GRADIENTESTIMATESFORSOLUTIONSTODIVERGENCEFORMELLIPTICEQUATIONSWITHPIECEWISECONSTANTCOEFFICIENTSINDIMENSIONN.ByKhaldounAl-YasiriADISSERTATIONSubmittedtoMichiganStateUniversityinpartialentoftherequirementsforthedegreeofMathematics-DoctorofPhilosophy2016ABSTRACTGRADIENTESTIMATESFORSOLUTIONSTODIVERGENCEFORMELLIPTICEQUATIONSWITHPIECEWISECONSTANTCOEFFICIENTSINDIMENSIONN.ByKhaldounAl-YasiriInaboundedC10domainˆRn,0<01,containstwosimplyconnectedandstrictlyconvexC10subdomains(inclusions)D1andD2thatsatisfyD1[D2ˆˆandD1\D2=f0g,westudythefollowingelliptictialequation8>>>>>>>><>>>>>>>>:div(a(x)ru)=0in;@u(x)=gon@;R@u=0;(1)wherea(x)=1+(k1)˜(D1[D2)(x);0<k<1;k6=1;andg2L20(@:=fg2L2(@:Z@g=0g:Theproblemarisesfromstudyingerre-inforcedcompositemediawithcloselyspacedinclusionsindimensionn=2.Fordimensionn=3,theproblemalsoappearinthestudyofsteadystatesolutionstoMaxwell'sequations.Theboundonthegradientyieldstheboundednessofthestraininerre-inforcedmaterialsandelectricalwhenthetwoinclusionsaretouchingeachother.Followingtheideasof[1],weassumethatD1liesinthehalfspacexn<0,andD2inthehalfspacexn>0.Thenwebeginbyseparatingtheinclusionsbyadistance>0,thatis,wesetD1=D12en;andD2=D2+2en;whereen=(00;1).Thenwestudytheapproximatetialequationcorrespondingtotheseparatedinclusionswhichis8>>>>>>>><>>>>>>>>:div(a(x)ru)=0in;@u(x)=gon@;R@u=0;(2)wherea(x)=1+(k1)˜(D1[D2)(x).Thesolutionoftheellipticequation(2)hasanintegralrepresentationintermsofpotentialfunctionsontheboundaryofeachsubdomain.Fromtherepresentationformula,wederiveuniformpiecewiseC1,0<<0,estimatesforthissolutionwhichareindependentofthedistancebetweenthesubdomains.Thatis,wetheestimatekukC1~nD1[D2+kukC1(D1)+kukC1(D2)CkgkL2(@;where~=fx2:dist(x;@>~gandCisindependentof.Ourresultextendstheearlierresultfordimensionn=2[1],buttheanalysisismuchmorecomplicated.Finalestimatesrelyondetailedanalysisnearthetouchingpointandcollectivecompactnessofsomeintegraloperators.ACKNOWLEDGMENTSIwouldliketoexpressmydeepestgratitudetomyadvisorDr.ZhengfangZhou,whopro-videsexceptionalguidanceandtremendoussupportduringmygraduatestudy.Withouthisguidance,thisthesiswouldnothavebeenpossible.Ifeelveryluckytohavetheopportunitytostudyunderthesupervisionofsuchaknowledgableandsmartmathematicianwithgreatpersonality.Healwaysgenerouslytimetomeetforquestionsanddiscussions,heisveryinspiringandpatientinenhancingmymathematicalunderstanding,andhisviewonmathematicshelpsmetoshapemyownunderstanding.IamalsoverythankfulforhisencouragementsandsuggestionswhenImeetwithHiselegantpersonalityandpersistentpassioninmathematics&lifewillhavealongtimeonme.IwouldliketothankDr.BaishengYan,Dr.CasimAbbas,Dr.MoxunTang,Dr.WillieWong.forservingasmembersofmydoctoralcommitteeandfortheirinvaluablesuggestions.IwouldliketothankDr.PatriciaLammforthegeneroussupportfromherNSFgrantDMS-126547.IwanttothankDr.ZhengfangZhouandDr.CasimAbbas,formerdirectorsofthegraduatefortheirconsistentsupportandhelpduringmygraduatestudy.MythanksalsogotoDr.keithPromislow,ChairoftheDepartment,forhishelpandguidanceduringmygraduatestudy.IamgratefultotheentirefacultyandintheDepartmentofMathematicswhohavetaughtmeandassistedmeinonewayoranother.MyspecialthanksgotoDr.ZhengfangZhou,Dr.CasimAbbas,Dr.VeraZeidan,Dr.IgnacioUriarte-Tuero,Dr.SheldonNewhousefortheirhelpandsupport.AndIwouldliketoexpressmyappreciationtoMs.BarbaraMiller,Ms.LeslieAitcheson,Ms.BrittaAltinselandMs.AmiMcMurphyandMs.DebieivLecatofortheirassistanceinprovidinginformationandhelpduringmygraduatestudy.Also,Iwanttakethisopportunitytothankmyfellowgraduatestudentsandfriendsforprovidingkindhelp,sharingwithmetheirexpertiseknowledge,andsharingthechallengesandhappinesstogether.ParticularthanksgotoXinYangandJonathanBohn.FinallyIwouldliketothankmyparentsandmybrothersandsistersfortheirendlessloveandalwaysstandingbesidesmeovertheyears.vTABLEOFCONTENTSLISTOFFIGURES...................................viiiChapter1Introduction...............................1Chapter2LayerPotentialsforsystemoftwoinclusions...........202.1Notationsandassumptions............................202.2ExistenceandUniquenessTheorem.......................212.3Approximationoftialequation......................242.4LayerPotentials..................................302.5Representationformula..............................382.6Piecewiseoldercontinuityforthemotialequation.......48Chapter3Decompositionofthesystemofintegralequations........523.1BehaviorofT,>0...............................533.2AuxiliaryfunctionsandBasicproperties.....................583.3DecompositionofL2...............................743.4DecompositionofT................................82Chapter4olderestimatefortheoperatorJ2................864.1Preliminaries...................................884.2L1normforJ2,>0...........................944.3oldercontinuityofJ2,>0..........................1004.3.1BoundednessofR2.............................1014.3.2BoundednessofR3.............................1064.4ConvergenceofJ2................................124Chapter5UniformoldercontinuityforI2..................1265.1L1-normforI2,>0..............................1285.1.1L1-normforI1,>0...........................1285.1.2L1-normforI2,>0...........................1305.2olderestimateforI2,>0..........................1315.2.1olderestimateforI1,>0.......................1315.2.2olderestimateforI2,>0.......................1435.3ConvergenceofI2................................146Chapter6Theconvergenceandthemainresults................1536.1Preliminaries...................................1536.2CollectiveCompactnessandconvergenceofK2................1586.3PointwiseConvergenceofT...........................1626.4Themainresults..................................173viBIBLIOGRAPHY....................................177viiLISTOFFIGURESFigure1.1:Thetouchingsubdomains.......................6Figure1.2:Theapproximatesurfaces.......................14Figure2.1:Thetouchinginclusions........................21Figure2.2:Theseparatedinclusions........................26Figure3.1:Theapproximatesurfaces.......................77viiiChapter1IntroductionThepurposeofthisworkistostudygradientestimatesforsolutionsofdivergenceformellipticequationswithpiecewiseconstantcotsindimensionn2.Theproblemarisesfromstudyingerre-inforcedcompositemediawithcloselyspacedinclusions.AcompositemediumdescribedbyaboundeddomaininRn(n2),whichincludesmanyinclusions(subdomains)Dj,j=1;;m.ThephysicalcharacteristicsofthemediumaresmoothineachinclusionDjaswellasinn(D1[[Dm),buttheyarepossiblydiscontinuousacrosstheirboundaries@Dj,j=1;m.Weusetheexamplefromtheintroductionsof[8,30]tomotivatetheproblemfordi-mensionn=2.TheboundeddomainˆR2modelsthecross-sectionofaer-reinforcedcomposite.LetD1andD2betwosubdomainsrepresentthecross-sectionoftheers,andn(D1[D2)representtheregionsurroundingtheers.Assumethattheshearmodulus(modulusofrigidity)oftheersis1<k<1,andis1forthematerialsinthesurround-ingarea.Usingastandardmodelofanti-planeshear,weobtainthefollowingtialequationdiv1+(k1)˜D1[D2ru=0in;(1.1)withappropriateboundarydata,forinstance,u=gon@:1Thefunctionurepresentstheoutofplaneelasticdisplacementandrurepresentsthestrain.Fordimensionn=3,theproblemalsoappearinthestudysteadystatesolutionstoMaxwell'sequations.LetEbetheelectricandDbetheelectricdisplacementinconductingisotropicmediumˆR3.TheEandDsatisfythetialequationscurlE=0;D=E;anddiv(D)=0in;wheretherealvaluedfunctionrepresentstheconductivity(dielectricity)ofthematerials.Therefore,thereexistsapotentialfunctionu(representingthevoltage)suchthatE(x)=u(x)in;whereassumedtobesimplyconnecteddomaininR3.Hence,D(x)=(x)ru(x);anddivru=0in:Nowsupposethataconductortheregionthatconsistsoftmaterials,say,D1,D2andn(D1[D2).Weassumethatconductivity(x)equalstokinD1[D2and(x)equalsto1innD1[D2.Thecurrenton@isrepresentedbythenormalderivative@u@.Therefore,foragivenNeumannboundarydatagon@weobtainthe2followingtialequation8>>>>>>>><>>>>>>>>:div1+(k1)˜D1[D2ru=0in;@u(x)=gon@;R@g=0:(1.2)Atthispoint,wecanemphasizetheaimofstudyingsuchclassofdivergenceellipticequations,whichistostudythebehaviorofthestrain(electricruwhentheinclu-sionsapproacheachother(touch)aswellaswhentheshearmodulus(conductivity)oftheinclusionskdegenerate(k=0ork=1).E.BonnetierandM.Vogelius[8]haveshownthatthesolutionuofproblem(1.1)indimensionn=2isinW1;1forany0<k<1whentheinclusionsD1andD2aretwodisks.Y.Y.LiandM.Vogelius[30]studiedmoregeneralclassofelliptictialequations,whereisaboundeddomaininRnwithaC1boundary,0<<1,containingmdisjointsubdomainsDk,1km,eachwithaC1boundarysuchthat=[mk=1Dk(forprecisegeometricpictureofthesubdomainssee[30]).ThematrixA(x)=(aij(x))isuniformlyellipticmatrixandA(x)isCineachsubdomainDk,k=1;;m(butpossiblydiscontinuousacrosstheboundaries@Dk).Foravectorvaluedfunctiong=(g1;;gn)thatisCineachsubdomainDk,k=1;;m(butalsopossiblydiscontinuousacrosstheboundaries@Dk),denotegktobethefunctiongrestrictedtothesubdomainDk.Theyfoundthatforh2L1ifu2H1isasolutionto@iaij@ju=h+@igiin;(1.3)3thenthefollowingestimateholdsmax1kmkukC10(Dk\)CkukL1+khkL1+max1kmgkC0(Dk);(1.4)whereCisindependentofthedistancebetweentheinclusions,=fx2:dist(x;@>g,and0<0and0<n(1+).TheyalsohavestudiedspecialcaseinR2whenisadiskcenteredattheoriginwithradiusRanditcontainstwounitdisksD1andD2centered,respectively,at(0;1)and(0;1).Thatis,theboundaries@D1and@D2touchattheorigin.Theyconsideredthefollowingtialequation8>>><>>>:@ia(x)@iu=0in;u=gon@;whereg2H12(@anda(x)=8>>><>>>:1ifx2nD1[D20<a0<1ifx2D1[D2:ForR>2,theyfoundbyusingtheconformalmappingthefollowingboundskDuk1C(l)inD1andD2;8l;jjl;kDuk1C(k;)inD3\;8l;jjl;4whereD3=nD1[D2.Y.Y.LiandL.Nirenberg[29]generalizedtheresultof[30]intosystemsofellipticentialequationsandtheyfoundthatuthesameboundin(1.4)withslightlybetterregularity0where0<0minf2(1+)g:G.CittiandF.Ferrari[9]followed[30]withslightlytmoandtheyfoundtheoptimalregularityfortheproblem(1.3)whentheinclusionsarestrictlyseparated.Thatis,theyshowedthatthegradientruisC0,ineachcomponent,for0minfgwhentheinclusionsarestrictlydisjoint.J.Mateu,J.OrobitgandJ.Verdera[22]approachedtheproblemdiv(Aru)=0,det(A)=1viaBeltramiequationandtheyusedCalderon-Zygmundoperatorsandquasiconformalmap-ping.IftheinclusionsDmdonottouch,theyshowedthatruisC0,for0<minfgineachinclusionandtheirC0normisindependentofthedistancebetweentheinclusions.Recently,H.Ammari,E.Bonnetier,F.Triki,M.S.Vogelius[1]havestudiedproblem(1.5)indimensionn=2andtheyfoundthatruispiecewiseuniformlyboundedinC0normforany0<independentofthedistancebetweenthesubdomains.Inthiswork,wewillgeneralizetheresultsof[1]intohigherdimensionsn2bymimickingtheirideaswithnecessarymoBeforewestartoutlineourwork,wementionsomeresultswhentheconductivitykdegenerates.Inthecaseofk=1(perfectconductivity)ork=0(insulatedconductivity)thegradientmayblow-upastheinclusionsapproacheachother(touch).In[12,13]ithasbeenshownthattherateoftheblow-upfortheperfectconductivityproblemis1=2fordimensionn=2,(jlnj)1fordimensionn=3and1forhigherdimensionsn4,whereisthe5X0xnD2D121Figure1.1:Thetouchingsubdomainsdistancebetweentheinclusions.Fortheinsulatedconductivityproblemtheyhavefound1=2asanupperboundofthegradient.H.Kang,M.LimandK.Yun[20]studiedproblem(1.1)indimensionn=2whentheinclusionsaretwodisksseparatedbyadistance.Theydecomposedthesolutionuto(1.1)asu=g+b;wherergissingularfunctiondependingonthedistancebetweentheinclusionswhilerbisboundedfunctionregardlessofthedistance.Theyfoundanexplicitformulaforthesingularpartrgthatdescribesthebehavioroftheruwhichmayblow-upatrate1=2whenk=0ork=1.H.Ammari,G.Ciraolo,H.Kang,H.LeeandK.Yun[17]generalizedtheresultsof[20]toanystrictlyconvexsimplyconnectedC1subdomainsD1andD2inR2.Formoreresultsindimensionn=2whentheinclusionsarecircles,see[16,18,19].Now,wereturntoourproblemindimensionn2.LetˆRnbeaboundedsmooth6domaincontainingtheorigin0.For0<01,letD1andD2betwoC10simplyconnectedandstrictlyconvexsubdomains(inclusions)containedinsatisfyingD1[D2ˆˆandD1\D2=f0g.Furthermore,weassumethatD1liesinthehalfspacexn<0,andD2inthehalfspacexn>0.WedenotebyjtobetheboundaryofDj,j=1;2.Westudythefollowingtailequation8>>>>>>>>>>>><>>>>>>>>>>>>:div(a(x)ru)=0in;@u(x)=gon@;R@u=0;R@g=0;(1.5)wherea(x)=1+(k1)˜(D1[D2)(x):Clearly,Lax-Milgramtheoremshowsthattheproblem(1.5)hasauniquesolutionu2H1foragiveng2L2(@Followingtheideasof[1],webeginbyseparatingtheinclusionsbyadistance>0,thatis,wesetD1=D12en;andD2=D2+2en;whereen=(00;1).Thenwestudytheapproximatetialequationcorrespondingtotheseparatedinclusionswhichis78>>>>>>>>>>>><>>>>>>>>>>>>:div(a(x)ru)=0in;@u(x)=gon@;R@u=0;R@g=0;(1.6)whereaisthecorrespondingpiecewiseconstantcocients,thatis,a(x)=1+(k1)˜(D1[D2)(x):Itiswellknownthatequation(1.6)hasauniquesolutionuinH1foragivenNeumannboundarydatag2L20(@:=fg2L2(@:R@gd˙(x)=0g.WeproveinTheorem2.5thesolutionto(1.6),u,approachesuniformlyinH1tou,thesolutionofequation(1.5),asapproaches0.Nowwestatethemainresultofourworkinthefollowingtheoremwhichcoincideswiththeearlierresultfordimensionn=2[1].Theorem1.1.Let~>0and0<<0.Thesolutionto(1.6)kukC1~nD1[D2+kukC1(D1)+kukC1(D2)CkgkL2(@;where~=fx2:dist(x;@>~gandCisindependentof.Inordertoprovethemainresult,wethesingleanddoublelayerpotentialsforthe8Laplacianoperator,respectively,byS@'(x):=Z@x;y)'(y)d˙(y);x2Rn;(1.7)andD@'(x):=Z@@x;y)@(y)'(y)d˙(y);x2Rnn@:(1.8)wherex;y)isthefundamentalsolutiontoLaplaceequationu(x)=0inRn.Itisknownfrom[23],thesolutionofproblem(1.6)canbeuniquelyrepresented(seeTheorem2.14)asu(x)=H(x)+S1'1(x)+S2'2(x);x2;(1.9)whereSjisthesinglelayerpotentialonthesurfacej,j=1;2:andHisaharmonicfunctioninthathasthefollowingformH(x)=S@g(x)+D@(uj@)(x);x2:Whilethepotentials'1and'2solvethefollowingsystemofintegralequations8>>><>>>:K1'1(x)@S2'2(xen)=@Hx2en;x21;@S1'1x+en+K2'2(x)=@Hx+2en;x22;(1.10)whereinthissystem=k+12(k1),andKidenotestheoperatorKi'i(x)=1!nZi(xy)(x)jxyjn'i(y)d˙(y);i=1;2:9Fromtheclassicpotentialtheory,weeasilyshowthatKj,j=1;2:arecompactoperatorsfromC1)intoC2)forany<0.Ifwedefor'1;'22C1)C2)theoperatorT0B@'1'21CA:=0B@K1L2L1K21CA0B@'1'21CA;(1.11)whereL2'2(x)=@S2'2xen;x21;(1.12)andL1'1(x)=@S1'1(x+en;x22:(1.13)Thenwecanrewritethesystem(1.10)inthefollowingformT0B@'1'21CA=0B@@Hy1(x;)@Hy2(x;)1CA;(1.14)wherey1(x;)=x2en;x21;andy2(x;)=x+2en;x22:SinceHisharmonicfunctiononRn,thenweeasilyprovethatHisuniformlyboundedin10C0normonthesurfacesj,j=1,2.Thatis,form=1;2;,thefollowingboundsholdk@HkC0j)CkgkL2(@;(1.15)whereCisindependentof.Fromtheregularityofthesinglelayerpotential(seeLemma2.18)andtherepresentationformula(1.9),wehaveforany0<0<<0and>0thefollowingbound2Xj=1kuk1Dj+kuk1~nD1[D2C 2Xi=1'i0i+kgkL2(@!;(1.16)where~=fx2:dist(x;@>~gandCisdependingon,0,0,,k,and~butindependentof.Weclearlyobservefrom(1.16)thegradientruisCineachcomponentD1,D2and~nD1[D2ifthepotentials'jareuniformlyboundedinCj),j=1;2:,forany<0.Inotherwords,Inordertoobtainpiecewiseoldercontinuityofru,weneedtoshow2Xi=1'ii)CkgkL2(@;8<0;(1.17)whereCisindependentof.Byinvolving(1.14),thebound(1.17)followsifweprovetheoperatorTisinvertibleinC1)C2)anditsinverseisunifromlyboundedonC1)C2),forany<0.Fromelementarypotentialtheorywhen>0,theoperatorTisinvertibleasanoperatoronC1)C2).Tobemoreprecise,TcanbewrittenasT=I+K,whereKisacompactoperatoronC1)C2).ThenweuseFredholmtheorytoshowthe11invertibilityoftheoperatorsT.Recallthat,ifwehaveafamilyofboundedlinearoperators,say,TonaBanachspace,say,X.LetusassumethatTconvergesinnormtoanoperatorT02L(X)(thespaceofallboundedlinearoperatorsonX).Noticethat,if(T0)12L(X),thenthereexist0>0besuchthatforany0thereexistuniformlybounded(T)12L(X),inwhichcase(T)1(T0)1!0:Naively,weconcludefromthisobservationthatwewouldobtaintheexistenceofuniformlybounded(T)12C1)C2)ifweprovedtheexistenceof(T0)12C1)C2)(weconsiderT0asalimitingoperatorcorrespondingtoT)andTconvergesinnormintoT0.Therefore,weaskwhetherwecanhaveconvergenceinnormforthefamilyTintotheirlimitingoperatorT0.Ammari,H.andBonnetier,E.andTriki,F.andVogelius,M.S.[1]haveprovedthatsuchconvergenceinnormcannothold.Infact,theyshowedfortheircaseindimensionn=2thelimitingoperatorsL0j;j=1;2:(thelimitingoperatorscorrespondingtothecompactoperatorsLj;j=1;2:)arenotcompactoperatorsonCforany<0.ThusLj;j=1;2:cannotconvergeinnormtotheirlimits.Therefore,theaboveobservationcannotbeusedtoobtaintheuniformlyboundedoperators(T)1.Indimensionn=2[1]theauthorsusedthenotionofcollectivelycompactoperatorsthatrequirejustpointwiseconvergence[3].Thatis,ForcollectivelycompactoperatorsKn,n=1;2;:::suchthatKn!K(pointwiseconvergence)and(IK)1exists.ThenforsomenNtheoperators(IKn)1existandareboundeduniformly,insuchacase(IKn)1!(IK)1:12Therefore,inordertoobtaintheexistenceofuniformlyboundedoperators(T)1inC1)C1)for<0byusingthenotionofcollectivelycompactoperators,weneedtoshowtwomajorthingswhichare:1.TheoperatorTcanbewrittenasT=I~where~arecollectivelycompactand~convergespointwisetosomelimitingoperator~0.2.ThelimitingoperatorT0=I~0isinvertible.Noticethat,whenapproaches0,thekerneloftheoperatorTbecomessingularatx=0.Infact,theoperatorsL2andL1becomesingularatx=0whenapproaches0.Toovercomethesingularity,wefollow[1]anddecomposetheoperatorsLj,j=1;2.WeonlyshowitforL2butthedecompositionofL1followssimilarly.Fix0andlet00.Wetwoauxiliaryfunctions 1and 2globallyonRn1(seeLemma3.4)suchthat8>>><>>>: j= j;x0j=1;2; j1;Rn1C j10;j=1;2:(1.18)forany<0,=0,andx0=(x1;:::;xn1)2Rn1.Forthenof j;j=1;2seetheRemark2.1.Let>0and'2C2).Forx21,jxj<R0,wesetx=(x0; 1(x0))andthenwetheapproximatesurface2=fy=(y0; 2(y0))jy02Rn1g:1322x02211Figure1.2:TheapproximatesurfacesLetS2Ebethesinglelayerpotentialontheapproximatesurface2forsomepotentialEthatonthesurface2besuchthatE'on2\B():Wealsoforthepointx=(x0; 1(x0))21thehyper-planex02=fy2Rnjyn= 2(x0)g:(1.19)Again,letSx02Ebethesinglelayerpotentialonthehyper-surfacex02forthepo-tentialE(weremarkherethatthepotentialEisalsowellonthehyper-planex02seeSection3.3).ForafunctiononRnthatissupportedintheballB(R0)andidenticallyequalsto1intheballB(0),wewritetheoperatorL2inthefollowingformL2'(x)=(x)L2'(x)+1(x)L2'(x):14SincethesingularpartoftheoperatorL2isL2,wedecomposeitasfollowsL2'(x)=K2'(x)+1!nq1+r 