “ 2 : (B.42) Inaddition,directcalculationyields L b; 0 ‹ z 2 U œ b “ @ 2 z U b : (B.43) 166 AppendixC GeometricEvolutionCo-dimension2 C.1Appendix:DetailedexpansionoftheFCHequationusing innervariables Westartbyexpandingeachtermontheright-handsideof( 4.3 )toobtain " 2 z " Ñ © z " 2 ‹ @ 2 s ‹ z Ñ “ Ñ © z “ O ‹ " 3 “ ; (C.1) " 2 ~ u z ~ u 0 " ‹ z ~ u 1 Ñ © z ~ u 0 “ " 2 ‹ z ~ u 2 Ñ © z ~ u 1 @ 2 s ~ u 0 ‹ z Ñ “ Ñ © ~ u 0 “ O ‹ " 3 “ ; (C.2) W œœ ‹ ~ u “ W œœ ‹ ~ u 0 “ "W œœœ ‹ ~ u 0 “ ~ u 1 " 2 W œœœ ‹ ~ u 0 “ ~ u 2 1 2 W ‹ 4 “ ‹ ~ u 0 “ ~ u 2 1 (C.3) " 3 W œœœ ‹ ~ u 0 “ u 3 W ‹ 4 “ ‹ ~ u 0 “ ~ u 1 ~ u 2 1 6 W ‹ 5 “ ‹ ~ u 0 “ ~ u 3 1 O ‹ " 4 “ (C.4) W œ ‹ ~ u “ W œ ‹ ~ u 0 “ "W œœ ‹ ~ u 0 “ ~ u 1 " 2 W œœ ‹ ~ u 0 “ ~ u 2 1 2 W œœœ ‹ ~ u 0 “ ~ u 2 1 (C.5) " 3 W œœ ‹ ~ u 0 “ ~ u 3 W œœœ ‹ ~ u 0 “ ~ u 1 ~ u 2 1 6 W ‹ 4 “ ‹ ~ u 0 “ ~ u 3 1 O ‹ " 4 “ : (C.6) Next,collectingthetermsbyordersof " yields ‹ " 2 W œœ ‹ u “ 1 “ ‹ z W œœ ‹ ~ u 0 ““ (C.7) " ‹ Ñ © z W œœœ ‹ ~ u 0 “ ~ u 1 1 “ (C.8) " 2 ‰ @ 2 s ‹ z Ñ “ Ñ © z W œœœ ‹ ~ u 0 “ ~ u 2 1 2 W ‹ 4 “ ‹ ~ u 0 “ ~ u 2 1 ’ (C.9) O ‹ " 3 “ ; (C.10) ‹ " 2 u W œ ‹ u ““ ‹ z ~ u 0 W œ ‹ ~ u 0 ““ (C.11) 167 " ‹ z ~ u 1 Ñ © z ~ u 0 W œœ ‹ ~ u 0 “ ~ u 1 “ (C.12) " 2 ‰ z ~ u 2 Ñ © z ~ u 1 @ 2 s ~ u 0 ‹ z Ñ “ Ñ © ~ u 0 W œœ ‹ ~ u 0 “ ~ u 2 1 2 W œœœ ‹ ~ u 0 “ ~ u 2 1 ’ (C.13) O ‹ " 3 “ : (C.14) C.2Appendix:Calculationofthesolvabilitycondition, ˝ "t Recallthattheoperators L p;m givenin( 2.81 )areself-adjointintheR-weightedinnerproduct,introduced in( 2.74 ).WecalculateeachtermoftheR-weightedinnerproductof ‹ Q 1 ;U p “ L R ,where Q 1 1 Ñ © z p; 1 d Ñ © z L 1 p ‹ W œ ‹ U p ““ 1 W œœœ ‹ U p “ p; 2 Ñ © z U p (C.15) d W œœœ ‹ U p “ L 2 p ‹ W œ ‹ U p ““ Ñ © z U p 1 Ñ © z U p : CalculationoftheR-innerproductofthetermin( C.15 )yields ‹ 1 Ñ © z p; 1 ;@ z 1 U p “ L R S 2 ˇ 0 S ª 0 1 ‹ 1 cos 2 sin “ œ p; 1 ‹ R “ U œ p ‹ R “ cos RdR (C.16) 1 1 S 2 ˇ 0 cos 2 S ª 0 œ p; 1 U œ p RdR ˇ 1 1 S ª 0 œ p; 1 U œ p RdR; usingintegrationbypartsandidentity( C.39 ),equation( C.16 )reducesto ˇ 1 1 S ª 0 œ p; 1 U œ p RdR ˇ 1 1 S p; 1 ‹ U œ p R “ œ dR ˇ 1 1 S p; 1 L p ‰ 1 2 RU œ p ’ RdR (C.17) ˇ 1 1 2 S U œ p R 2 dR ˇ 1 1 S ^ URdR ˇ 1 1 S 1 ; where S 1 isthetotalmass,in( 4.70 ).Similarly, ‹ 1 Ñ © z p; 1 ;@ z 2 U “ L R S 2 ˇ 0 S ª 0 › 1 ‹ 1 cos 2 sin “ œ p; 1 ‹ R “ ” U œ ‹ R “ sin RdR ˇ 1 2 S 1 : (C.18) Nextwecalculatetheinnerproductofthe2 nd termof Q 1 ,( C.15 ),with @ z i U p .Recallthat W œ ‹ U p “ z U p and,usingidentity( C.42 ),wehave L 1 p ‹ W œ ‹ U p ““ L 1 p ‹ z U p “ 1 2 RU œ p .Therefore,thesecondtermtakes theform d Ñ © z ‹ 1 2 RU œ “ andtheinnerproductreducesto ‹ d Ñ © z ‹ 1 2 RU œ “ ;@ z 1 U “ L R S 2 ˇ 0 S ª 0 d 2 ‹ 1 cos 2 sin “‹ U œ RU œœ “ U œ cos RdR (C.19) 168 d 1 2 S 2 ˇ 0 S ª 0 ‹ U œ RU œœ “ U œ cos 2 RdR ˇ d 1 2 S ª 0 ‹ U œ RU œœ “ U œ RdR (C.20) ˇ d 1 2 S ª 0 ‹ U œ “ 2 RdR ˇ d 1 2 S ª 0 1 2 ‹‹ U œ “ 2 “ œ R 2 dR (C.21) ˇ d 1 2 S 4 ˇ d 1 2 S ª 0 ‹ U œ “ 2 RdR ˇ d 1 2 S 4 ˇ d 1 2 S 4 0(C.22) where S 4 . Nextwecalculatetheinnerproductofthe3 nd termof Q 1 ,( C.15 ),with @ z i U p . ‹ 1 W œœœ ‹ U p “ p; 2 Ñ © z U p ;@ z 1 U p “ L R ˇ 1 1 S ª 0 W œœœ ‹ U p “‹ U œ p “ 2 p; 2 RdR ˇ 1 1 S ª 0 L 2 ‹ 1 2 RU œ p “ p; 2 RdR (C.23) ˇ 1 1 S ª 0 1 2 U œ p R 2 dR ˇ 1 1 S ª 0 ^ U p RdR ˇ 1 1 S 1 : (C.24) The4 th term- ‹ d W œœœ ‹ U p “ L 2 p ‹ W œ ‹ U p ““ Ñ © z U p ;@ z 1 U p “ L R ˇ d 1 S ª 0 W œœœ ‹ U p “‹ U œ p “ 2 L 2 p ‹ W œ ‹ U p ““ RdR (C.25) ˇ d 1 S ª 0 L 2 p ‹ 1 2 RU œ p “ L 2 p ‹ W œ ‹ U p ““ RdR (C.26) ˇ d 1 S ª 0 1 2 U œ p W œ ‹ U p “ R 2 dR (C.27) ˇ d 1 S ª 0 1 2 U œ p ‹ U œœ p 1 R U œ p “ R 2 dR (C.28) ˇ d 1 S ª 0 1 2 U œ p U œœ p R 2 dR ˇ d 1 S ª 0 1 2 ‹ U œ p “ 2 RdR (C.29) ˇ d 1 S ª 0 1 2 ‹ 1 2 ‹ U œ p “ 2 “ œ R 2 dR ˇ d 1 2 S 2 (C.30) ˇ d 1 S ª 0 1 2 ‹ U œ p “ 2 RdR ˇ d 1 2 S 2 (C.31) ˇ d 1 2 S 4 ˇ d 1 2 S 4 0(C.32) Usingthefactthat W œ ‹ U p “ U œœ p 1 R U œ p .Andtheinnerproductofthelasttermyields ‹ 1 Ñ © z U p ;@ z 1 U p “ L R ˇ 1 1 S ª 0 ‹ U œ p “ 2 RdR ˇ 1 1 S 4 (C.33) wecansummarizeit ‹ Q 1 ;@ z i U p “ L R 2 ˇ ~ B 1 i S 1 1 ˇ i S 4 : (C.34) 169 C.3Appendix:UsefulIdentities Thefollowingbasictrigonometricidentitiesmaycomeuseful cos ‹ ˇ “ cos ‹ “ ; (C.35) sin ‹ ˇ “ sin ‹ “ : (C.36) Thefollowingareusefuloperatoridentitiesfor L p;m .Recallthatthespaces Z m areorthogonal.We calculate L p ‹ 1 2 RU œ p “ L p ‹ 1 2 RU œ p “ 1 2 L p; 0 ‹ RU œ p “ 1 2 „ 2 U œœ p RU œœœ p U œ p R U œœ p RW œœ ‹ U p “ U œ p ‚ (C.37) 1 2 ™ Œ Œ Œ Œ Œ Œ fl 2 U œœ p 2 U œ p R RL p; 1 U œ p 0 ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ RU œœœ p U œœ p U œ p R RW œœ ‹ U p “ U œ p fi Š Š Š Š Š Š Ł (C.38) Thisyieldstheidentity L p ‹ 1 2 RU œ p “ 1 2 L p; 0 ‹ RU œ p “ z U ‹ U œœ p U œ p R “ (C.39) Toobtainthenextidentity,wetialequation( 2.75 )twicew.r.t R U ‹ 4 “ p 1 R U œœœ p 2 U œœ p R 2 2 U œ p R 3 W œœ ‹ U p “ U œœ p W œœœ ‹ U p “‹ U p “ 2 (C.40) Nextwecalculate L p ‹ z U p “ L p ‹ U œœ p U œ p R “ L p; 0 ‹ U œœ p U œ p R “ U ‹ 4 “ p 1 R U œœœ p 2 U œœ p R 2 2 U œ p R 3 W œœ ‹ U p “ U œœ p 1 R „ U œœœ p U œœ p R U œ p R 2 W œœ ‹ U p “ U œ p ‚ (C.41) usingidentity( C.40 )weseethatboxedtermssumupto W œœœ ‹ U p “‹ U p “ 2 ,andthesecondboxedterms sumupto 1 R L p; 1 U œ p 0.Thisyieldsthesecondidentity L p ‹ z U p “ L p; 0 ‹ U œœ p U œ p R “ W œœœ ‹ U p “S © U p S 2 W œœœ ‹ U p “‹ U œ p “ 2 : (C.42) 170 AppendixD PearlingCo-dimension1 D.1Calculationsoftheexpansionof L b Considerthe2 nd variationof F L b 2 F u 2 › " 2 W œœ ‹ u “ 1 ”› " 2 W œœ ‹ u “ ” › " 2 u W œ ‹ u “ ” W œœœ ‹ u “ d W œœ ‹ u “ : (D.1) Usingtheexpansionof u b ,givenin( 5.5 ),andconsideringtheTaylorexpansionof W ‹ u b “ anditsderiva- tives, L b takestheform L b " 2 W œœ ‹ U b “ "W œœœ ‹ U b “ u 1 " 2 ‰ W œœœ ‹ U b “ u 2 1 2 W ‹ 4 “ ‹ U b “ u 2 1 ’ 1 (D.2) " 2 W œœ ‹ U b “ "W œœœ ‹ U b “ u 1 " 2 ‰ W œœœ ‹ U b “ u 2 1 2 W ‹ 4 “ ‹ U b “ u 2 1 ’ " 2 U b " 3 u 1 W œ ‹ U b “ "W œœ ‹ U b “ u 1 " 2 ‰ W œœ ‹ U b “ u 2 1 2 W œœœ ‹ U b “ u 2 1 ’ W œœœ ‹ U b “ "W ‹ 4 “ ‹ U b “ u 1 d W œœ ‹ U b “ "W œœœ ‹ U b “ u 1 " 2 ‰ W œœœ ‹ U b “ u 2 1 2 W ‹ 4 “ ‹ U b “ u 2 1 ’ higherorderterms : Notethat " 2 U b W œ ‹ U b “ "HU œ b " 2 ˘ ˘ ˘ ˘: 0 G U b since U b isthehomoclinicsolutionanditisindependentof s . Usingtheofthefulloperator L b ,in( 2.45 ),equation( D.2 )reducesto L b L b "W œœœ ‹ U b “ u 1 " 2 ‰ W œœœ ‹ U b “ u 2 1 2 W ‹ 4 “ ‹ U b “ u 2 1 ’ 1 (D.3) L b "W œœœ ‹ U b “ u 1 " 2 ‰ W œœœ ‹ U b “ u 2 1 2 W ‹ 4 “ ‹ U b “ u 2 1 ’ 171 " HU œ b " 2 u 1 W œœ ‹ U b “ u 1 " ‰ W œœ ‹ U b “ u 2 1 2 W œœœ ‹ U b “ u 2 1 ’ W œœœ ‹ U b “ "W ‹ 4 “ ‹ U b “ u 1 d W œœ ‹ U b “ "W œœœ ‹ U b “ u 1 " 2 ‰ W œœœ ‹ U b “ u 2 1 2 W ‹ 4 “ ‹ U b “ u 2 1 ’ higherorderterms : Collecting L b inordersof " yields L b L 2 b (D.4) " L b X ‹ W œœœ ‹ U b “ u 1 “ ‹ W œœœ ‹ U b “ u 1 1 “ L b ‹ L b u 1 HU œ b “ W œœœ ‹ U b “ d W œœ ‹ U b " 2 L b X ‰ W œœœ ‹ U b “ u 1 1 2 W ‹ 4 “ ‹ U b “ u 2 1 ’ ‰ W œœœ ‹ U b “ u 1 1 2 W ‹ 4 “ ‹ U b “ u 2 1 ’ L b ‹ W œœœ ‹ U b “ u 1 1 “ W œœœ ‹ U b “ u 1 ‰ L b u 2 1 2 W œœœ ‹ U b “ u 2 1 ’ W œœœ ‹ U b “ ‹ L b u 1 HU œ b “ W ‹ 4 “ ‹ U b “ u 1 d W œœœ ‹ U b “ u 1 O ‹ " 3 “ : D.2Calculating M ‹ L 2 b 0 j ; 0 k “ L 2 ‹ “ ‹ L b 0 j ; L b 0 k “ L 2 ‹ (D.5) S S l ~ " l ~ " ‹ L b; 0 "H@ z " 2 G “ 0 j ‹ L b; 0 "H@ z " 2 G “ 0 k Jdzds (D.6) S S l ~ " l ~ " L b; 0 0 j L b; 0 0 k Jdzds (D.7) " „ S S l ~ " l ~ " L b; 0 0 j H@ z ‹ 0 k “ Jdzds S S l ~ " l ~ " H@ z ‹ 0 j “ L b; 0 ‹ 0 k “ Jdzds ‚ " 2 ™ fl S S l ~ " l ~ " L b; 0 0 j G ‹ 0 k “ Jdzds S S l ~ " l ~ " G ‹ 0 j “ L b; 0 ‹ 0 k “ Jdzds S S l ~ " l ~ " H@ z ‹ 0 j “ H@ z ‹ 0 k “ Jdzds fi Ł O ‹ " 3 “ "P k P j S k j J 0 ds S l ~ " l ~ " ‹ 0 0 “ 2 dz " º " ‹ P j P k “ S S l ~ " l ~ " j k H œ 0 0 Jdzds (D.8) " 2 º "" 2 ‹ P j k P k j “ S S l ~ " l ~ " j k 2 0 Jdzds " 2 S S l ~ " l ~ " H 2 j k ‹ œ 0 “ 2 Jdzds O ‹ " 3 “ 172 since 0 0 0 ~ J 1 ~ 2 ,wehave œ 0 ‹ 0 0 “ œ ~ J 1 ~ 2 0 0 ‹ ~ J 1 ~ 2 “ œ ‹ 0 0 “ œ ~ J 1 ~ 2 1 2 0 0 ‹ ~ J 3 ~ 2 “ ~ J œ (D.9) andusingtheidentity ~ J œ "H ~ J weget œ 0 ‹ 0 0 “ œ ~ J 1 ~ 2 1 2 0 0 ‹ ~ J 3 ~ 2 “ "H ~ J ‹ 0 0 “ œ ~ J 1 ~ 2 1 2 "H 0 0 ‹ ~ J 1 ~ 2 “ ‹ 0 0 “ œ ~ J 1 ~ 2 1 2 "H 0 (D.10) and ‹ œ 0 “ 2 ‹‹ 0 0 “ œ “ 2 ~ J 1 "H ‹ 0 0 “ œ 0 0 ~ J 1 1 4 " 2 H 2 ‹ 0 0 “ 2 ‹ ~ J 1 “ (D.11) ‹ L 2 b 0 j ; 0 k “ L 2 ‹ “ (D.12) "P k P j S k j J 0 ds S l ~ " l ~ " ‹ 0 0 “ 2 dz (D.13) " º "P j S H k j j 0 ds ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘: 0 S l ~ " l ~ " ‹ 0 0 “ œ 0 0 dz 1 2 " 2 º "P j S H 2 k j J 0 ds S l ~ " l ~ " ‹ 0 0 “ 2 dz " º "P k S H k j j 0 ds ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘: 0 S l ~ " l ~ " ‹ 0 0 “ œ 0 0 dz 1 2 " 2 º "P k S H 2 k j J 0 ds S l ~ " l ~ " ‹ 0 0 “ 2 dz " 2 º " ‹ P j " 2 k P k " 2 j “ S j k J 0 ds S l ~ " l ~ " ‹ 0 0 “ 2 dz " 2 S H 2 j k J 0 ds S l ~ " l ~ " ‹‹ 0 0 “ œ “ 2 dz " 3 S H 3 j k j 0 ds S l ~ " l ~ " ‹ 0 0 “ œ 0 0 dz 1 4 " 4 S H 4 j k J 0 ds S l ~ " l ~ " ‹ 0 0 “ 2 dz O ‹ " 3 “ ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ "P 2 k " 2 R ‹ H k “ 2 J 0 ds R l ~ " l ~ " ‹‹ 0 0 “ œ “ 2 dz O ‹ " 2 º " “ if k j; " 2 R H 2 k j J 0 ds R l ~ " l ~ " ‹‹ 0 0 “ œ “ 2 dz O ‹ " 2 º " “ if k x j Calculating M 1 yields ‹ L 1 0 j ; 0 k “ L 2 ‹ “ S S l ~ " l ~ " ‹ W œœœ ‹ U b “ u 1 “ 0 j L b 0 k Jdzds (D.14) S S l ~ " l ~ " ‹ W œœœ ‹ U b “ u 1 1 “ 0 k L b ‹ 0 j “ Jdzds S S l ~ " l ~ " ‹ HU œ b L b u 1 “ W œœœ ‹ U b “ 0 j 0 k Jdzds d S S l ~ " l ~ " W œœ ‹ U b “ 0 j 0 k Jdzds º "P k S S l ~ " l ~ " W œœœ ‹ U b “ u 1 j k 2 0 Jdzds (D.15) º "P j S S l ~ " l ~ " ‹ W œœœ ‹ U b “ u 1 1 “ k j 2 0 Jdzds 173 S S l ~ " l ~ " HU œ b W œœœ ‹ U b “ j k 2 0 Jdzds S S l ~ " l ~ " L b ‹ u 1 “ W œœœ ‹ U b “ j k 2 0 Jdzds d S S l ~ " l ~ " W œœ ‹ U b “ j k 2 0 Jdzds using 0 ~ J 1 ~ 2 0 0 weget ‹ L 1 0 j ; 0 k “ L 2 ‹ “ º "P k S k j J 0 ds S l ~ " l ~ " W œœœ ‹ U b “ u 1 ‹ 0 0 “ 2 dz (D.16) º "P j S k j J 0 ds S l ~ " l ~ " ‹ W œœœ ‹ U b “ u 1 1 “‹ 0 0 “ 2 dz S H 0 k j J 0 ds ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘: 0 S l ~ " l ~ " U œ b W œœœ ‹ U b “‹ 0 0 “ 2 dz (D.17) " S H 1 k j J 0 ds S l ~ " l ~ " U œ b W œœœ ‹ U b “‹ 0 0 “ 2 zdz S j k j 0 ds S l ~ " l ~ " L b ‹ u 1 “ W œœœ ‹ U b “‹ 0 0 “ 2 dz (D.18) d S j k J 0 ds S l ~ " l ~ " W œœ ‹ U b “‹ 0 0 “ 2 dz O ‹ " 3 “ ¢ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¤ R l … " l … " W œœœ ‹ U b “‹ 0 0 “ 2 L b; 0 u 1 dz d R l … " l … " W œœ ‹ U b “‹ 0 0 “ 2 dz º "P k ‹ 1 2 “ " R H 1 2 k ds R l … " l … " W œœœ ‹ U b “ U œ b ‹ 0 0 “ 2 zdz O ‹ " 2 “ if k j; " R H 1 k j ds R l … " l … " W œœœ ‹ U b “ U œ b ‹ 0 0 “ 2 zdz O ‹ " 3 “ if k x j (D.19) D.3Simplifyingtheexpressionfor M k;k Recallthat, M k;k isgivenby M 0 k;k P 2 k S l … " l … " W œœœ ‹ U b “ L b; 0 u 1 d W œœ ‹ U b 0 0 “ 2 dz: (D.20) Usingthefollowingidentities L b; 0 u 1 1 1 d ‰ z 2 U œ ’ ; (D.21) W œœ ‹ U “ 0 œœ 0 L b; 0 0 œœ 0 0 0 ; (D.22) 174 weget d S W œœ ‹ U “ 2 0 S ‹ 1 ' 1 d z 2 U œ “ W œœœ ‹ U “ 2 0 d S W œœ ‹ U “ 2 0 (D.23) 1 S b ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ S 1 ' 1 W œœœ ‹ U “ 2 0 S d z 2 U œ W œœœ ‹ U “ 2 0 1 S b d S W œœ ‹ U “ 2 0 S d z 2 ‹ W œœ ‹ U ““ œ 2 0 : (D.24) Integratingbypartswehave d S W œœ ‹ U “ 2 0 S ‹ 1 ' 1 d z 2 U œ “ W œœœ ‹ U “ 2 0 1 S b (D.25) d S ‰ W œœ ‹ U “ 2 0 1 2 W œœ ‹ U “ 2 0 W œœ ‹ U “ z œ ’ 1 S b d S ‰ 1 2 W œœ ‹ U “ 2 0 W œœ ‹ U “ z 0 œ 0 ’ (D.26) 1 S b d S ‰ 1 2 0 z œ 0 “’ W œœ ‹ U “ 0 (D.27) Usingidentity( D.22 )yields d S W œœ ‹ U “ 2 0 S ‹ 1 ' 1 d z 2 U œ “ W œœœ ‹ U “ 2 0 1 S b d S ‹ 1 2 0 z œ 0 “‹ œœ 0 0 0 “ (D.28) 1 S b d S ‰ 1 2 0 œœ 0 1 2 0 2 0 z œ 0 œœ 0 0 z 0 œ 0 ’ (D.29) 1 S b d S 1 2 0 œœ 0 1 2 0 d S 2 0 (D.30) d S z œ 0 œœ 0 0 d S z 0 œ 0 Usingintegrationbypartswecanshowthat S z œ 0 œœ 0 1 2 SS œ 0 SS 2 2 ; S z 0 œ 0 1 2 SS 0 SS 2 2 (D.31) therefore, d S W œœ ‹ U “ 2 0 S ‹ 1 ' 1 d z 2 U œ “ W œœœ ‹ U “ 2 0 1 S b d S 1 2 0 œœ 0 1 2 0 d S 2 0 (D.32) d S z œ 0 œœ 0 0 d S z 0 œ 0 175 1 S b 1 2 d SS œ 0 SS 2 2 1 2 0 d SS 0 SS 2 2 1 2 d SS œ 0 SS 2 2 1 2 0 d SS 0 SS 2 2 (D.33) 1 S b 0 SS 0 SS 2 2 d : (D.34) Hence,thediagonalterms M k;k taketheform M k;k P 2 k 1 S b d 0 SS 0 SS 2 2 : (D.35) D.4Usefulidentitiesandinequalities TheoremD.4.1 older'sInequality) . Suppose f > L p ‹ R d “ ;g > L q ‹ R d “ and 1 p 1 q 1 with 1 B p;q;r B ª .Then SS fg SS 1 B SS f SS p SS g SS q : (D.36) TheoremD.4.2 (Generalizedolder'sInequality) . Suppose f > L p ‹ R d “ ;g > L q ‹ R d “ and 1 p 1 q 1 r with 1 B p;q;r B ª .Then SS fg SS r B SS f SS p SS g SS q : (D.37) 1. Anexpressionforthederivativeof 0 œ 0 ‹ 0 0 “ œ ~ J 1 ~ 2 "H 0 (D.38) 2. L b; 0 z 0 z 0 0 2 œ 0 (D.39) 3. L b; 0 œ 0 0 œ 0 œœœ 0 W œœœ ‹ U “ U œ 0 (D.40) 4. L b; 0 ‹ z œ 0 “ zL b; 0 