1(x0)2J2'(x)+I2'(x):(1.20)For~x=xen;theoperatorsK2,J2andI2areasfollowsK2'(x)=@S2'(xen)+@S2E(~x);(1.21)J2'(x)=!nq1+r 1(x0)2@Sx02E(~x)@S2E(~x);andI2'(x)=!nq1+r 1(x0)2@Sx02E(~x);where(x)isthenormalunitvectorontheapproximatesurface1atthepointx,thatis,(x)=1q1+r 1(x0)20B@ 1(x0)11CA:(1.22)Remark1.2.Since 2= 2aroundtheorigin,thenweseetheoperatorsK2arenotsingularintheneighborhoodoftheorigin.Alsowenoticethatawayfromtheorigin,thekernelfunctionsthatarecorrespondingtotheoperatorsK2areuniformlybounded.ThenwecaneasilyshowthattheoperatorsfK2gformafamilyofcollectivelycompactoperatorsfromC2)intoC1).Remark1.3.IntheoftheoperatorJ2,wehaveinvolvedtheoperator@Sx02E(~x)15inordertotaketheadvantageoftheence 2(x0) 2(y0)togettheoldercontinuityfortheoperatorJ2.AfterdecomposingtheoperatorsLj,theoperatorTin(1.11)maynowbedecomposedasfollowsT0B@'1'21CA=0B@'1'21CA+C0B@'1'21CA;(1.23)where=0BBBBB@!nr1+r 1(x0)2J2+I2!nr1+r 2(x0)2J1+I11CCCCCA;(1.24)andC=0B@K1K2K2K21CA+(1)0B@0L2L101CA:(1.25)WeproveinTheorems(4.1)and(5.1)forany00and<0thefollowingbound!nq1+r 12J2+I2L(C2);C1))121+C()(1+0):Similarly,wewillgetthebound!nq1+r 22J1+I1L(C1);C2))121+C()(1+0):16Since1+02<jj;thenwereadilyseetheinvertibilityoftheoperatorsandtheuniformbound1Cjj121+C()(1+0);00:(1.26)Asaconsequenceoftheuniformbound(1.26),wehavethepointwiseconvergence10B@'1'21CA!100B@'1'21CAinC01)C02);0<:WeshowinTheorem6.12thattheoperatorsfC1garecollectivelycompactonC1)C2),andC1!C010pointwiseinLC1)C2)asapproaches0.Consequently,weobtainthecompactnessoftheoperatorfC010gonC1)C2).InTheorem6.13weshowthattheoperatorT0isinvertible.Thus,sinceT0and0areinvertibleandT0=I+C0100:Thenweseethat0T01=I+C0101:Therefore,wegetthepointwiseconvergenceI+C1!I+C010inC1)C2):ThusfromLemma(6.7)wehavethatI+C11existwhenistlysmalland17theyareuniformlyboundedwithrespecttointheoperatornorm.Therefore,weobtainthepointwiseconvergenceI+C11!I+C0101:(1.27)SinceT=I+C1,thenbyusingthebound(1.26)weobtain(T)1=1I+C11;areuniformlynormboundedandsatisfy(T)1!T01inC01)C02);0<:Therefore,thebound(1.17)hasbeenprovedandtheTheorem1.1follows.Thatis,wehaveproved2Xj=1kukC1(Dj)+kukC1~nD1[D2CkgkL2(@:(1.28)Sinceu!u0inH1;(seeTheorem2.5);and(T)1!T01inC01)C02);0<:18Thenwehavethatthesolutionu0,to(1.5),canberepresentedbyu0(x)=H0(x)+S1'01(x)+S2'02(x);x2;(1.29)whereH0isharmonicinsideanddebyH0(x)=S@g(x)+D@(u0j@)(x);x2;(1.30)andthepair('01;'02)2C1)C2)istheuniquesolutiontothelimitingsystemcorrespondingto(1.14).Thedissertationisorganizedasfollows:InChapter2,weintroduceallnecessaryingre-dientsfortherepresentationformulatothesolutionofequation(1.6).Chapter3isdevotedtothedecompositionoftheoperatorsLj,j=1;2:aswellastheoperatorT.InChapter4,weproveforany00and<0thefollowingboundholds!nq1+r 12J2L(C2);C1))C():WhileinChapter5weprovethat!nq1+r 12I2L(C2);C1))121+C()(1+0):Finally,inChapter6weshowtheinvertibilityoftheoperatorT0andthentheexistenceofuniformlyboundednessoftheoperators(T)1.19Chapter2LayerPotentialsforsystemoftwoinclusions2.1NotationsandassumptionsLetn2.WedenotebyB(r)theopenballinRncenteredattheoriginofradiusrandbyB0(r)theopenballinRn1.Forx=(x1;;xn)2Rn,letx0=(x1;;xn1)whereweoftenregardx0asapointinRn1.LetˆRnbeaboundedsmoothdomaincontainingtheorigin0.For0<01,letD1andD2betwoC10simplyconnectedandstrictlyconvexdomains(inclusions)containedinsuchthatD1[D2ˆˆandD1\D2=f0g.RecallthatwesayDjisaC10domainifeachpointofthe@Djhasaneighborhoodinwhich@DjisthegraphofC10functionofn1variablesx1;::::;xn1,j=1;2.Furthermore,assumethatD1liesinthehalfspacexn<0,andD2inthehalfspacexn>0.Remark2.1.SinceD1andD2arebothC10domains,thenaroundthetouchingpointx=0,1and2wouldbeparametrizedbyx0; 1(x0)andx0; 2(x0)respectively,where jareC10functionsandj:=@Dj,j=1;2.Thegraphof 1liesbelowthex0-hyperplane,whilethegraphof 2liesabovethex0-hyperplane.20x0xnD2D121Figure2.1:ThetouchinginclusionsLetH1:=W1;2betheusualSobolevspaceandC10isthespaceofsmoothfunctionswithcompactsupportinWeoftenusethespaceL20(@:=f'2L2(@:R@'d˙(x)=0g.2.2ExistenceandUniquenessTheoremLetˆRnbeaboundeddomaininRnwithsmoothboundaryandconsideraC10boundeddomainDˆˆLetubeasolutiontotheNeumannproblem8>>>>>>>><>>>>>>>>:div(1+(k1)˜D)ru=0in;@u(x)=gon@;R@ud˙(x)=0;(2.1)whereg2L20(@and˜(D)isthecharacteristicfunctionofD.2.2.Wesaythatu2H1isaweaksolutionto(2.1)ifR@ud˙(x)=0and21thefollowingidentityholds:Za0(x)ru(x)r(x)dx=Z@g(x)(x)d˙(x);82C1:(2.2)Theexistenceofthesolutionto(2.1)isverybasicandweshowitinthefollowingtheorem.Theorem2.3.Foranyg2L20(@,thereexistsauniqueu2H1solves(2.1).Proof.LetH1=fu2H1:R@ud˙(x)=0g.ThenH1isaclosedsubspaceofH1Infact,letunbeasequenceinH1suchthatun!uinH1ThenbyusingthetracetheoremitfollowsthatZ@ud˙=Z@(unu)d˙Z@junujd˙CkunukH1:ThereforeR@ud˙=0.ThusH1isaclosedsubspaceoftheHilbertspaceH1Consequently,H1isaHilbertspace.WeabilinearformonH1asfollowsB:H1H1!R:B[u;v]=Zak(x)DuDvdxforu;v2H1;whereak=1+(k1)˜(D)besuchthatmkakkL1MforsomepositiveconstantsmandM.Let`beaboundedlinearfunctionalonH1by`(v)=Z@gvd˙forv2H1;22ItisclearthatjB[u;v]jkukH1kvkH1forsome>0.WeshowthatB[u;u]Ckuk2H1forsomeC>0.B[u;u]=ZakjDuj2dxmkruk2L2:ByusingPoincare'sinequality(2.7),itfollowsthatB[u;u]Ckuk2H1:(2.3)ByapplyingtheLax-MilgramTheorem,weauniquefunctionu2H1satisfyingB[u;v]=`(v)forallv2H1NowweshowB[u;v]=Z@gvd˙forv2H1:Letv2H1byprojectiontheoremonHilbertspaces,thereareuniquev2H1andc2Rsuchthatv=v+c.SinceR@gd˙=0,thenitfollowsthatZ@gv=Z@gv+cZ@g=Z@gv=B[u;v]:Thatis,B[u;v]=B[u;v]:23Thusthereexistsuniqueu2H1suchthatB[u;v]=`(v)forallv2H1Consequently,uistheuniqueweaksolutionto(2.1).2.3Approximationoftialequation.For0<k<+1andk6=1,westudythetialequation8>>>>>>>><>>>>>>>>:div(a0(x)ru0)=0in;@u0(x)=gon@;R@u0=0;(2.4)wherea0(x)=1+(k1)˜(D1[D2)(x);˜isthecharacteristicfunctionofD1[D2andg2L20(@Thetialequation(2.4)hasbeenintroducedandverywellstudiedin[1]forthedimensionn=2.Wewillusetheideaof[1]tostudythebehaviorofthesolutiontothetialequation(2.4)nearthetouchingpointforanydimensionn2.For>0,wesetD1=D12en;D2=D2+2en;whereen=(00;1)andwedenotebyathecorrespondingpiecewiseconstantcots,24thatis,a(x)=1+(k1)˜(D1[D2)(x):Letubethesolutiontothefollowingmotialequation8>>>>>>>><>>>>>>>>:div(a(x)ru)=0in;@u(x)=gon@;R@u=0:(2.5)WeclearlyseethatthefunctionuisharmonicinsideandoutsidetheinclusionsD1,D2andthejumpconditionsu+=u;@u+@=k@u@on@Di;j=1;2:(2.6)Whereu+denotesthesolutionto(2.5)outsidetheinclusionD1[D2,whileudenotesthesolutionto(2.5)insideD1[D2andistheoutwardunitnormaltoDj,j=1;2.Next,weshowthattheapproximatesolutionuto(2.5)convergestou0,thesolutionto(2.4),inH1asapproaches0.Forthispurpose,weneedthefollowingmoPoincare'sinequality.TheproofissimilartotheusualPoincare'sinequality[28].Proposition2.4.LetbeaboundedandconnecteddomaininRn,withaC1boundary@.Foru2H1thereexistsaconstantC,dependingonnand,suchthatkuk2L2C kruk2L2+Z@ud˙2!:(2.7)25x0xnD2D1Figure2.2:TheseparatedinclusionsProof.Wearguebycontradiction.Iftheinequality(2.7)werefalse,thenonewouldasequenceoffunctionsfujg2H1besuchthatuj2L2>j ruj2L2+Z@ujd˙2!:(2.8)Wenormalizeujbywj=ujujL2(j=1;2;):(2.9)Then(2.8)impliesrwj2L2+Z@wjd˙2<1j(j=1;2;):(2.10)ClearlyweseethatthefunctionsfwjgareboundedinH1ThenbySobolevembedding26theorem,thereexistsasequencefwjkgˆfwjgandafunctionw2L2suchthatwjk!winL2:(2.11)ToshowthatwhasweakderivativeinL2let˚2C10thenweuse(2.10)toobtainthefollowingZwr˚dx=limjk!1Zwjkr˚dx=limjk!1Zrwjk˚dx=0:Consequentlyw2H1withrw=0.Thuswisconstantbecauseisconnected.Ontheotherhand,themapw0!Z@w0d˙2;iscontinuousonH1Then(2.10)impliesthatw=0;inwhichcasekwkL2=0.Thiscontradictionprovestheinequality(2.7).NowwearereadytostateandprovetheconvergenceofuinH1Theorem2.5.Thesolutionof(2.5)approachestothesolutionof(2.4)inH1asapproacheszero.Thatislim!0kuu0kH1=0:(2.12)Proof.First,weprovethata=1+(k1)˜D1[D2!a0inLpforanyp<1.Let=[4j=1j,where1=D1nD1\D1,2=D1nD1\D1,3=D2nD2\D2,and274=D2nD2\D2.Thenkaa0kpLp=Zjaa0jpdx=Zjaa0jpdx=Zjk1jpdx=jk1jp)!0as!0;whereisthendimensionalLebesguemeasure.Thusa!a0inLpforanyp<1.FromtheoftheweaksolutionwehavethefollowingZ(arua0ru0)r=0;82H1:ThenwehaveZa0(ruru0)rdx=Z(aa0)rurdx:(2.13)Choosing=uu0in(2.13),wegetthefollowingZa0jruru0j2dx=Z(aa0)ru(ruru0)dxkaa0kLpkrukLqkruru0kL2:Thelastinequalitymakessenseifu2W1;qforsomeq>2.Wewillprovethislater.SinceC1<ka0kL1<C2forsomepositiveconstantsC1andC2,weobtainthefollowingkruru0kL2Ckaa0kLpkrukLq;28Alsosincekaa0kLp!0as!0,thenwehavekruru0kL2!0as!0:(2.14)ByapplyingPoincare'inequality(2.7)foruu0,wehavekuu0k2L2Ckruru0k2L2:Thuskuu0kL2!0as!0:(2.15)Then(2.14)and(2.15)implykuu0kH1!0as!0:Nowweproveourclaimthatu2W1;qforsomeq>2.Todothis,ittoproveu2W1;q(V)forasubsetVthatsatisfyingD1[D2ˆˆVˆˆLetUbeasubsetsothatVˆˆUˆand'beafunctionsuchthat'1inVandsupp'ˆU.~u='u,thenweseethat8>><>>:divar~u=finU;~u=0on@U;(2.16)wheref=div(ur')+(rur')andwehaveusedthata1onUnV.Since(ur')2Lqand(rur')2L2ˆW1;q,thenf2W1;q(U)forsomeq>2.Byusing29Meyer'stheorem[7],wehave~u2W1;q0(U)andk~ukW1;q0(U)CkfkW1;q(U):Consequently,u2W1;q(V).2.4LayerPotentialsInthissectionwestudytwointegraloperatorsthatcalledlayerpotentials.Theseoperatorsplayanimportantroleinthederivationofthedecompositiontheoremforthesolutionofthetransmissionproblem(2.4).TheseoperatorsareverywellstudiedforC2-domainsin[10,24,25,26,27].Also[11,15,31]gavesomebasicresultsfortheseoperatorsforC1domains.InvertiblityandcomapctnessforLipschitzdomainsaregivenin[2,32].WebeginbythefundamentalsolutiontotheLaplaceequationindimensionn.Wedenoteby!ntheareaofunitsphereindimensionn.Lemma2.6.[14]AfundamentalsolutiontotheLaplaceequationu=0isgivenbyx;y)=8>>><>>>:12ˇlnjxyjifn=2;1(2n)!njxyj2nifn>2:(2.17)WethelayerpotentialsforL2densityfunctions.GivenaboundedC1domaininRn,n2,wedenoterespectivelythesinglelayeranddoublelayerpotentialsofafunction'2L2(@asS@'andD@',whereS@'(x):=Z@x;y)'(y)d˙(y);x2Rn;(2.18)30andD@'(x):=Z@@x;y)@(y)'(y)d˙(y);x2Rnn@:(2.19)ForafunctionuonRnn@wedenoteu(x)=limt!0+u(x(x));x2@;(2.20)and@@u(x)=limt!0+ru(x(x))(x);x2@;(2.21)ifthelimitexists.Here(x)istheoutwardunitnormalto@atx.Forsimplicity,sometimesweuse@u(x)insteadof@@u(x)andxinsteadof(x).Westatethejumprelationsforthedoubleandsinglelayerpotentials.Lemma2.7.[2]LetbeaboundedC1domaininRn.For'2L2(@S@'+(x)=S@'(x)x2@;(2.22)@S@'(x)=12I+K@'(x)x2@;(2.23)D@'(x)=12I+K@'(x)x2@;(2.24)whereK@isdbyK@'(x)=1!nZ@(yx)(y)jxyjn'(y)d˙(y);(2.25)31andK@istheL2adjointofK@,i.e.,K@'(x)=1!nZ@(xy)(x)jxyjn'(y)d˙(y):(2.26)ThecompactnessoftheoperatorsK@andK@forC2domainshasbeenshownin[10,25],withsomemoweshowthecompactnessforC1domains.Theorem2.8.LetbeaboundedC1domaininRn,theoperatorsK@;K@:C(@!C(@arecompactoperatorsforany0<<1.Proof.First,weshowthattheoperatorK@:C(@!C(@;isboundedoperatorforany<.Thatis,forany'2C(@weshowkK@'kCk'k1;whereCisdependingon@,andn.ItiseasytoshowthefollowinginequalitiesholdforC1domain,j(xy)(x)jCjxyj1+8x;y2@;(2.27)32andj(x)(y)jCjxyj8x;y2@;(2.28)whereCisdependingon@andn.Alsobyusingthemeanvaluetheorem,itisnottoshowthatforanyx1;x2;y2@with2jx1x2jjx1yj,thefollowinginequalityholds1jx1yjn1jx2yjnCjx1x2jjx1yjn+1;(2.29)whereCisdependingonnonly.FortheuniformboundednessoftheoperatorK@,weeasilyseethatjK@'(x)jCk'k1Z@jxyj1+nd˙(y):ThereforewehavejK@'(x)jCk'k1:ThatiskK@'k1Ck'k1:(2.30)Toestablisholdercontinuity,letuseforx2@andr>0,theportionSx;rasfollowsSx;r:=fy2@:jxyj<rg:(2.31)33Byusing(2.28),itfollowsthat(x)(x)(y)Cjxyj:Noticethat,(x)(y)=1(x)(x)(y):ThuswecandR2(0;1]besuchthatforanyx;y2@withjxyjR,thefollowingholds(x)(y)12:(2.32)WeassumeRistlysmallbesuchthatSx;Risconnectedforeachx2@Thusbythehelpof(2.32),Sx;Rcanbebijectivelyprojectedintothetangentplaneto@atthepointx.Thesurfaceelementd˙(y)onSx;Randthesurfaceelementd~˙(y)onthetangentplanearerelatedbyd˙(y)=d~˙(y)(x)(y)12d~˙(y):(2.33)Letx1;x22@besuchthatjx1x2jR4andletr=4jx1x2j.WeestimateovertheportionSx;rtoobtainKSx1;r'(x1)KSx1;r'(x2)Ck'k1 ZSx1;rjx1yj1+nd˙(y)+ZSx2;2rjx2yj1+nd˙(y)!:34Thusbyusingpolarcoordinates,weobtainKSx1;r'(x1)KSx1;r'(x2)Ck'k1Z2r0ˆ1dˆ:Thatis,KSx1;r'(x1)KSx1;r'(x2)Ck'k1jx1x2j:(2.34)Now,weestimateovertheportionSx1;RnSx1;r.Noticethatfory2Sx1;RnSx1;rand4jx1x2j=r,wehave2jx1x2j<jx1yj.Observethefollowing(x1y)(x1)jx1yjn(x2y)(x2)jx2yjn=(x1y)(x1)(x2y)(x1)jx1yjn+(x2y)(x1)(x2)jx1yjn+(x2y)(x2)jx1yjn(x2y)(x2)jx2yjn:Thereforebyusing(2.27),(2.28),(2.29)andnoticingthatjx2yjjx2x1j+jx1yj<12jx1yj+jx1yj=32jx1yj;wehave(x1y)(x1)jx1yjn(x2y)(x2)jx2yjnCjx1x2jjx1yjn1+Cjx1x2jjx1yjn:Thusafterconvertingtothepolarcoordinates,wehavethefollowingZ~SR;r(x1y)(x1)jx1yjn(x2y)(x2)jx2yjnd˙(y)Cjx1x2jZRr4ˆ()1dˆ;35where~SR;r=Sx1;RnSx1;r.ThuswehaveZ~SR;r(x1y)(x1)jx1yjn(x2y)(x2)jx2yjnd˙(y)Cjx1x2j:ThatisK~SR;r'(x1)K~SR;r'(x2)Ck'k1jx1x2j:(2.35)Similarly,estimatingovertheportion@nSx1;R,wehaveZ@nSx1;R(x1y)(x1)jx1yjn(x2y)(x2)jx2yjnd˙(y)CZ@nSx1;R jx1x2jjx1yjn+jx1x2jjx1yjn1!d˙(y):ThereforeZ@nSx1;R(x1y)(x1)jx1yjn(x2y)(x2)jx2yjnd˙(y)Cjx1x2j:ThatisK@nSx1;R'(x1)K@nSx1;R'(x2)Ck'k1jx1x2j:(2.36)Bycombining(2.34),(2.35),and(2.36),weobtainjK@'(x1)K@'(x2)jCk'k1jx1x2j;8<:(2.37)36Thenbycombining(2.30)and(2.37),itfollowsthatkK@'kCk'k1:ForcompactnessoftheoperatorK@,letf'ngbeaboundedsequenceinC(@Thatisk'nk~C:(2.38)Thenclearlyweseethatj'n(x)j~C;8x2@;andj'n(x)'n(y)j~Cjxyj;8x;y2@;ThusbyArzela-AscoliTheorem,f'nghasconvergentsubsequencef'njg.Thatis,'nj!'0inC(@:ThuswehaveK@'nj!K@'0inC(@:ThereforeweconcludethatK@iscompactoperator.ThecompactnessoftheoperatorK@followssimilarly.37Theorem2.9.[11]LetbeaboundedC1domaininRn,thenwehaveK@(1)=12.Thatis1!nZ@(yx)(y)jxyjnd˙(y)=12:Remark2.10.SinceK@istheL2adjointoperatorofK@,thenwehaveZ@K@'d˙(x)=Z@'K@(1)d˙(x)=12Z@'d˙(x)8'2L2(@:Theorem2.11.[2,32]LetbeaboundedC1domaininRn,theoperatorK@:C(@!C(@isinvertibleonforjj>12and0<<1.Thenormalderivativeofthedoublelayerpotentialiscontinuousacrosstheboundaryasweseeinthefollowingtheorem.Theorem2.12.[15]LetbeaboundedC1domaininRn,then@+D@'and@D@'exist.Moreover,@+D@'=@D@';8'2C(@:2.5Representationformula.Beforewegivetherepresentationformulato(2.1),weshowthattheHarmonicfunctionsinanunboundeddomainRnnB(R)withdecayoforderjxj1nsatisfytheGreenidentityasweseeinthefollowingLemma.38Lemma2.13.LetR>>1belargeandubeaharmonicfunctioninRnnBR(0)thathasdecayoforderjxj1ninRnnBˆ(0),whereˆ>4R.ThenthefollowingidentityholdsZ@BR(0)u@ud˙(x)=ZRnnBR(0)jruj2dx:Proof.SinceuisharmonicintheannulusBˆ(0)nBR(0),thenwehaveZBˆ(0)nBR(0)jruj2dx=Z@Bˆ(0)nBR(0)u@ud˙(x);thatis,ZBˆ(0)nBR(0)jruj2dx=Z@Bˆ(0)u@ud˙(x)Z@BR(0)u@ud˙(x):(2.39)Fromtheassumptions,weknowthatu=O(ˆ1n)on@Bˆ(0).Nowweshowthat@u=O(ˆn)on@Bˆ(0).Choosex02@Bˆ(0),thenclearlyweseethatuisharmonicinBˆ4(x0).LetvbeaharmonicfunctionintheballB1(x0),thenbyPoisson'sformulawehavethatv(x)=1!nZ@B1(x0)1jxx0j2jxyjnv(y)d˙(y);x2B1(x0):Bytiation,weobtainthefollowingrv(x)jx=x0=1!nZ@B1(x0)r 1jxx0j2jxyjn!jx=x0v(y)d˙(y);x2B1(x0):Therefore,jrv(x0)jCkvkL1B1(x0):(2.40)39Assumingv(x)=u(ˆ4x)andusing(2.40),itfollowsthatjru(x0)jCˆkukL1Bˆ=4(x0):(2.41)From(2.41)andthedecayconditionofuinBˆ=4(x0),itfollowsthat@u=O(ˆn)on@Bˆ(0).ThenweconcludethatZ@Bˆ(0)u@ud˙(x)=O(ˆ12n):Thusthelemmafollowsbylettingˆ!1in(2.39).Wegivearepresentationformulaforthesolutionto(2.1).ThisformuladependsonthesubdomainDandthepair(uj@;g).Theproofwasgivenin[2,23].Theorem2.14.LetbeaboundeddomainwithsmoothboundaryandletDbeasubdomiancompactlyembeddedinwithC10boundaryandconductivity0<k6=1<1.Thesolutionto(2.1)canbeuniquelyrepresentedasu(x)=H(x)+S@D'(x);x2;(2.42)whereHisaharmonicfunctiongivenbyH(x)=S@g(x)+D@f(x);x2;f:=uj@;(2.43)and'2L20(@D)theintegralequationk+12(k1)IK@D'=@H@j@Don@D:(2.44)40Proof.Considerthefollowingproblem8>>>>>>>>>>>><>>>>>>>>>>>>:rak(x)rh=0inRnn@;hh+=fon@;@h@+h=gon@;h(x)=O(jxj1n)ifjxj!1;(2.45)whereak=1+(k1)˜(D).LetV1(x):=S@g(x)+D@f(x)+S@D'(x)forx2Rn:WebeginbyshowingV1isaweaksolutionto(2.45)inthesenseofthefollowing2.15.Wesayvisaweaksolutionto(2.45)ifthefollowingidentitieshold:Zak(x)rvrdx=082C10;andZRnnrvr~dx=08~2C10(Rnn:LetusverifythejumpandthedecayconditionsforV1.BythecontinuityofthesinglelayerpotentialS@gacross@smoothnessofS@D'on@andthejumpconditionD@f(x)=12I+K@f(x);x2@;41itfollowsthatV1(x)V+1(x)=f(x)forx2@:Similarly,bythecontinuityofthenormalderivativeofthedoublelayerpotentialD@facross@smoothnessofS@D'on@andthejumprelationforthenormalderivativeofthesinglelayerpotential@S@g(x)=12I+K@g(x);x2@;wehave@V1@+V1=gon@:Fromtheofsinglelayerpotential,wehaveS@g(x)=Z@x;y)g(y)d˙(y):Sinceg2L20(@wegetS@g(x)=Z@x;y)x;y0)]g(y)d˙(y);fory02Sincejx;y)x;y0)jCjxj1nwhenjxj!1andy2forsomeconstantC.ThereforeS@g(x)=O(jxj1n)asjxj!1:42Similarly,wehaveS@D'(x)=O(jxj1n)asjxj!1:Obviouslyfromtheonofthedoublelayerpotential,weseethatD@f(x)=O(jxj1n)asjxj!1:ThereforeV1(x)=Ojxj1nwhenjxj!1:NowweshowV1isaweaksolutionto(2.45)inthesenseofthe2.15.SinceV1isharmonicinRnnthenitisclearthatZRnnrV1r~dx=0;8~2C10Rnn:For2C10wehaveZak(x)rV1rdx=ZnDrV1rdx+kZDrV1rdx:SinceV1isharmonicinsidenDandalsoisharmonicinsideD,thenitfollowsthatZak(x)rV1rdx=kZ@D@V1d˙Z@D@+V1d˙:Thatis,Zak(x)rV1rdx=kZ@D@H+@S@D'd˙Z@D@H+@+S@D'd˙:43Notethatifk@H(x)+@S@D'(x)=@H(x)+@+S@D'(x)forx2@D;(2.46)thenwehaveZak(x)rV1rdx=0;82C10:Bysubstitutingthejumpconditions(2.23)inequation(2.46),weobtainthefollowing(k1)@H(x)+k12I+K@D'(x)12I+K@D'(x)=0forx2@D;thatis@H=k+12(k1)IK@D'on@D:(2.47)ThuswehaveshownthatZak(x)rV1rdx=0;82C10;ifandonlyif(2.47)holds.ThereforeV1isaweaksolutionto(2.45).Now,weV2(x)=8>>><>>>:u(x)ifx2;0ifx2Rnn:ThenV2isalsoaweaksolutionto(2.45).Thereforeinordertoprovetherepresentationformula(2.42),ittoshow(2.45)hasuniquesolutioninW1;2loc(Rnn@Forthat,44supposew2W1;2loc(Rnn@isaweaksolutionto(2.45)withf=g=0.ThusweclearlyseethatwiscontinuousonRnandharmonicinRnnD.Thuswisaweaksolutionto(2.45)intheentiredomainRn.ForalargeR,wehaveZBR(0)jrwj2dx1+kkZBR(0)1+(k1)˜(D)jrwj2dx=1+kkZ@BR(0)w@wd˙(byofweaksolution)=1+kkZRnnBR(0)jrwj2dx0(byLemma(2.13)):ThisinequalityholdsforallRandhencewisconstant.Sincew(x)!0aty,weconcludethatw0.Toprovetheuniquenessoftherepresentationformula(2.42),supposethatH0isharmonicinandH+S@D'=H0+S@D'0inThenS@D(''0)isharmonicinandhence@S@D(''0)=@+S@D(''0)on@D:Itfollowsfrom(2.23)that''0=0on@DandthenH=H0.ByusingTheorem2.14,theharmonicpartsH0andHofu0anducanberespectivelyrepresentedasH0(x)=S@g(x)+D@(u0j@)(x);x2;(2.48)H(x)=S@g(x)+D@(uj@)(x);x2:(2.49)ThefollowinglemmashowsthatHisuniformlyboundedindependentlyofinanynormaswellasHapproachesH0whenapproaches