œ 0 2 œœ 0 z ‹ 0 @ 2 z “ œ 0 W œœœ ‹ U “ U œ 0 2 œœ 0 : (D.41) 5. L b; 0 ‹ 0 D s; 2 k “ 0 0 D s; 2 k 2 œ 0 ‹ D s; 2 “ z k 0 ‹ D s; 2 “ zz k (D.42) 176 AppendixE PearlingCo-dimension2 E.1Appendix:Self-adjointoperators Considerthe L 2 ‹ “ innerproductin( 2.73 )andthetwooperators ~ L p @ 2 z W œœ ‹ u “ " 2 s and L p @ 2 z z W œœ ‹ u “ " 2 G .Theco-dimeaniontwoLaplacianoperatoris not self-adjointinthisinnerproduct, ‹ s f;g “ L 2 ‹ “ S S l … " l … " gJ 1 0 © s ‹ g 1 J 0 © s “ fJ 0 ~ Jdzds S S l … " l … " g 1 J 0 © s ‹ g ~ J “ © s fdzds (E.1) S S l … " l … " © s ‹ g 1 J 0 © s ‹ g ~ J ““ fdzds x ‹ f; s g “ L 2 ‹ “ (E.2) althoughitisself-adjointintheproduct.However, G isself-adjointinthe L 2 ‹ “ innerproduct. ‹ G f;g “ L 2 ‹ “ S S l … " l … " gJ 1 © s ‹ g 1 J © s “ fJdzds S S l … " l … " g 1 J © s ‹ g “ © s fdzds (E.3) S S l … " l … " © s ‹ g 1 J © s ‹ g ““ fdzds ‹ f; G g “ L 2 ‹ “ : (E.4) Asfortherestofthetermsintheoperator, ‹ @ 2 z f;g “ L 2 ‹ “ S S l … " l … " f œœ gJdzds S S l … " l … " f ‹ gJ “ œœ dzds S S l … " l … " fg œœ J 2 fg œ J œ fgJ œœ dzds (E.5) ‹ f;@ 2 z g “ L 2 ‹ “ 2 " S S l … " l … " fg œ dzds " S S l … " l … " fg œ Jdzds " 2 S S l … " l … " fg 2 Jdzds; (E.6) 177 " ‹ z f;g “ L 2 ‹ “ " S S l … " l … " œ gJdzds " S S l … " l … " f ‹ gJ “ œ dzds S S l … " l … " f œ J "fgJ œ fg œ dzds (E.7) " ‹ f; z g “ L 2 ‹ “ " S S l … " l … " fg œ Jdzds " 2 S S l … " l … " fg 2 Jdzds; (E.8) weseethateachofthetermsseparatelyisnotself-adjointinthe L 2 ‹ “ -innerproduct,butthesumofthem is ‹ @ 2 z f z f;g “ L 2 ‹ “ ‹ f;@ 2 z g z g “ L 2 ‹ “ : Therefore,ourfulloperator L p isself-adjointinthe L 2 ‹ “ -innerproductwherethe ~ L p operatorisnot. E.2Calculationsoftheexpansionof L Weconsiderthe2 nd variationof F L p 2 F u 2 › " 2 W œœ ‹ u “ 1 ”› " 2 W œœ ‹ u “ ” › " 2 u W œ ‹ u “ ” W œœœ ‹ u “ d W œœ ‹ u “ : (E.9) UsingtheexpansionoftheLaplacianinlocalcoordinates,givenin( 2.65 ),writing u p using( 6.5 )andTaylor expand W ‹ u p “ anditsderivatives. L p z W œœ ‹ U p “ "D z " 2 @ 2 G " ‹ W œœœ ‹ U p “ u 1 1 “ " 2 ‰ W œœœ ‹ U p “ u 2 1 2 W ‹ 4 “ ‹ U p “ u 2 1 ’ X (E.10) z W œœ ‹ U p “ "D z " 2 @ 2 G " ‹ Ñ © z W œœœ ‹ U p “ u 1 “ " 2 ‰ W œœœ ‹ U p “ u 2 1 2 W ‹ 4 “ ‹ U p “ u 2 1 ’ z U p W œ ‹ U p “ "D z U p " z u 1 " 2 D z u 1 " 2 @ 2 G U p "W œœ ‹ U p “ u 1 " 2 ‰ W œœ ‹ U p “ u 2 1 2 W œœœ ‹ U p “ u 2 1 ’ X W œœœ ‹ U p “ "W ‹ 4 “ ‹ U p “ u 1 d W œœ ‹ U p “ "W œœœ ‹ U p “ u 1 " 2 ‰ W œœœ ‹ U p “ u 2 1 2 W ‹ 4 “ ‹ U p “ u 2 1 ’ higherorderterms ; Recallthat L p L p "D z " 2 @ 2 G , L p isin( 2.78 ),and U p istheradialsymmetricsolutionof equation( 2.75 ).Then, L p L p " ‹ W œœœ ‹ U p “ u 1 1 “ " 2 ‰ W œœœ ‹ U p “ u 2 1 2 W ‹ 4 “ ‹ U p “ u 2 1 ’ X (E.11) L p " ‹ W œœœ ‹ U p “ u 1 “ " 2 ‰ W œœœ ‹ U p “ u 2 1 2 W ‹ 4 “ ‹ U p “ u 2 1 ’ "D z U p " L p u 1 " 2 ‰ W œœ ‹ U p “ u 2 1 2 W œœœ ‹ U p “ u 2 1 ’ X W œœœ ‹ U p “ "W ‹ 4 “ ‹ U p “ u 1 178 d W œœ ‹ U p “ "W œœœ ‹ U p “ u 1 " 2 ‰ W œœœ ‹ U p “ u 2 1 2 W ‹ 4 “ ‹ U p “ u 2 1 ’ higherorderterms ; Rewriting L p inordersof " wehave L p L 2 p " L 1 O ‹ " 2 “ ; (E.12) where L 1 ‹ W œœœ ‹ U p “ u 1 1 “ X L p L p X ‹ W œœœ ‹ U p “ u 1 “ ‹ L p u 1 D z U p “ W œœœ ‹ U p “ d W œœ ‹ U p “ : (E.13) E.3Calculating M 0 Toobtainanexplicitexpressionfor M 0 wecalculatetheinnerproductsgivenin( 6.22 ).Westartwiththe innerproductinvolving L 2 p :Recallthat L p ,givenin( 2.77 ),isself-adjointinthe L 2 ‹ “ innerproduct.Then ‹ L 2 p 0 j ; 0 k “ L 2 ‹ “ ‹ L p 0 j ; L p 0 k “ L 2 ‹ “ (E.14) ‹‹‹ L " 2 @ 2 s “ " Ñ © z " 2 ‹ z Ñ “ Ñ © z “ 0 j ; ‹‹ L " 2 @ 2 s “ " Ñ © z " 2 ‹ z Ñ “ Ñ © z “ 0 k “ L 2 ‹ “ (E.15) O ‹ " 3 “ ‹‹ L " 2 @ 2 s “ 0 j ; ‹ L " 2 @ 2 s “ 0 k “ L 2 ‹ “ (E.16) " › ‹‹ L " 2 @ 2 s “ 0 j ; Ñ © z 0 k “ L 2 ‹ “ ‹ Ñ © z 0 j ; ‹ L " 2 @ 2 s “ 0 k “ L 2 ‹ “ ” O ‹ " 2 “ "P k P j ‹ 0 j ; 0 k “ L 2 ‹ “ " º " ‹ P j P k “‹ j k 0 ; Ñ © z 0 “ L 2 ‹ “ O ‹ " 2 “ (E.17) whereforthesecondequalityweusedtheexpansionoftheLaplacian,givenin( ?? ).Since psi 0 isaradial functionwehave ‹ j k 0 ; Ñ © z 0 “ L 2 ‹ “ S k j ds S l ~ " 0 0 © z 0 Jdz (E.18) changingtopolarcoordinates,equation( E.18 )takestheform ‹ j k 0 ; Ñ © z 0 “ L 2 ‹ “ S k j ds ‹ 0 ; 0 “ ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ S 2 ˇ 0 ‹ cos ; sin “ S ª 0 0 œ 0 dz O ‹ " “ (E.19) Thatis,theterm " º " ‹ P j P k “‹ j k 0 ; Ñ © z 0 “ L 2 ‹ “ isactually O ‹ " 2 º " “ andnegligible.Usingthe orthonormalityoftheco-dimeaniontwoLaplacianeigenmodeswecanrewriteequation( E.17 )inthefollowing 179 way ‹ L 2 p 0 j ; 0 k “ L 2 ‹ “ ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ "P 2 k O ‹ " 2 º " “ if k j; O ‹ " 2 º " “ if k x j (E.20) Thenexttermin( 6.22 )involvesthe L 1 operator,givenin( 6.9 ). ‹ L 1 0 j ; 0 k “ L 2 ‹ “ ‹ A “ ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ ‹‹ W œœœ ‹ U p “ u 1 1 “ X L p 0 j ; 0 k “ L 2 ‹ “ ‹ B “ ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ ‹ L p X ‹ W œœœ ‹ U p “ u 1 “ 0 j ; 0 k “ L 2 ‹ “ (E.21) ‹ C “ ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ ‹‹ L p u 1 D z U p “ 0 j ; 0 k “ L 2 ‹ “ ‹ D “ ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ ‹ d W œœ ‹ U p “ 0 j ; 0 k “ L 2 ‹ “ (E.22) wecalculate( E.21 )termbyterm: (A) ‹‹ W œœœ ‹ U p “ u 1 1 “ X L p 0 j ; 0 k “ L 2 ‹ “ º "P j ‹‹ W œœœ ‹ U p “ u 1 1 “ 0 j ; 0 k “ L 2 ‹ “ (E.23) " ‹‹ W œœœ ‹ U p “ u 1 1 “ Ñ © z 0 j ; 0 k “ L 2 ‹ “ O ‹ " 2 “ ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ º "P j ‹‹ W œœœ ‹ U p “ u 1 1 “ 0 k ; 0 k “ L 2 ‹ “ O ‹ " “ if k j; " ‹‹ W œœœ ‹ U p “ u 1 1 “ Ñ © z 0 j ; 0 k “ L 2 ‹ “ O ‹ " 2 “ if k x j (E.24) (B) ‹ L p X ‹ W œœœ ‹ U p “ u 1 “ 0 j ; 0 k “ L 2 ‹ “ ‹‹ W œœœ ‹ U p “ u 1 “ 0 j ; L p 0 k “ L 2 ‹ “ (E.25) ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ º "P j ‹ W œœœ ‹ U p “ u 1 0 k ; 0 k “ L 2 ‹ “ O ‹ " “ if k j; " ‹ 0 j ;W œœœ ‹ U p “ u 1 Ñ © z 0 k “ L 2 ‹ “ O ‹ " 2 “ if k x j (E.26) 180 (C) ‹‹ L p u 1 D z U p “ 0 j ; 0 k “ L 2 ‹ “ ¢ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¤ ‹‹ Lu 1 ‹ Ñ © z U p ““ W œœœ ‹ U p “ 0 k ; 0 k “ L 2 ‹ “ O ‹ " “ if k j; ‹‹ Ñ © z U p “ W œœœ ‹ U p “ 0 j ; 0 k “ L 2 ‹ “ if k x j " ‹ Ñ ‹ © z ‹ u 1 U p ““ W œœœ ‹ U p “ 0 j ; 0 k “ L 2 ‹ “ O ‹ " 2 “ (E.27) Recallthat U p ; 0 0 and W œœœ ‹ U p “ areallradialfunctions,andusing( A.2 ),thecalculationoftheinner- product ‹‹ Ñ © z U p “ W œœœ ‹ U p “ 0 k ; 0 k “ L 2 ‹ “ (boxed)becomes ‹‹ Ñ © z U p “ W œœœ ‹ U p “ 0 j ; 0 k “ L 2 ‹ “ S S ` ~ " ` ~ " ‹ Ñ © z U p “ W œœœ ‹ U p “‹ 0 0 “ 2 j k dzds (E.28) S S 2 ˇ 0 S ª 0 ‹ Ñ ‹ cos ; sin “ @ R U p “ W œœœ ‹ U p “‹ 0 0 “ 2 j k RdRds (E.29) S Ñ j k ds ‹ 0 ; 0 “ ³¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹·¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹µ S 2 ˇ 0 ‹ cos ; sin “ S ª 0 ‹ @ R U p “ W œœœ ‹ U p “‹ 0 0 “ 2 RdR (E.30) 0 : (E.31) Itfollowsfrom( E.28 )thatequation( E.27 )reducesto ‹‹ L p u 1 D z U p “ 0 j ; 0 k “ L 2 ‹ “ ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ ‹‹ Lu 1 “ W œœœ ‹ U p “ 0 k ; 0 k “ L 2 ‹ “ O ‹ " “ if k j; " ‹ Ñ ‹ © z ‹ u 1 U p ““ W œœœ ‹ U p “ 0 j ; 0 k “ L 2 ‹ “ O ‹ " 2 “ if k x j (E.32) summarizingthecalculationofeachoftheterms,andreturningto( E.21 )weseethat ‹ L 1 0 j ; 0 k “ L 2 ‹ “ (E.33) ¢ ¨ ¨ ¨ ¨ ¨ ¦ ¨ ¨ ¨ ¨ ¨ ¤ ‹‹‹ Lu 1 “ W œœœ ‹ U p “ d W œœ ‹ U p ““ 0 j ; 0 k “ L 2 ‹ “ O ‹ º " “ if k j; " ‹‹‹ 2 W œœœ ‹ U p “ u 1 1 “ Ñ © z 0 Ñ ‹ © z u 1 U p “ W œœœ ‹ U p “ 0 “ j ; 0 k “ L 2 ‹ “ O ‹ " 2 “ if k x j E.4Simplifyingtheexpressionfor M k;k Recallthat, M k;k isgivenby M 0 k;k P 2 k S l … " 0 W œœœ ‹ U b “ Lu 1 d W œœ ‹ U b 0 0 “ 2 dz; (E.34) 181 where u 1 isin( 6.6 )andinpolarcoordinates,usingtheidentity L 1 ‹ W œ ‹ U p ““ L 1 ‹ z U p “ 1 2 RU œ p takestheform Lu 1 1 ' 1 d L 1 ‹ W œ ‹ U p ““ 1 ' 1 d 1 2 RU œ p : (E.35) Forfuturecalculationswehavethefollowingidentity W œœ ‹ U p “ 0 0 ‹ 0 0 “ œœ L 0 0 0 ‹ 0 0 “ œœ p; 0 0 0 : (E.36) Sinceallthefunctionin( E.34 )areradialfunctions,wechangetopolarcoordinates M 0 k;k P 2 k 2 ˇ S ª 0 W œœœ ‹ U b “ Lu 1 d W œœ ‹ U b 0 0 “ 2 RdR; (E.37) andplugging( E.35 )into( E.37 )yields M 0 k;k P 2 k 2 ˇ 1 S 2 ˇ d S ª 0 W œœœ ‹ U b “ 1 2 U œ p R W œœ ‹ U b 0 0 “ 2 RdR; (E.38) where S istheshapefactorin( 6.33 ).Considerthe d termin( E.37 ) S ª 0 W œœœ ‹ U b “ 1 2 U œ p R W œœ ‹ U b 0 0 “ 2 RdR S ª 0 W œœ ‹ U b œ 1 2 ‹ 0 0 “ 2 R 2 dR S ª 0 W œœ ‹ U b “‹ 0 0 “ 2 RdR (E.39) S ª 0 W œœ ‹ U b “‹ 0 0 “‹ 0 0 “ œ R 2 dR ( ( ( ( ( ( ( ( ( ( ( S ª 0 W œœ ‹ U b “‹ 0 0 “ 2 RdR (E.40) ( ( ( ( ( ( ( ( ( ( ( S ª 0 W œœ ‹ U b “‹ 0 0 “ 2 RdR S ª 0 ‹ 0 0 “ œœ ‹ 0 0 “ œ R 2 dR S ª 0 p; 0 ‹ 0 0 “‹ 0 0 “ œ R 2 dR (E.41) S ª 0 1 2 ‹‹‹ 0 0 “ œ “ 2 “ œ R 2 dR p; 0 S ª 0 1 2 ‹‹ 0 0 “ 2 “ œ R 2 dR (E.42) S ª 0 ‹‹‹ 0 0 “ œ “ 2 “ RdR p; 0 S ª 0 ‹‹ 0 0 “ 2 “ RdR: (E.43) Thesecondandthelastequalitiesfollowfromintegrationbyparts,andthethirdequalityfollowsfrom identity( E.36 ).Plugging( E.43 )into( E.37 )yields M 0 k;k P 2 k 2 ˇ 1 S d − TT ‹ 0 0 “ œ TT 2 L R p; 0 TT ‹ 0 0 “ TT 2 L R ‘ : (E.44) 182 E.5Usefulidentitiesandinequalities TheoremE.5.1 (Young'sInequality) . Suppose f > L p ‹ R d “ ;g > L q ‹ R d “ and 1 p 1 q 1 r 1 with 1 B p;q;r B ª .Then SS f ⁄ g SS r B SS f SS p SS g SS q : (E.45) TheoremE.5.2 older'sInequality) . Suppose f > L p ‹ R d “ ;g > L q ‹ R d “ and 1 p 1 q 1 with 1 B p;q;r B ª .Then SS fg SS 1 B SS f SS p SS g SS q : (E.46) E.6Appendix:Detailedcalculationsofoperatorbounds Firstweconsiderthefollowingoperator @ 2 G D z .Usingthesofeachoftheoperators,givenin( 2.67 ), and( 2.66 ),wecanwrite @ 2 G D z v @ 2 G „ 1 ~ J p © z v ‚ Ñ ~ J p › @ 2 G ‹ © z v “ ” „ @ 2 G Ñ ~ J p ‚ © z v 2 @ s „ Ñ ~ J p ‚ @ s © z v (E.47) Takingthe L 2 ‹ “ -normyields TT ~ @ 2 G D z v TT 2 L 2 ‹ “ WW Ñ ~ J p › @ 2 G ‹ © z v “ ” „ @ 2 G Ñ ~ J p ‚ © z v 2 @ s „ Ñ ~ J p ‚ @ s © z v WW L 2 ‹ “ (E.48) B WW Ñ ~ J p › @ 2 G ‹ © z v “ ” WW 2 L 2 ‹ “ WW„ @ 2 G Ñ ~ J p ‚ © z v WW 2 L 2 ‹ “ 4 WW @ s „ Ñ ~ J p ‚ @ s © z v WW 2 L 2 ‹ “ (E.49) B WW Ñ ~ J p WW L ª TT› @ 2 G ‹ © z v “ ”TT 2 L 2 ‹ “ WW„ @ 2 G Ñ ~ J p ‚WW L ª SS © z v SS 2 L 2 ‹ “ (E.50) 4 WW @ s „ Ñ ~ J p ‚WW L ª SS @ s © z v SS 2 L 2 ‹ “ B " 4 c 1 SS v SS L 2 ‹ “ c 2 SS v SS L 2 ‹ “ 4 WW @ s „ Ñ ~ J p ‚WW L ª TT @ s ‹ @ 2 s “ TT l 2 l 2 TT @ 2 s © z v TT 2 L 2 ‹ “ (E.51) B " 4 C SS v SS L 2 ‹ “ ; (E.52) wheretheinequalityisthetriangleinequality,forthesecondinequalityweuseolder,thethird inequalityfollowsfromLemma 6.3.1 ,combinedwiththeassumptionthat > W 2 ; ª . 183 REFERENCES blankspace 184 REFERENCES [Alikakosetal.,1999] Alikakos,N.,Bates,P.,andChen,X.(1999).Periodictravelingwavesandlocating oscillatingpatternsinmultidimensionaldomains. TransactionsoftheAmericanMathematicalSociety , 351(7):2777{2805. [AlikakosandKatzourakis,2011] Alikakos,N.andKatzourakis,N.(2011).Heteroclinictravellingwavesof gradientsystems. TransactionsoftheAmericanMathematicalSociety ,363(3):1365{1397. [Alikakosetal.,1994] Alikakos,N.D.,Bates,P.W.,andChen,X.(1994).Convergenceofthecahn-hilliard equationtothehele-shawmodel. ArchiveforRationalMechanicsandAnalysis ,128(2):165{205. [Alikakosetal.,2000] Alikakos,N.D.,Bates,P.W.,Chen,X.,andFusco,G.(2000).Mullins-sekerkamotion ofsmalldropletsonaboundary. JournalofGeometricAnalysis ,10(4):575{596. [AlikakosandFusco,1993] Alikakos,N.D.andFusco,G.(1993).Thespectrumofthecahn-hilliardoperator forgenericinterfaceinhigherspacedimensions. IndianaUniversityMathematicsJournal ,42(2):637{674. [Anderson,1975] Anderson,R.(1975).Photocurrentsuppressioninheterojunctionsolarcells. Applied PhysicsLetters ,27(12):691{693. [Andreussietal.,2012] Andreussi,O.,Dabo,I.,andMarzari,N.(2012).Revisedself-consistentcontinuum solvationinelectronic-structurecalculations. TheJournalofChemicalPhysics ,136(6):064102. [BatesandChen,2002] Bates,P.W.andChen,F.(2002).Spectralanalysisandmultidimensionalstability oftravelingwavesfornonlocalallen{cahnequation. JournalofMathematicalAnalysisandApplications , 273(1):45{57. [BatesandChen,2006] Bates,P.W.andChen,F.(2006).Spectralanalysisoftravelingwavesfornonlocal evolutionequations. SIAMJournalonMathematicalAnalysis ,38(1):116{126. [BatesandHan,2005a] Bates,P.W.andHan,J.(2005a).Thedirichletboundaryproblemforanonlocal cahn{hilliardequation. JournalofMathematicalAnalysisandApplications ,311(1):289{312. [BatesandHan,2005b] Bates,P.W.andHan,J.(2005b).Theneumannboundaryproblemforanonlocal cahn{hilliardequation. JournalofentialEquations ,212(2):235{277. [Bellskyetal.,2013] Bellsky,T.,Doelman,A.,Kaper,T.J.,andPromislow,K.(2013).Adiabaticstability undersemi-stronginteractions:theweaklydampedregime. arXivpreprintarXiv:1301.4466 . [BudinandSzostak,2011] Budin,I.andSzostak,J.W.