0inanycompactsubsetofItwasprovedin[1]fordimensionn=2andtheproofisstillvalidforanydimensionn2.45Lemma2.16.Let0>0andWˆˆ,suchthatD1[D2ˆW,forall<0.Thenforallm=0;1;2;3;:::thereexistsC=C(n;m;k;;d)whered=dist(@;W)suchthatkHkCm(W)CkgkL2(@8<0;(2.50)and,lim!0kHH0kCm(W)=0:(2.51)Proof.From(2.48)and(2.49),weseethatHH0=D@(uj@)D@(u0j@);whereD@(uj@)=1!nZ@(yx)yjxyjnu(y)d˙(y);andD@(u0j@)=1!nZ@(yx)yjxyjnu0(y)d˙(y):SinceWˆˆthenitisclearthatkHH0kCmWCkuu0kL2(@:Bytracetheorem,wehavekHH0kCmWCkuu0kH1:(2.52)46FromTheorem2.5,itfollowsthatlim!0kHH0kCmW=0:Nowweshowtheuniformbound(2.50).FromtheofH,weclearlyseethatkHkCmWCkgkL2(@+kukL2(@:(2.53)Byapplyingtracetheorem,wegetkHkCmWCkgkL2(@+kukH1:(2.54)ToprovekukH1isuniformlyboundedbykgkL2(@,weusethebilinearforminequality(2.3)forHtodeducekuk2H1CZ@gud˙(x)CkgkL2(@kukH1C14kgk2L2(@+kuk2H1:ChoosingC<1inthelastinequality,itfollowsthatkukH1CkgkL2(@(2.55)whereCisindependentof.Thusbysubstituting(2.55)in(2.54),theuniformboundfollows.472.6Piecewiseoldercontinuityforthemodif-ferentialequation.For>0,wepotentialfunctions'1and'2respectivelyontheboundariesofD1andD2by'1(x)=@u+@ux2enforx21;(2.56)and'2(x)=@u+@ux+2enforx22:(2.57)TosimplifythenotationofthesinglelayerpotentialontheboundariesofD1andD2,wewriteS1'1(x)=Z1xy)'1(y)d˙(y);andS2'2(x)=Z2xy)'2(y)d˙(y);Itwasgivenin[1]therepresentationformulafortheapproximatedsolutionuto(2.5)whichworksforanydimensionn2.Theorem2.17.Thesolutionofproblem(2.5)canbeuniquelyrepresentedasu(x)=H(x)+S1'1(x)+S2'2(x);x2;(2.58)whereH(x)=S@g(x)+D@(uj@)(x);x2;48and'1and'2solvethefollowingsystemofintegralequationsK1'1(x)@S2'2(xen)=@Hx2en;x21;@S1'1x+en+(K2)'2(x)=@Hx+2en;x22;(2.59)inthissystem=k+12(k1),andKidenotestheoperatorKi'i(x)=1!nZi(xy)(x)jxyjn'i(y)d˙(y);i=1;2:Proof.For2C10,wehaveZ1+(k1)˜D1[D2rH+S1'1+S2'2rdx=ZnD1[D2rH+S1'1+S2'2rdx+kZD1[D2rH+S1'1+S2'2rdx=Z@D1[@D2@@+H+S1'1+S2'2+k@@H+S1'1+S2'2d˙=Z@D1@@+H+S1'1+S2'2+k@@H+S1'1+S2'2d˙+Z@D2@@+H+S1'1+S2'2+k@@H+S1'1+S2'2d˙=(k1)Z@D1@Hk+12(k1)I+K1'1+@S2'2d˙+(k1)Z@D2@Hk+12(k1)I+K2'2+@S1'1d˙by(2.23):If'1;'2solves(2.59),thenwegetZ1+(k1)˜D1[D2rH+S1'1+S2'2rdx=0;82C10:49Thereforeuisweaksolutionof(2.5).ThefollowingLemmaisveryusefulanditwillbeusedtogivethepiecewiseolderestimateforthemotialequation(2.5).Theproofcanbefoundin[15,21].Lemma2.18.LetˆRnbeaboundeddomainandletDˆˆbeaC10subdomain.ThederivativesofasinglelayerpotentialS@D'(x)=Z@Dx;y)'(y)d˙(y);x2;witholderdensityfunction'2C(@D),0<<0canbeuniformlyextendedinaoldercontinuousfashionfromnDintonDandfromDintoDwithlimitingvalues@S@D'j(x)=12I+K@D'(x);x2@D:(2.60)Furthermore,wehavefor0<theestimatekS@D'k10D+kS@D'k10nD)Ck'k@D:(2.61)ApplyingLemma2.18forthepotentials'1and'2thatsolve(2.59),weimmediatelyhaveforany0<0<<0thefollowingestimateSi'i10Di+Si'i10nDiC'i(i);i=1;2;(2.62)whereCisindependentof.Fromtherepresentationformula(2.58),Lemma(2.16)andtheestimate(2.62),weobtain50Lemma2.19.Letubethesolutionto(2.5)andlet'1;'2bethesolutionto(2.59).Foranysmall~>0,let~denotestheset~=fx2:dist(x;@>~g.Thenforany0<0<<0wehavetheboundkuk10D1+kuk10D2+kuk10~nD1[D2C 2Xi=1'ii+kgkL2(@!;(2.63)forsomeconstantCdependingon,0,0,,k,and~butindependentof.AsaconsequenceofthisLemma,weobtainthedesiredpiecewiseoldercontinuityforruoneachcomponentofiftherighthandsideof(2.63)canbeshowntobeuniformlyboundedindependentlyof.Thatis,ifweprove2Xi=1'ii)CkgkL2(@;(2.64)whereCisindependentof,thenweobtainthedesiredboundforru.Theuniformbound(2.64)dependsonthesolvabilityofthesystem(2.59)andthiswillbeourtaskinthenextchapters.51Chapter3Decompositionofthesystemofintegralequations.Ourgoalistosolvethesystem(2.59)withboundsuniformin.Tothatend,letustheoperatoronC1)C2)byT0B@'1'21CA:=0B@K1L2L1K21CA0B@'1'21CA;whereL2'2(x)=@S2'2xen;x21;(3.1)andL1'1(x)=@S1'1(x+en;x22:(3.2)Thusthesystem(2.59)canbewrittenasT0B@'1'21CA=0B@@Hy1(x;)@Hy2(x;)1CA;(3.3)wherey1(x;)=x2en;x21;52andy2(x;)=x+2en;x22:FromLemma2.16wehavethattherighthandsideof(3.3)isuniformlyboundedinC0normonthesurfacesj,j=1;2.Thatisk@HkC0j)CkgkL2(@:Thereforetoshowtheuniformbound(2.64),weneedtoshowthattheoperatorTisinvertibleasanoperatoronC1)C2)anditsinverseisuniformlyboundedinC1)C2),forany<0.ThusthepurposeofthischapteristostudythebehavioroftheTandhowtoovercomethesingularityofitskernelwhenapproaches0.3.1BehaviorofT,>0.Recallthat=k+12(k1),where0<k<1andk6=1.Thatis,weconcludethatjj>12.Theorem3.1.For>0,TiscontinuouslinearoperatoronC1)C2),invertiblewithboundedinverseforany0<<0andforanyjj>12.Proof.For>0,weevidentlyseethatTisboundedlinearoperatoronC1)C2).ForI=0B@OO1CAandK=0B@K1L2L1K21CA;TcanbewrittenasT=I+K:SinceKisacompactoperatoronC1)C2),thenTisFredholmoperator.Therefore53theinvertibilityofTfollowsifweshowthatTisinjective.Let('1;'2)2C1)C2)besuchthatT0B@'1'21CA=0B@001CA.Thatis,8>>><>>>:K1'1(x)@S2'2(xen)=0;x21;@S1'1(x+en)+K2'2(x)=0;x22:(3.4)Orequivalently,8>>><>>>:K1'1x+2en@S2'2(x2en)=0;x2@D1;@S1'1(x+2en)+K2'2(x2en)=0;x2@D2:(3.5)ConsiderthefunctionwonRnbyw=S1'1+S2'2:(3.6)Weclaimthatwisaweaksolutionofthefollowingproblem8>>><>>>:div1+(k1)˜(D1[D2)rw=0;x2Rn;w(x)=O(jxj1n);jxj!1:(3.7)Toprovetheclaim,let2C10(BR(0))whereBR(0)˙˙Forak=1+(k1)˜(D1[D2),54weseethefollowingZBR(0)akrwrdx=kZD1rS1'1(x+2en)+S2'2(x2en)rdx+kZD2rS1'1(x+2en)+S2'2(x2en)rdx+ZBR(0)n(D1[D2)rS1'1(x+2en)+S2'2(x2en)rdx:Usingthedivergencetheorem,itfollowsZBR(0)akrwrdx=kZ@D1@S1'1(x+2en)+@S2'2(x2en)d˙(x)+kZD2@S1'1(x+2en)+@S2'2(x2en)d˙(x)Z@D1@+S1'1(x+2en)+@S2'2(x2en)d˙(x)ZD2@S1'1(x+2en)+@+S2'2(x2en)d˙(x):Utilizingthejumpconditionsforthesinglelayerpotential(2.23),wehaveZBR(0)akrwrdx=(k1)Z@D1(K1)'1(x+2en)@S2'2(x2en)d˙(x)+(k1)Z@D2(K2)'2(x2en)@S1'1(x+2en)d˙(x):Byapplying(3.4),weobtainthefollowingZBR(0)akrwrdx=0:55Inordertogetthedecayconditionforw,itisenoughtoshowthefollowingidentityZ1'1(+en)d˙=Z2'2(en)d˙=0;(3.8)since,ifRj'jd˙(y)=0,thenSj'j=Zjx;y)'jd˙(y)=Zjx;y)x;y0)'jd˙(y)Cjjxj1n;(3.9)wherey02Dj.Sincejx;y)x;y0)jCjxj1nifjxj!1andy2j:WeonlyshowR1'1(+en)d˙=0andsameworkwouldholdforR2'2(en)d˙=0.SinceS2'2(en)isharmonicinD1,thereforewegetZ1@S2'2(xen)d˙(x)=0:Thereforebyusing(3.4),itfollowsthat0=Z1(K1)'1(x+en)@S2'2(xen)d˙(x):Thatis,Z1(K1)'1(x+en)d˙(x)=0:56Thusbytheduality(remark2.10),weobtainthefollowingZ11(x+en)K1(1)'1(x+en)d˙(x)=(12)Z1'1(x+en)d˙(x)=0:Sincejj>12,thenitfollowsthatZ1'1(x+en)d˙(x)=0:Weshowthatw0inRn.ForR>>1,wehaveZBR(0)jrwj2dx1+kkZBR(0)1+(k1)˜(D1[D2)jrwj2dx=1+kkZ@BR(0)w@wd˙(x)(byofweaksolution)=1+kkZRnnBR(0)jrwj2dx(bylemma2.13)0:ThuswisconstantinRn.Usingthedecayconditionaty,itfollowsthatw0inRn,thatisS1'1(x+2en)+S2'2(x2en)=0;x2Rn:Consequently,weconcludethatS1'1issmoothacross1andS2'2issmoothacross2.Thereforeapplyingthejumpconditionsforthesinglelayerpotential(2.23),wehave'1(x)=@+S1'1(x)@S1'1(x)=0;x21;'2(x)=@+S2'2(x)@S2'2(x)=0;x22:57ThusTisinjective.3.2AuxiliaryfunctionsandBasicproperties.Asin[1],westudythebehaviorofTanditsinverseasapproaches0andweusethesamenotationsthattheyusedintheirpaper.WhenthesubdomainsD1andD2approachthetouchingpointx=0,thekerneloftheoperators@S2'2(xen)=Z2(xeny)jxenyjn'2(y)d˙(y);x21;and@S1'1(xen)=Z2(xeny)jxenyjn'1(y)d˙(y);x22;becomesingularatx=0.Fordimensionn=2[1]thesingularitywasovercomebydecomposingTasasum+Cwhere,foraxed>0tlysmall,theoperatorcontainsthesingularpartofT(i.e,theidentityplusapieceoftheagonalterms),andCiscompact.Theirideaworksforanydimensionn2withsomenecessarytechnicalmoSowearegoingtomimictheirideatoovercomethesingularityforgeneraldimensionn2.Weasmallparameter0<0<1sothat12<1+02<jj;(3.10)where=k+12(k1)andrecallingthattheconstantkistheconductivityinthesubdomainsD1andD2whiletheconstant1istheconductivityinn(D1\D2).LetR0=2(1+10),58r1=diam(D1),r2=diam(D2)and%=20(1+r1+r2)R0wherediam(Di)isthediameterofthesubdomainDi,i=1;2.Werescalethedomainbyassumingx=%xtomakesurethatthesurface1[2withintheballB(2R0)meetsxn-axisat0only.LetbeasmoothfunctioninRnthatsupportedintheballB(R0)besuchthat8>>>>>>>><>>>>>>>>:01;1;x2B(0);krk10:(3.11)Wealsoassumethat0isientlysmallsothataroundthetouchingpointx=0,thesurfaces1and2canbeparametrizedby8>>><>>>:x00!x=x0; 1(x0)21;y00!y=(y0; 2(y0))22;(3.12)forsomeC10parametricfunctions 1and 2.3.2.[1]AclosedandboundedsurfaceˆRniscalledofregularityC1ifitcanbecoveredbyalocalsetofcharts j:x2B0jˆRn1! j;1(x);; j;n(x)ˆRn;whereB0j;1jm,areopenballsinRn1and j;i;1inareC1BjfunctionswithRank(r j)=Rankr 1;;r n=n1.Wesaythatacontinuousfunctionf59isofregularityC0ifforanyofthelocalchartsf jC0Bj:=supx0;y02Bj;jx0y0j<1f j;1(x0);; j;n(x0)f j;1(y0);; j;n(y0)jx0y0jC:ThenormonC0isdbykfk0=maxkfkL1;max1jmf jC0Bj:Lemma3.3.[1]Given0<0<1forwhich(3.12)holdswith(3.12)asoneoflocalcoordinatechart,andgiven0<<1;thereexistsanoperatorE:C2)!C(Rn1);suchthatforany'2C2),wehave8>>>>>>>><>>>>>>>>:kE'k;Rn1(1+0)k'k;2E'(y0)='y0; 2(y0);y00;supp(E')ˆB20:(3.13)Proof.For'2C2),let~'beafunctiononRn1thatby~'(y0)=8>>><>>>:'(y0; 2(y0));ify00;'y; 2(y);ify0>0;wherey=0(y0)and(y0)isthenormalunitvectory0jy0j.Weshow~'isColder60continuousonRn1.Lety01;y022Rn1.Case1:Wheny010andy020.Wetriviallyseethat~'(y01)~'(y02)y01y02='(y01; 2(y01))'y02; 2(y02)y01y02k'k:Case2:Wheny010andy02>0.Wehave~'(y01)~'(y02)y01y02='(y01; 2(y01))'y2; 2y2y01y02:Sincey01y2y01y02,thenweobtain~'(y01)~'(y02)y01y02'(y01; 2(y01))'y2; 2y2y01y2k'k:Case3:Wheny01>0andy02>0.Weshowthatjy1y2jy01y02:(3.14)Theaboveinequalitywouldbetrivialifthedotproducty01y020.Thenitisenoughtoshow(3.14)wheny01y02>0.Withoutlossofgeneralitywemayassumethaty02y01andlet~y2=y012(y0).Weclearlyseethatjy1y2j=0y01y01~y2<y01~y2y01y02:Wherewenaivelycanseethatvalidityoflastinequalityifweassumeafterrotationthat61y01=(0;0;;0;y1;n1).Nowweformthefollowing~'(y01)~'(y02)y01y02='y1; (y1)'y2; (y2)y01y02:Bytakingaccountof(3.14),weobtain~'(y01)~'(y02)y01y02'y1; (y1)'y2; (y2)y1y2k'k:Thuswehaveshownthatk~'kk'k:(3.15)Next,letˆ2C1(Rn1)besuchthat0ˆ1and8>>>>>>>><>>>>>>>>:ˆ(y0)=1;ify00;krˆk0;supp(ˆ)2B20:(3.16)E'(y0)=ˆ(y0)~'(y0).ItisclearthatkE'k1k'k1.62NowweshowC-normforE'.Fory01;y012Rn1wehavesupjy01y02j<1y016=y02E'(y01)E'(y02)y01y02=supjy01y02j<1y016=y02ˆ(y01)~'(y01)ˆ(y02)~'(y02)y01y02supjy01y02j<1y016=y02"kˆk1~'(y01)~'(y02)y01y02+k~'k1ˆ(y01)ˆ(y02)y01y02#k'k+0k'k(1+0)k'k:ThereforekE'k;Rn1(1+0)k'k;2.NextweetwoC1auxiliaryfunctions 1and 2thatgloballyonRn1besuchthat j1;Rn1=O();j=1;2forevery<0and=0.Lemma3.4.Let<0,=0and0<2<0.ThereexistC1-functions 1and 2donRn1sothat8>>><>>>: j= j;x0j=1;2; j1;Rn1C j10;j=1;2:(3.17)whereCisindependentof.Proof.Recallthatthefunctions j2C10(B00)arelocallytosatisfy j(x0)=r j(x0)=0onlyatx0=0,j=1;2and 2isnon-negativewhile 1isnon-positive.Thenforanyx02B0(0),wehave j(x0)Cx01+0 j10;j=1;2:(3.18)63For<120,let2C1c(Rn1)beafunctionthatsatisfyingthefollowing8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:01;(x0)=1;ifx0(x0)=0;ifx02krk1=O(1);r21=O(2);(3.19)wherer2istheHessianmatrixof.Forany0<<0,wede 2(x0)= 2(x0)(x0)+1(x0)1+0k 2k10; 1(x0)= 1(x0)(x0)1(x0)1+0k 1k10:(3.20)Clearlyweseethat jaregloballyC10and j(x0)= j(x0)forx02B0(),j=1;2.Thenweonlyneedtoshowthebound j1;Rn1Ck 2k10;8<0;j=1;2:(3.21)Weprove(3.21)forj=2andthesameargumentswouldholdwhenj=1.First,weshowtheL1normfor 2.Inthecasex0,wehave 2(x0)= 2(x0)andthenbyusing(3.18)weobtainthefollowing 2(x)Ck 2k10jxj1+0Ck 2k101+0Ck 2k10:64Thatis 2(x0)Ck 2k10:(3.22)Forthecase<x0<2,wehave 2(x0)= 2(x0)(x0)+(1(x0))1+0k 2k10:Againbyusing(3.18)itfollowsthat 2(x0)Ck 2k10x01+0+21+0k 2k10:Thuswehave 2(x0)Ck 2k10:(3.23)Forthelastcasex02,wehave 2(x0)=1+0k 2k10.Thentriviallyweget 2(x0)Ck 2k10:(3.24)Combining(3.22),(3.23),and(3.24),itfollowsthat 21Ck 2k10:(3.25)65WeproceedtoshowtheboundednessofL1normforr 2.Forthatwehaver 2(x0)=r 2(x)(x0)+ 2(x)r(x0)1+0k 2k10r(x):Whenx0,wehaver 2(x0)=r 2(x0)(x0)andsincer 2(0)=0,thenwegetr 2(x0)=r 2(x0)r 2(0)k 2k10jxj0k 2k100:Thereforer 2(x0)Ck 2k10:(3.26)Forthecase<x0<2,weseethefollowingr 2(x0)r 2(x0)+C 2(x0)+C0k 2k10:(3.27)Since 2(0)=r 2(0)=0,thenitfollowsthatr 2(x0)Ck 2k10:(3.28)Finally,whenjxj2,thenr 2(x)=0andtriviallywegettheboundr 2(x0)Ck 2k10:(3.29)66Thuswehaveshowntheboundr 21Ck 2k10:(3.30)Fortheolderestimateofr 2,wehavesevencasesandweelaboratethesecasesasfollows:1.Ifx0andy0,wehaver 2(x0)r 2(y0)jx0y0j=r 2(x0)r 2(y0)jx0y0jx0y00k 2k10:Thusweconcludethatr 2(x0)r 2(y0)jx0y0jCk 2k10:(3.31)2.If<x0<2and<y0<2,wehaver 2(x0)r 2(y0)jx0y0jr 2(x0)(x0)r 2(y0)(y0)jx0y0j+ 2(x0)r(x0) 2(y0)r(y0)jx0y0j+1+0k 2k10r(x0)r(y0)jx0y0j:Weestimateeachtermintherighthandsideoftheaboveinequalityseparately.LetjAj=r 2(x0)(x0)r 2(y0)(y)jx0y0j:67ThenweseethefollowingjAjr 2(x0)(x0)r 2(y0)(x0)jx0y0j+r 2(y0)(x0)r 2(y0)(y)jx0y0jx0y00r 2(x0)r 2(y0)jx0y0j0+r 2(y0)(x0)(y0)jx0y0j:Thusforsome˘0liesonthelinethatjoiningx0andy0,andsincer 2(0)=0,wehavejAjCk 2k10+k 2k10y00x0y01r(˘0):ThereforejAjCk k10:(3.32)Similarly,letjBj= 2(x0)r(x0) 2(y0)r(y0)jx0y0j:ThenwehavejBj 2(x0)r(x0) 2(y0)r(x0)jx0y0j+ 2(y0)r(x0) 2(y0)r(y0)jx0yj:Forsome˘01thatjoiningx0andy0,weobtainthefollowingboundjBjC 2(x0) 2(y0)jx0y0j+ 2(y0)x0y0r2(˘01)jx0y0j:68Againforsome˘02thatjoiningx0toy0,weobtainthefollowingboundjBjCr 2(˘02)x0y0jx0y0j+C2k 2k10y01+0x0y01:ThatisjBjCk 2k10:(3.33)Forthelastterm,wehaveforsome˘0thatjoiningx0andy0thefollowingbound1+0k 2k10r(x0)r(y0)jx0y0j1+0k k10x0y01r2(˘0):Thatis1+0k 2k10r(x0)r(y0)jx0y0jCk 2k10:(3.34)3.Forthecasex02andy02,wehaver 2(x0)=0andr 2(y0)=0.Thedesiredboundfollowstrivially.4.Ifx0and<y0<2,wehaver 2(x0)=r 2(x0)andr 2(y0)=r 2(y0)(y0)+ 2(y0)r(y0)1+0k 2k10r(y0):69Thenwehavethefollowingboundr 2(x0)r 2(y0)jx0y0jr 2(x0)r 2(y0)(y0)jx0y0j+ 2(y0)r(y0)jx0y0j+1+0k 2k10r(y0)jx0y0jSincer(x0)=0,thenweclearlyseethatthelasttwotermsontherighthandsideoftheaboveinequalitysatisfythedesiredbound.Forthetermwehave(x0)=1thenweformthefollowingboundr 2(x0)r 2(y0)(y0)jx0y0jr 2(x0)(x0)(y0)jx0y0j+(y0)r 2(x0)r 2(y0)jx0y0j:Thusforsome˘0liesonthelinesegmentthatjoiningx0andy0,weobtaintheboundr 2(x0)r 2(y0)(y0)jx0y0jx0y01x00k 2k10r(˘0)+x0y00k 2k10:Thereforeweobtaintheboundr 2(x0)r 2(y0)(y0)jx0y0jCk 2k10:5.Whenx0andy02,wehaver 2(x0)=r 2(x0)andr 2(y0)=0.Thuswehavetheboundr 2(x0)r 2(y0)jx0y0j=r 2(x0)jx0y0jCx00k 2k10:70Thatisr 2(x0)r 2(y0)jx0y0jCk 2k10:(3.35)6.When<x0<2and2<y0<3,wehaver 2(y0)=0.Thenwehaver 2(x0)r 2(y0)jx0y0j=r 2(x0)(x0)+ 2(x)r(x0)1+0k 2k10r(x0)jx0y0j:Since(y0)=r(y0)=0,thenweobtainthefollowingboundr 2(x0)r 2(y0)jx0y0jr 2(x0)(x0)r 2(x0)(y0)jx0y0j+ 2(x0)r(x0)r(y0)jx0y0j+1+0k 2k10r(x0)r(y0)jx0y0j:Thusforsome˘01and˘02thatlieonthelinesegmentjoiningx0andy0weobtainthefollowingboundr 2(x0)r 2(y0)jx0y0jx00k 2k10x0y0r(˘01)jx0y0j+Cx01+0k 2k10x0y0r2(˘02)jx0y0j+1+0k 2k10x0y0r2(˘02)jx0y0j:Thereforewehavethedesiredboundr 2(x0)r 2(y0)jx0y0jCk 2k10:(3.36)717.If<x0<2andy03,thenr 2(y0)=0andwehavethefollowingr 2(x0)r 2(y0)jx0y0jr 2(x0)jx0y0j+ 2(x0)r(x0)jx0y0j+1+0k 2k10r(x0)jx0y0j:Thatisr 2(x0)r 2(y0)jx0y0jCk 2k10:(3.37)Thuswehaveshownthefollowingboundsupx0;y02Rn1x06=y0r 2(x0)r 2(y0)jx0y0jCk 2k10:(3.38)ThereforetheLemmafollowsbycombining(3.25),(3.30)and(3.38).ThefollowingLemmawillbeneededforthedecompositionoftheoperatorL2anditwillbeusedasatooltoextendtheintegraloverthesurface2tothewholespaceRn1.Itwasstatedin[1]fordimensionn=2.Theproofiselementary.Lemma3.5.Let0<<0,for'2C2)we˚(y0)=E'(y0)q1+r 2(y0)2;y02Rn1:Then˚2C(Rn1)andk˚k;Rn1(1+C())(1+0)k'k2;(3.39)where0istheregularityofthesurface2,E'isdinLemma3.3andC()!0as72approaches0.Proof.First,weseethat˚iswonRn1becauseE'hascompactsupportinB(20).FromLemma3.3,wehavetheboundkE'k;Rn1(1+0)k'k2:(3.40)AlsofromLemma3.4wehaveforany<0that 21=O()where=0,thenforanyx02Rn1,itfollowsthat˚(x0)E'(x0)1+C():Thatisk˚k11+C()(1+0)k'k:(3.41)Nextforx0;y02Rn1,weestimate˚(x0)˚(y0)jx0y0j=E'(x0)q1+r 2(x0)2E'(y0)q1+r 2(y0)2jx0y0jE'(x0)E'(y0)q1+r 2(x0)2jx0y0j+E'(y0)q1+r 2(x0)2q1+r 2(y0)2jx0y0j:Byusing(3.40),weeasilyseethatE'(x0)E'(y0)q1+r 2(x0)2jx0y0j1+C()(1+0)k'k:(3.42)73Forthesecondterm,letI2=E'(y0)q1+r 2(x0)2q1+r 2(y0)2jx0y0j:ThenwehaveI2E'(y0)r 2(x0)r 2(y0)r 2(x0)+r 2(y0)jx0y0jq1+r 2(x0)2+q1+r 2(y0)2:Thatis,I2E'(y0)r 2(x0)+r 2(y0)q1+r 2(x0)2+q1+r 2(y0)2r 2(x0)r 2(y0)jx0y0j:ThereforeweobtainthefollowingboundI2C(1+0)k'k 210:(3.43)Thusthebound(3.39)followsbycombining(3.41),(3.42)and(3.43).3.3DecompositionofL2InthissectionweelaborateamethodtodecomposetheoperatorL2inordertoovercomethesingularitywhenx=0and=0.Themethodwasintroducedin[1]fordimensionn=2.WestartwiththefollowingLemmawhichwillbeusedfrequentlyduringourwork.74Lemma3.6.Forn2,wehaveZRn1dz0jz0j2+1n2=!n2;(3.44)where!nisthesurfaceareaofunitsphere@B(1)inRn.Proof.Thesurfacearea!nisrepresentedinpolarcoordinatesbythefollowingform!n=Zˇ˚1=0:::Zˇ˚n2=0Z2ˇ˚n1=0(sin˚1)n2:::(sin˚n3)2(sin˚n2)d˚n1d˚n2d˚1:Thenwecanseethat!n=!n1Zˇ0(sin˚1)n2d˚1;(3.45)where!n1isthesurfaceareaofunitsphere@B(1)inRn1.Thusbyusingthepolarcoordinates,thelefthandsideof(3.44)becomesZRn1dz0jz0j2+1n2=!n1Z10rn2r2+1n2dr:(3.46)Substitutingr=tan˚1intherighthandsideof(3.46),wehaveZRn1dz0jz0j2+1n2=!n1Zˇ20(sin˚1)n2d˚1=!n2Tobegin,let>0and'2C2).Forx21,jxj<R0,wesetx=(x0; 1(x0))and75wetheapproximatesurface2=fy=(y0; 2(y0))jy02Rn1g:TheoperatorL2isbyL2'(x)=@S2'(xen):Letbethefunctionin(3.11).ThenwedecomposetheoperatorL2asfollowsL2'(x)=(x)L2'(x)+1(x)L2'(x):(3.47)Noticethattheterm(1(x))L2'(x)hassmoothkernel.Whereastheterm(x)L2'(x)issingularwhenx=0and=0.Tosimplifythenotation,wedeL2(x):=(x)L2'(x);x21;'2C2):NextwedecomposetheoperatorL2asfollows:L2(x)=L2(x)+(x)@S2E'P(xen)(x)@S2E'P(xen);(3.48)wherePistheprojectionmapfrom2ontoRn1,thatis8>>><>>>:P:2!Rn1;Py0; 2(y0)=y0;(3.49)7622x02211Figure3.1:TheapproximatesurfacesandS2isthesinglelayerpotentialontheapproximatesurface2.FromthesmoothnessofS2on1wehave(x)@S2E'P(xen)=(x)!nZ2(xyen)(x)jxyenjnE'P(y)d˙(y);(3.50)where(x)isthenormalunitvectorontheapproximatesurface1atthepointx,thatis,(x)=1q1+r 1(x0)20B@ 1(x0)11CA:(3.51)77Werewrite@S2forthepotentialE=E'Patthepoint~x=xenasfollows(x)@S2E(~x)=~C(x0)(x)ZRn1(y0x0)r 1(x0)+ 1(x0) 2(y0)jx0y0j2+( 1(x0) 2(y0))2n=2˚(y0)dy0;(3.52)where˚isdinLemma3.5by˚(y0)=E'(y0)q1+r 2(y0)2and~C(x0)=1!nr1+r 1(x0)2.Now,we(x)K2'(x)=(x)@S2'(xen)+(x)@S2E(~x):(3.53)Wewillprovelaterthatoperatorsin(3.53)formafamilyofcompactoperatorsfromC2)toC1).Alsobyusingtheofthefunction˚wecanrewritetheoperatorsK2asfollows(x)K2'(x)=(x)!nZ2(xyen)(x)jxyenjn'(y)d˙(y);+~C(x)ZRn1(y0x0)r 