(2011).Physicalunderlyingthetransition fromprimitivetomoderncellmembranes. ProceedingsoftheNationalAcademyofSciences ,108(13):5249{ 5254. [CahnandHilliard,1958] Cahn,J.W.andHilliard,J.E.(1958).Freeenergyofanonuniformsystem.i. interfacialfreeenergy. TheJournalofChemicalPhysics ,28(2):258{267. [Canham,1970] Canham,P.B.(1970).Theminimumenergyofbendingasapossibleexplanationofthe biconcaveshapeofthehumanredbloodcell. JournalofTheoreticalBiology ,26(1):61{81. [Charleuxetal.,2012] Charleux,B.,Delaittre,G.,Rieger,J.,andDAgosto,F.(2012).Polymerization- inducedself-assembly:fromsolublemacromoleculestoblockcopolymernano-objectsinonestep. Macro- molecules ,45(17):6753{6765. 185 [Chavel,1984] Chavel,I.(1984). EigenvaluesinRiemannianGeometry ,volume115.Academicpress. [Chen,1994] Chen,X.(1994).Spectrumfortheallen-chan,chan-hillard,andequationsfor genericinterfaces. CommunicationsinPartialentialEquations ,19(7-8):1371{1395. [DaiandPromislow,2013] Dai,S.andPromislow,K.(2013).Geometricevolutionofbilayersunderthe functionalizedcahn{hilliardequation.In ProceedingsoftheRoyalSocietyofLondonA:Mathematical, PhysicalandEngineeringSciences ,volume469,page20120505.TheRoyalSociety. [DaiandPromislow,2015] Dai,S.andPromislow,K.(2015).Competitivegeometricevolutionofam- phiphilicinterfaces. SIAMJournalonMathematicalAnalysis ,47(1):347{380. [DalMasoetal.,2014] DalMaso,G.,Fonseca,I.,andLeoni,G.(2014).Secondorderasymptoticde- velopmentfortheanisotropiccahn{hilliardfunctional. CalculusofVariationsandPartialential Equations ,pages1{27. [deMottoniandSchatzman,1990] deMottoni,P.andSchatzman,M.(1990).Developmentofinterfacesin r n . ProceedingsoftheRoyalSocietyofEdinburgh:SectionAMathematics ,116(3-4):207{220. [DeMottoniandSchatzman,1995] DeMottoni,P.andSchatzman,M.(1995).Geometricalevolutionof developedinterfaces. TransactionsoftheAmericanMathematicalSociety ,347(5):1533{1589. [DischerandEisenberg,2002] Discher,D.E.andEisenberg,A.(2002).Polymervesicles. Science , 297(5583):967{973. [Doelmanetal.,2014] Doelman,A.,Hayrapetyan,G.,Promislow,K.,andWetton,B.(2014).Meanderand pearlingofsingle-curvaturebilayerinterfacesinthefunctionalizedcahn{hilliardequation. SIAMJournal onMathematicalAnalysis ,46(6):3640{3677. [Evans,2010] Evans,L.(2010). PartialentialEquations:SecondEdition .AmericanMathematical Society. [Gavishetal.,2011] Gavish,N.,Hayrapetyan,G.,Promislow,K.,andYang,L.(2011).Curvaturedriven wofbi-layerinterfaces. PhysicaD:NonlinearPhenomena ,240(7):675{693. [Gavishetal.,2012] Gavish,N.,Jones,J.,Xu,Z.,Christlieb,A.,andPromislow,K.(2012).Variational modelsofnetworkformationandiontransport:applicationstopionomermembranes. Polymers ,4(1):630{655. [GilbargandTrudinger,2001] Gilbarg,D.andTrudinger,N.S.(2001). EllipticPartialentialEqua- tionsofSecondOrder ,volume224.SpringerScience&BusinessMedia. [Gomezetal.,2005] Gomez,E.D.,Rappl,T.J.,Agarwal,V.,Bose,A.,Schmutz,M.,Marques,C.M.,and Balsara,N.P.(2005).Plateletself-assemblyofanamphiphilicabcatetrablockcopolymerinpurewater. Macromolecules ,38(9):3567{3570. [GompperandSchick,1990] Gompper,G.andSchick,M.(1990).Correlationbetweenstructuralandinter- facialpropertiesofamphiphilicsystems. PhysicalReviewLetters ,65(9):1116. [Grisvard,1985] Grisvard,P.(1985). EllipticProblemsinNonsmoothDomains .ClassicsinAppliedMathe- matics.SocietyforIndustrialandAppliedMathematics(SIAM,3600MarketStreet,Floor6,Philadelphia, PA19104). 186 [GurtinandJabbour,2002] Gurtin,M.E.andJabbour,M.E.(2002).Interfaceevolutioninthreedimen- sionswithcurvature-dependentenergyandsurfaceInterface-controlledevolution,phasetransi- tions,epitaxialgrowthofelastic ArchiveforRationalMechanicsandAnalysis ,163(3):171{208. [HayrapetyanandPromislow,2014] Hayrapetyan,G.andPromislow,K.(2014).Spectraoffunctionalized operatorsarisingfromhypersurfaces. ZeitschriftfurAngewandteMathematikundPhysik ,pages1{32. [Helfrichetal.,1973] Helfrich,W.etal.(1973).Elasticpropertiesoflipidbilayers:theoryandpossible experiments. Z.Naturforsch.c ,28(11):693{703. [HomburgandSandstede,2010] Homburg,A.J.andSandstede,B.(2010).Homoclinicandheteroclinic bifurcationsinvector HandbookofDynamicalSystems ,3:379{524. [JainandBates,2004] Jain,S.andBates,F.S.