1(x0)+ 1(x0) 2(y0)jx0y0j2+( 1(x0) 2(y0))2n=2˚(y0)dy0:Fromtheparameterizationofthesurfaces1and2aroundtheorigin,wehavethatx=(x0; 1(x0))21forx0<,andsimilarly,y=(y0; 2(y0))22wheny0<.Thereforeweconcludethatforx0<andy0<thetwointegrandsintheof(x)K2coincide.Wefurtherforeachx=(x0; 1(x0))21thehyper-planex02=fy2Rnjyn= 2(x0)g:(3.54)78Thuswerewrite(x)L2'(x)asfollows(x)L2'(x)=(x)K2'(x)(x)@S2E(~x)+(x)@Sx02E(~x)(x)@Sx02E(~x);(3.55)whereSx02isthesinglelayerpotentialonthehyper-surfacex02anditsnormalderivativeforthepotentialEisby(x)@Sx02E(~x)=(x)!nZx02xy(x0)en(x)jxy(x0)enjnE(y(x0))d˙(y(x0));(3.56)wherey(x0)=(y0; 2(x0)).Tobemoreexplicit,(x)@Sx02E(~x)=~C(x)ZRn1(y0x0)r 1(x0)+ 1(x0) 2(x0)jx0y0j2+ 1(x0) 2(x0)2n2˚(y0)dy0:(3.57)Notethatwehaveperturbedbytheoperator(x)@Sx02E(~x)inordertogettheoldercontinuityfortheoperator(x)@S2E(~x).Thatis,wewillusetheadvantageofthe 2(x0) 2(y0)togettheoldercontinuityfortheterm(x)@Sx02E(~x)(x)@S2E(~x):Whiletheterm(x)@Sx02E(~x)willbeeasilyprovedisoldercontinuousbecauseitisproducedfromthesinglelayerpotentialoverthehyper-planex02.79For13x6=0or6=0and'2C2)wetheoperatorI2'(x)=ZRn1(+ 2(x0) 1(x0))r 1(x0)(y0x0)jx0y0j2+ 1(x0) 2(x0)2n2˚(y0)dy0:(3.58)whereasforx=0and=0,weI02'(0)asfollowsI02'(0)=lim!0ZRn1jy0j2+2n2˚(y0)dy0:(3.59)Usingthechangeofvariablez0=y0,wehaveI02'(0)=lim!0ZRn11jz0j2+1n2˚(z0)dz0:(3.60)Fromthedominatedconvergencetheorem,itfollowsthatI02'(0)=ZRn11jz0j2+1n2˚(0)dz0=!n2'(0);(3.61)where˚(0)='(0)and!n1isthesurfaceareaofunitsphere@B(1)inRn1.Againtosimplifythenotation,letJ2(x)=(x)@S2E(~x)+(x)@Sx02E(~x):80From(3.52)and(3.57),wehaveJ2(x)=~C(x0)(x)ZRn1+ 2(y0) 1(x0)r 1(x0)(y0x0)jx0y0j2+ 1(x0) 2(y0)2n2+ 2(x0) 1(x0)r 1(x0)(y0x0)jx0y0j2+ 1(x0) 2(x0)2n2˚(y0)dy0:Forx6=0or6=0and'2C2),wetheoperatorJ2'(x)=ZRn1(+ 2(y0) 1(x0))r 1(x0)(y0x0)jx0y0j2+ 1(x0) 2(y0)2n2(+ 2(x0) 1(x0))r 1(x0)(y0x0)jx0y0j2+ 1(x0) 2(x0)2n2˚(y0)dy0;(3.62)whereas,whenx=0and=0,wesetJ02'(0)=ZRn1 2(y0)y21+:::+y2n1+ 22(y0)n2˚(y0)dy0;(3.63)wherethisintegraliswell-becausewheny0<,wehave 2(y0)Cy01+0.Thuswewrite(x)L2as(x)L2'(x)=(x)K2'(x)+(x)!nq1+r 1(x0)2J2'(x)+I2'(x):(3.64)81UsingtheabovewehaveL2'(x)=(x)0@K2+1!nq1+r 1(x0)2J2+I21A'(x)+(1(x))L2'(x):(3.65)Inasimilarmanner,weoperators(x)K1,(x)J1and(x)I1fromC1)intoC2),thathelptodecomposetheoperatorL1.Thatis,wedecomposetheoperatorL1asfollows:For>0,x22,jxj<R0and'2C1)weformthefollowing:L1'(x)=(x)L1'(x)+(1(x))L1'(x)(3.66)=(x)0@K1+1!nq1+r 2(x0)2J1+I11A'(x)+(1(x))L1'(x):(3.67)3.4DecompositionofT.AfterdecomposingthediagonaloperatorsL2andL1,wemaynowdecomposetheoper-atorTfor>0asfollows:T0B@'1'21CA=0B@'1'21CA+C0B@'1'21CA;(3.68)82where=0BBBBB@!nr1+r 1(x0)2J2+I1!nr1+r 2(x0)2J1+I11CCCCCA;(3.69)andC=0B@K1K2K2K21CA+(1)0B@0L2L101CA:(3.70)Weclearlynoticethat,for'2C2)andx21thenormalderivativeofthesinglelayerpotential@S2'(x)=1!nZ2(xy)(x)jxyjn'(y)d˙(y)is0oldercontinuouswheneverjxj>0.Therefore,itisconvenienttothelimitofthecompactoperator(1)L2asanoperatorfromC2)intoC1)asfollowsL02'=lim!0L2'inC1\fjxj>0g):(3.71)WehighlightatthispointthattheoperatorL02isasanoperatorfromC2)intoC1\fjxj>0g).Furthermore,theoperatorL02iscompactoperatorbecauseitisthenormlimitofcompactoperators.NowweturntothelimitoftheoperatorsL2'(x),whenjxj0byusingtheoperators(x)K2,(x)J2and(x)I2thatintheprevioussection.Inother83words,for'2C2)andx21\fjxj0g,wed0@K02+1!nq1+r 1(x0)2J02+I021A'(x)(3.72)asalimitoperatorofL2'(x).Globally,weuse(3.71)and(3.72)tothelimitingoperatorcorrespondingtoL2.Thatis,for'2C2)andx21),wedL02'(x)=(x)0@K02+1!nq1+r 1(x0)2J02+I021A'(x)+1(x)L02'(x):(3.73)Similarly,wetheoperatorL01.Thatis,for'2C1)andx22),weL01'(x)=(x)0@K01+1!nq1+r 1(x0)2J01+I011A'(x)+1(x)L01'(x):(3.74)WewillshowlatertheoperatorL02isapointwiselimitofL2asanoperatorfromC2)intoC1)andsimilarywewillobtainthatL01isapointwiselimitofL1asanoperatorfromC1)intoC2).However,in[1]theauthorshaveproventhefollowingTheoremTheorem3.7.[1]TheoperatorsL01andL02arenotcompactonCforany0<<0.Consequently,weconcludethatthecompactoperatorsL:=L1;L2donotconvergeinnormtoL0:=L01;L02.84NowweusetheoperatorsL01andL02tothesystemT00B@'1'21CA=0B@K1L02L01K21CA0B@'1'21CA:(3.75)AlsowewillshowlaterthatT0isapointwiselimitofthecompactoperatorsTasanoperatorfromC1)C2)intoC1)C2).Byusing(3.73)and(3.74),wemaydecomposeT0asfollows:T0=0+C0;where0=0BBBBB@!nr1+r 1(x0)2J02+I01!nr1+r 2(x0)2J01+I011CCCCCA;(3.76)andC0=0B@K1K02K02K21CA+(1)0B@0L02L0101CA:(3.77)Aswehavementionedearlierthattheoperators(L1;L1)donotconvergeinnormto(L01;L02).ThustheoperatorsTdonotconvergeinnormtoT0.Consequently,eventhoughweshowthatthelimitingsystemT0isinvertablewecan'tgetdirectlythattheoperatorsTareinvertibleandtheirinverses(T)1areuniformlybounded.Recallthat,theuniformbounded(2.64)requiresuniformboundednessfor(T)1inC1)C2)for<0.Intheabsenceofnormconvergence,ourgoalistoproveuniformboundednessfor(T)1inC1)C2)for<0byusingthepointwiseconvergence.85Chapter4olderestimatefortheoperatorJ2WedevotethischaptertoprovetheuniformoldercontinuityoftheoperatorJ2whereisthefunctionin(3.11).Thework,whichisinanydimensionn2,dependsontheearlierworkfordimensionn=2[1],weusethesamedecompositionfortheoperatorJ2aswellasweusesamenotationsthatpresentedindimensionn=2fortheauxiliaryoperatorswithsomemothatarerequiredforthehigherdimension.Theorem4.1.Givenany0<<0andany0<0,thenJ2isacontinuouslinearoperatorfromC2)toC1),<0.Moreover,wehave!nq1+r 12J2L(C2);C1))C();8<0;whereistheregularityoftheauxilaryfunctions j,0istheregularityoftheboundariesj,j=1;2,andC()convergesto0asconvergesto0uniformlyin.Beforestartingtheproof,recallthatthefunctions 1and 2areinaneighbor-hoodoftheoriginandhaveregularityC10forsome0<01,whereastheauxiliaryfunctions 1and 2aregloballyonRn1andsatisfythebound j1C j10;j=1;2;86where=0.Thatis, j1C();j=1;2;whereC()approaches0whenapproaches0.TheoperatorJ2isexplicitlyin(3.62)and(3.63)butitisbtowriteitdownhereinordertoseethebehaviorofthekernelfunctionthatcorrespondingtotheoperatorneartheorigin.Forx6=0or6=0and'2C2),themainoperatorJ2isbyJ2'(x)=ZRn1+ 2(y0) 1(x0)r 1(x0)(y0x0)jx0y0j2+ 1(x0) 2(y0)2n2+ 2(x0) 1(x0)r 1(x0)(y0x0)jx0y0j2+ 1(x0) 2(x0)2n2˚(y0)dy0;(4.1)andforx=0and=0,theoperatorisbyJ02'(0)=ZRn1 2(y0)y21+:::+y2n1+ 22(y0)n2˚(y0)dy0;(4.2)where˚(x0)=E'(x0)q1+jr 2(x0)j2hascompactsupportinB0(R0),˚(0)='(0)andboundedbythenormk˚k1+C()(1+0)k'k:(4.3)Furthermore,thereexistsaconstantM>0suchthatforanyx=(x0;xn)21,thefunction0!˚(0+x0)issupportedinB0(M).WestartwithsomebasicpreliminariesthatareneededfortheproofoftheTheorem4.1.874.1PreliminariesInthissectionwepresentsomebasicbutusefulboundsfortheauxiliaryfunctions 2, 1andforsomelinearcombinationsof 2and 1withtheperturbationconstant.Fromtheoftheoperatorin(4.1)andin(4.2)weseethatthedenominatorofthekernelfunctiondependsonthefunctions 1and 2soitisagoodideatostartthelowerboundsofthesefunctions.Lemma4.2.Suppose0<<0.ThenthereexistsaconstantC>0,independentof(whereisassumedtobesmall),suchthatforanyx02Rn1,r (x0)C;k ik10 (x0)1+;i=1;2:(4.4)Proof.Weonlyfocuson 2,butthesameargumentsholdfor 1.Recalltheof 2from(3.20) 2(x)= (x)(x)+(1(x))1+0k kC10;whereisafunctionin(3.19).Foranyx0,˘02Rn1,weseethat 2(x0+˘0)= 2(x0)+r 2(x0)˘0+Z10ddt 2(x0+t˘0)r 2(x0)˘0dt;andclearlywehavethebound 2(x0+˘0) 2(x0)+r 2(x0)˘0+Z10r 2(x0+t˘0)˘0r 2(x0)˘0dt:88Usingtheoldercontinuityofr 2,itfollowsthat 2(x0+˘0) 2(x0)+r 2(x0)˘0+˘01+r 2Z10tdt 2(x0)+r 2(x0)˘0+11+˘01+k 2k10:AssumingC=11+k 2k10whichisevidentlyindependentoftlysmall),weobtain 2(x0+˘0) 2(x0)+r 2(x0)˘0+C˘01+:Thatis, 2(x0+˘0) 2(x0)r 2(x0)˘0C˘01+:Since 2(x0+˘0)0,wegetthat 2(x0)˘0C˘01+ 2(x0):(4.5)Letting˘0=1C(1+)1r 2(x0)11r 2(x0)in(4.5),wehavethefollowing1C(1+)1r 2(x0)1+1C1C(1+)(1+)r 2(x0)1+ 2(x0):Thereforewegetthedesiredboundr 2(x0)C 2(x0)1+:89Thefollowinglemma[1]isastraightforwardconsequenceofYoung'sinequalityandithasbeenusedfrequentlyforthelowerboundsforthedenominatorofthekernelfunctionthatcorrespondingtotheoperatorJ2.Lemma4.3.Foranyr;t0andforany01wehaver2+t2r1+t1:Proof.ByYoung'sinequality,wehavethefollowingr1+t11+2r2+12t2r2+t2Toappropriateuniformboundsforthekernelfunction,weneedsomebasicuniformboundsforalinearcombinationoftheauxiliaryfunctions 1and 2withtheperturbationconstant.Wecompilethemostfrequentlyusedboundsintheupcominglemma.Tosimplifythenotationandbecausethisworkisgeneralizationoftheearlierworkfordimensionn=2,weusesamenotationsforthecombinationoftheauxiliaryfunctionsthathasbeengivenin[1].Foranyx0,y0and0inRn1,wethefollowingquantitiesa:=a(x0)=+ 2(x0) 1(x0);(4.6)^a:=a(y0)=+ 2(y0) 1(y0);(4.7)b:=b(x0;0)=+ 2(0+x0) 1(x0);(4.8)^b:=b(y0;0)=+ 2(0+y0) 1(y0):(4.9)90Lemma4.4.Foranyx0,y0and0inRn1,thefollowingboundshold1.jbajr 20,alternatively;2.jbajr 20+1+r 2(x0)0;alternatively;3.jbajr 2(0+x0)0+r 20+1.4.b^bdr 2+r 1,whered=x0y0,alternatively;5.b^bdr 2(0+y0)+dr 1(y0)+d+1r 2+r 1,alterna-tively;6.b^bdr 2(0+x0)+dr 1(x0)+d+1r 2+r 1,7.ja^ajdr 2+r 1.Proof.1.Fromtheitionofaandb,wehavejbaj= 2(0+x0) 2(x0);andbyusingtheregularityof 2,thedesiredboundfollowsjbajr 20:2.Agianfromthedofaandb,wehavejbaj= 2(0+x0) 2(x0):Usingthemeanvaluetheoremforsome˘0liesonthelinesegmentthatjoining0and910,itfollowsthatjbajr 