(2004).Consequencesofnonergodicityinaqueousbinary peo-pbmicellardispersions. Macromolecules ,37(4):1511{1523. [KapitulaandPromislow,2013] Kapitula,T.andPromislow,K.(2013). SpectralandDynamicalStability ofNonlinearWaves .Springer. [Kato,1976] Kato,T.(1976). PerturbationTheoryforLinearOperators ,volume132.SpringerScience& BusinessMedia. [KraitzmanandPromislow,2014] Kraitzman,N.andPromislow,K.(2014). AnOverviewofNetworkBi- furcationsintheFunctionalizedCahn-HilliardFreeEnergy .Dynamics,GamesandScienceIII. [LinandYang,2002] Lin,F.andYang,X.(2002). GeometricMeasureTheory:AnIntroduction .Beijing: SciencePress. [Matyjaszewski,2012] Matyjaszewski,K.(2012).Atomtransferradicalpolymerization(atrp):currentstatus andfutureperspectives. Macromolecules ,45(10):4015{4039. [Peetetal.,2009] Peet,J.,Heeger,A.J.,andBazan,G.C.(2009).plasticsolarcells:self-assemblyof bulkheterojunctionnanomaterialsbyspontaneousphaseseparation. AccountsofChemicalResearch , 42(11):1700{1708. [Pego,1989] Pego,R.L.(1989).Frontmigrationinthenonlinearcahn-hilliardequation. Proceedingsofthe RoyalSocietyofLondon.A.MathematicalandPhysicalSciences ,422(1863):261{278. [Polking,1984] Polking,J.C.(1984).Asurveyofremovablesingularities.In SeminaronNonlinearPartial entialEquations ,pages261{292.Springer. [PromislowandWetton,2009] Promislow,K.andWetton,B.(2009).Pemfuelcells:amathematical overview. SIAMJournalonAppliedMathematics ,70(2):369{409. [PromislowandZhang,2013] Promislow,K.andZhang,H.(2013).Criticalpointsoffunctionalizedla- grangians. DiscreteandContinuousDynamicalSystems,A ,33:1{16. [Pruss,1984] Pruss,J.(1984).Onthespectrumof c 0 -semigroups. TransactionsoftheAmericanMathemat- icalSociety ,284(2):847{857. etal.,2013] L.P.,Ryan,A.J.,andArmes,S.P.(2013).Fromawater-immiscible monomertoblockcopolymernano-objectsviaaone-potraftaqueousdispersionpolymerizationformula- tion. Macromolecules ,46(3):769{777. 187 ogerandScatzle,2006] oger,M.andScatzle,R.(2006).Onamoconjectureofdegiorgi. Math- ematischeZeitschrift ,254(4):675{714. [SandstedeandScheel,2008] Sandstede,B.andScheel,A.(2008).Relativemorseindices,fredholmindices, andgroupvelocities. DiscreteandContinuousDynamicalSystemsA ,pages139{158. [Scherlisetal.,2006] Scherlis,D.A.,Fattebert,J.-L.,Gygi,F.,Cococcioni,M.,andMarzari,N.(2006).A electrostaticandcavitationmodelformoleculardynamicsinsolution. TheJournal ofChemicalPhysics ,124(7):074103. [TeubnerandStrey,1987] Teubner,M.andStrey,R.(1987).Originofthescatteringpeakinmicroemulsions. TheJournalofChemicalPhysics ,87(5):3195{3200. [Titchmarsh,1946] Titchmarsh,E.(1946). EigenfunctionExpansionAssociatedwithSecond-orderen- tialEquations .Oxford. [WilsonandGottesfeld,1992] Wilson,M.S.andGottesfeld,S.(1992).Highperformancecatalyzedmem- branesofultra-lowptloadingsforpolymerelectrolytefuelcells. JournaloftheElectrochemicalSociety , 139(2):L28{L30. [Zareetal.,2012] Zare,P.,Stojanovic,A.,Herbst,F.,Akbarzadeh,J.,Peterlik,H.,andBinder,W.H. (2012).Hierarchicallynanostructuredpolyisobutylene-basedionicliquids. Macromolecules ,45(4):2074{ 2084. [Zhangetal.,2013] Zhang,H.,Zhang,H.,Zhang,F.,Li,X.,Li,Y.,andVankelecom,I.(2013).Advanced chargedmembraneswithhighlysymmetricspongystructuresforvanadiumwbatteryapplication. En- ergy&EnvironmentalScience ,6(3):776{781. [Zhuetal.,2009] Zhu,J.,Ferrer,N.,andHayward,R.C.(2009).Tuningtheassemblyofamphiphilicblock copolymersthroughinstabilitiesofsolvent/waterinterfacesinthepresenceofaqueoussurfactants. Soft Matter ,5(12):2471{2478. [ZhuandHayward,2008] Zhu,J.andHayward,R.C.(2008).Wormlikemicelleswithmicrophase-separated coresfromblendsofamphiphilicabandhydrophobicbcdiblockcopolymers. Macromolecules ,41(21):7794{ 7797. [ZhuandHayward,2012] Zhu,J.andHayward,R.C.(2012).Interfacialtensionofevaporatingemulsion dropletscontainingamphiphilicblockcopolymers:ofsolventandpolymercomposition. Journal ofColloidandInterfaceScience ,365(1):275{279. [Zhuetal.,2012] Zhu,T.F.,Adamala,K.,Zhang,N.,andSzostak,J.W.(2012).Photochemicallydriven redoxchemistryinducesprotocellmembranepearlinganddivision. ProceedingsoftheNationalAcademy ofSciences ,109(25):9828{9832. [ZhulinaandBorisov,2012] Zhulina,E.andBorisov,O.(2012).Theoryofblockpolymermicelles:recent advancesandcurrentchallenges. Macromolecules ,45(11):4429{4440. 188