2(x0+˘0)0:Byaddingandsubtractingthequantityr 2(x0),weobtainjbajr 2(x0+˘0)r 2(x0)+r 2(x0)0:Usingtheoldercontinuityofr 2,thedesiredboundfollowsjbajr 20+1+r 2(x0)0:3.Forthealternativebound.Ifweaddandsubtractthequantityr 2(0+x0)insteadofr 2(x0),itfollowsthatjbajr 2(x0+˘0)r 2(0+x0)+r 2(0+x0)0:Noticingthat0˘00,Thedesiredboundfollowsjbajr 20+1+r 2(0+x0)0:4.Followstriviallyfromtheofband^b.5.Forthealternativeboundforb^b,let (x0)= 2(0+x0) 1(x0).Thenby92applyingthemeanvaluetheoremforsomet2(0;1),wehaveb^b= (x0) (y0)=r (˘0)(x0y0);where˘0=tx0+(1t)y0.Next,weaddandsubtractthequantityr (y0)togetthefollowingb^b=r (˘0)r (y0)+r (y0)(x0y0);equivalently,b^b=r 2(0+˘0) 2(0+y0) 1(˘0)+r 1(y0)+r 2(0+y0) 1(y0)(x0y0):Thusbytheoldercontinuityof 2and 1andnoticingthat˘0y0x0y0,thedesiredboundfollowsb^bd+1r 2+r 1+dr 2(0+y0)+dr 1(y0):6.Followssimilarly.7.Followstriviallyfromtheofaand^a.934.2L1normforJ2,>0InthissectionweshowtheuniformboundednessfortheoperatorJ2asanoperatorfrom2to1,thatis,for'2C2),andx21\B(0;R0),weshowthatJ2'(x)C()k'k;whereC()approaches0asapproaches0.Letk2bethekernelfunctioncorrespondingtotheoperatorJ2,thatis,k2(y0;x0)=+ 2(y0) 1(x0)r 1(x0)(y0x0)jx0y0j2+ 1(x0) 2(y0)2n2+ 2(x0) 1(x0)r 1(x0)(y0x0)jx0y0j2+ 1(x0) 2(x0)2n2;andtheninshortJ2canbewritteninasimpleformJ2'(x)=ZRn1k2(y0;x0)˚(y0)dy0:Usingthechangeofvariables0=y0x0andrecallingthatthefunction˚(+x0)hascompactsupportinB0(M),itfollowsthatJ'(x)=Zj0j<Mk2(0;x0)˚(0+x0)0:94Involvingtheitionsofa(x0)andb(x0;0)thatin(4.6)and(4.8)respectively,thekernelfunctionk2hastheequivalentformk2(0;x0)=b(x0;0)0r 1(x0)hj0j2+b2(x0;y0)in2a(x0)0r 1(x0)hj0j2+a2(x0)in2:(4.10)Innaivelyway,wehavetheboundJ2'(x)k˚k1Zj0j<Mb(x0;0)0r 1(x0)hj0j2+b2(x0;0)in2a(x0)0r 1(x0)hj0j2+a2(x0)in20:Weproceedtosimplifytheintegrandbyusingtheleastcommondenominator.WenotethatJ2'(x)k˚k1Zj0j<M(02+a2)n2(ba)+(02+a2)n2(02+b2)n2a0r 1(x0)(j0j2+b2)n2(j0j2+a2)n20k˚k1Zj0j<Mjbaj(j0j2+b2)n2:=A+k˚k1Zj0j<M(02+a2)n2(02+b2)n2a0r 1(x0)(j0j2+b2)n2(j0j2+a2)n20:=B:ForAterm,weusebound(3)fromLemma4.4toobtainthefollowinginequalityAk˚k1Zj0j<Mr 201+j0j2+b2n=2+r 2(x0+0)0j0j2+b2n=20:=A1+A2;whereA1=k˚k1Zj0j<Mr 201+j0j2+b2n=20;95andA2=k˚k1Zj0j<Mr 2(x0+0)0j0j2+b2n=20:Naively,wecanboundA1byA1k˚k1r 2Zj0j<M01+n:WeeasilyconcludetheboundA1C(M;)k˚k1C():(4.11)ForA2,werecallthatb=+ 2(0+x0) 1(x0)anduseLemma4.3toconcludetheboundA2k˚k1Zj0j<Mr 2(x0+0) 2(x0+0)(1)n201(1+)n20;thenweuselemma(4.2)toboundr 2(x0+0),thatis,weobtainthefollowingboundA2C;k 2k10k˚k1Zj0j<M 2(x0+0)1+(1)n201(1+)n20:Choosing11+2n<<1,weclearlyapproachthedesiredboundbound,i.e,wegetA2C;k 2k10k˚k1 21+(1)n21Zj0j<M01(1+)n20:96ThuswehaveA2C;n;M;k 2k10k˚k1C():(4.12)Finally,combining(4.11)and(4.12),wehavethedesiredboundforAwhichisAC;n;M;k 2k10k˚k1C():(4.13)ContinuinginthesamefashiontoboundthetermBuniformly,whereB=k˚k1Zj0j<M(02+a2)n2(02+b2)n2a0r 1(x0)(j0j2+b2)n2(j0j2+a2)n20:Byusingthemeanvaluetheorem,wehave(02+a2)n2(02+b2)n2n202+z2n21b2a2;forsomez2thatliesinthelinesegmentjoininga2tob2.Withoutlossofgeneralitywemayassumethata2<z2<b2.Onotherhand,sameargumentswouldholdifb2<z2<a2.ThereforwegetthefollowingboundBCk˚k1Zj0j<Ma0r 1(x0)jbajjb+ajj0j2+b2j0j2+a2n20:97UsingSchwartzinequalitytoboundthequantitya0r 1(x0)jb+aj,itfollowsthatBCk˚k1Zj0j<Mjbaja0r 1(x0)2+jb+aj2j0j2+b2j0j2+a2n20:Noticethefollowinga0r 1(x0)2+jb+aj26b2+02r 121:SoweconcludethefollowingBCk˚k1Zj0j<Mjbajb2+02r 121j0j2+b2j0j2+a2n20:(4.14)Sinceb2+02r 121j0j2+b21+r 121;thenBcanbeboundedbyBC1+r 121k˚k1Zj0j<Mjbajj0j2+a2n20:Usingthebound(2)formLemma4.4,thefollowingboundfollowsBC1+r 121k˚k1Zj0j<Mr 201+j0j2+a2n2+r 2(x0)0j0j2+a2n20:98ThuseasilywehavetheboundBC1+r 121k˚k1Zj0j<Mr 20+1n+r 2(x0)0j0jn2(1+)an2(1)0;whereweusedLemma4.3forthesecondintegrandwiththechoice12n1+<<1.AgainforthesecondintegrandweuseLemma4.2andthetrivialinequality 2(x0)atoobtainthefollowingBC1+r 121k˚k1 C()M+Zj0j<M 2(x0)m10m20!;wherem1=1+n2(1)andm2=1n2(1+).Clearlym1>0andtheradialfunction0m2isintegrableinthedomain0<M.ThuswehaveBC(;M;n)1+r 121k˚k1C():(4.15)ThereforethedesireduniformboundforJ2followsbycombining(4.13)and(4.15),thatis,J2'(x)C();(4.16)whereC()approaches0asapproaches0.994.3oldercontinuityofJ2,>0.For'2C2),letx;y21\B(0;R0)andsetd=x0y0.ToshowtheoldercontinuityoftheoperatorJ2,weformthefollowingJ2'(x)J2'(y)=Zj0j<Mk2(0;x0)˚(0+x0)k2(0;y0)˚(0+y0)0:(4.17)Weusethesamedecompositionforthecasen=2[1],thatis,weformthefollowingJ2'(x)J2'(y)=Zj0j<Mk2(0;x0)˚(0+x0)˚(0+y0)0+Zj0j<Mk2(0;x0)k2(0;y0)˚(0+y0)˚(y0)0+Zj0j<Mk2(0;x0)k2(0;y0)˚(y0)0:=R1+R2+R3:ThefactthatwesplitupR2andR3istechnical.Naively,onemaytrytonotdothesplittingandestimatedirectlytheirsum(cancelingoutthefactorsof˚(y0)),butthiswon'tgivethedesiredestimatethatwewant.WeeasilyboundR1asfollowsjR1jk˚kZj0j<Mx0y0k2(0;x0)0:100Intheprevioussection4.2wehaveshownthatZj0j<Mk2(0;x0)0C():ThusweconcludethatjR1jC()dk˚k:(4.18)TheworkforthetermsR2andR3aretedioussowewilldealwiththemseparatelyinthenextsubsections.4.3.1BoundednessofR2.ForR2wehaveR2=Zj0j<Mk2(0;x0)k2(0;y0)˚(0+y0)˚(y0)0;wherek2(0;x0)=b0r 1(x0)hj0j2+b2in2a0r 1(x0)hj0j2+a2in2;andk2(0;y0)=^b0r 1(y0)hj0j2+^b2in2^a0r 1(y0)hj0j2+^a2in2:WerewriteR2asfollowsR2=S1+S2+S3+S4;101whereS1=Zj0j<dk2(0;x0)˚(0+y0)˚(y0)0;S2=Zj0j<dk2(0;y0)˚(0+y0)˚(y0)0;S3=Zd<j0j<M0B@b0r 1(x0)hj0j2+b2in2^b0r 1(y0)hj0j2+^b2in21CA˚(0+y0)˚(y0)0;andS4=Zd<j0j<M0B@a0r 1(x0)hj0j2+a2in2^a0r 1(y0)hj0j2+^a2in21CA˚(0+y0)˚(y0)0:ForS1,weusetheoldercontinuityof˚toobtainthefollowingjS1jk˚kZj0j<d0k2(0;x0)0:ThenweclearlyhavethefollowingboundjS1jk˚kdZj0j<Mk2(0;x0)0:ThusbyusingtheL1-normforJ2(4.16),weobtainthedesiredboundforS1.Thatis,jS1jC()k˚kd:(4.19)102Clearly,thesameboundholdsforS2.ThusweconcludethatjS1j+jS2jC()k˚kd:(4.20)WestillhavetoshowtheuniformboundednessforS3andS4.ForS3wehaveS3=Zd<j0j<M0B@b0r 1(x0)j0j2+b2n2^b0r 1(y0)j0j2+^b2n21CA˚(0+y0)˚(y0)0;thentriviallybyusingtheleastcommondenominator,itfollowsthatjS3j=Zd<j0j<M(b0r 1(x0)02+^b2n202+b2n2j0j2+b2n2j0j2+^b2n20r 1(y0)0r 1(x0)j0j2+^b2n2(^bb)j0j2+^b2n2)˚(0+y0)˚(y0)0Usingtheoldercontinuityof˚andr 1andthemeanvaluetheoremforsomez2thatliesonlinesegmentofb2and^b2,wecanformthefollowingdecompositionjS3jCk˚kZd<j0j<M0b0r 1(x0)02+z2n21b2^b2j0j2+b2n2j0j2+^b2n20:=S3;1+Ck˚kZd<j0j<Md0+1r 1j0j2+^b2n20:=S3;2+Ck˚kZd<j0j<M^bb0j0j2+^b2n20:=S3;3:103ForS3;1weassume^b2<z2<b2andagainthesameworkholdswhenb2<z2<^b2.WeobtainthefollowingS3;1Ck˚kZd<j0j<Mb^bb+^bb0r 1(x0)0j0j2+b2j0j2+^b2n20:Usingthebound(4)fromLemma4.4,itfollowsthatS3;1Ck˚kdr 2+r 1Zd<j0j<Mb+^bb0r 1(x0)0j0j2+b2j0j2+^b2n20:Sinceb+^bb0r 1(x0)3b2+02r 1(x0)2;weconcludethefollowingS3;1k˚kC()dZd<j0j<Mb2+02r 1(x0)20j0j2+b2j0j2+^b2n20:Thusweapproachthedesiredbound,thatis,wehaveS3;1k˚k1+r 121C()dZd<j0j<M0n0:Thatis,wehaveprovedthatS3;1C(n;)k˚k1+r 121C()d:(4.21)104WeeasilyboundS3;2asfollows.WehaveS3;2=Ck˚kdZd<j0j<M0+1r 1j0j2+^b2n20:Trivially,wehavetheboundS3;2Ck˚kr 1dZd<j0j<M0+1n0:ThereforewehaveS3;2C(M;)k˚kC()d:(4.22)ForthelasttermS3;3weusethebound(4)fromLemma4.4togetS3;3Ck˚kC()dZd<j0j<M0n0:ThusS3;3C(M;)k˚kC()d:(4.23)Combining(4.21),(4.22)and(4.23),thedesiredestimateforS3follows,thatis,wegetjS3jC(n;M;)k˚kC()d:(4.24)ObservethatS4hasthesameboundasS3doesbecausethequantitiesb^bandja^ajhavethesameupperboundwhichisdr 2+r 2.Thereforewegatherallthe105aboveuniformboundstoendwiththedesireduniformboundforR2whichisjR2jC(n;M;)k˚kC()d:(4.25)4.3.2BoundednessofR3.Evenweareworkinginhigherdimensionsn2,weformthesamedecompositionthathasbeenpresentedforthedimensionn=2[1]butweneedsometechnicalmotodealwiththeexponentn2.ForthetermR3=Zj0j<Mk2(0;x0)k2(0;y0)˚(y0)0:WeformthefollowingR3=˚(y0)Zj0j<M b0r 2(0+x0)j0j2+b2n2^b0r 2(0+y0)j0j2+^b2n2!0:=˚(y0)T1+˚(y0)Zj0j<M ^aj0j2+^a2n2aj0j2+a2n2!0:=˚(y0)T2+˚(y0)Zj0j<M 0r 2(0+x0)0r 1(x0)j0j2+b2n20r 2(0+y0)0r 1(y0)j0j2+^b2n2!0:=˚(y0)T3+˚(y0)Zj0j<M 0r 1(x0)j0j2+a2n20r 1(y0)j0j2+^a2n2!0:=0:NoticethatforthetermT1wehaveused0r 2(0+:)insteadof0r 1(:)inordertoobtainanexacttialformasyouwillseelater.Alsonoticethatthelastintegralis106vanishing.WestartwiththemosttechnicaltermwhichisT1,thatis,webeginwithT1=Zj0j<M b0r 2(0+x0)j0j2+b2n2^b0r 2(0+y0)j0j2+^b2n2!0:Weremarkherethattheworkforthistermisdependingonthedimensionn.Sinceforthedimensionn=2ithasalreadyprovenin[1]thatjT1jC()d;sowerestrictourworkforn3.NowletusboundT1uniformlywhenn=3.ForsuchcasewehaveT1=Zj0j<M b0r 2(0+x0)j0j2+b232^b0r 2(0+y0)j0j2+^b232!0:Sinceb=+ 2(0+x0) 1(x0),thenr0b=r 2,wherer0meansthatthegradientistakenwithrespecttothevariable0.Byutilizingthepolarcoordinates1=rcosand2=rsin,thenweclearlyseethatr@rb=0r0b.ThusitfollowsthatZj0j<Mb0r0bj0j2+b2320=Z2ˇ0ZM0b(r;;x0)r@rbrr2+b2(r;;x0)32dr=Z2ˇ0b(r;;x0)pr2+b2(r;;x0)jr=Mr=0:Noticingthatb(0;;x0)=+ 2(x0) 1(x0)>0,thereforewehaveZj0j<Mb0r0bj0j2+b2320=Z2ˇ0b(M;;x0)pM2+b2(M;;x0)+2ˇ:(4.26)107SimilarlywehaveZj0j<M^b0r0^bj0j2+^b2320=Z2ˇ0b(M;;y0)pM2+b2(M;;y0)+2ˇ:(4.27)Using(4.26)and(4.27),wehavethefollowingjT1jZ2ˇ0b(M;;x0)pM2+b2(M;;x0)+b(M;;y0)pM2+b2(M;;y0):Weusethemeanvaluetheoremforsome˘0liesonthelinesegmentthatjoiningx0toy0toobtainthefollowingjT1jZ2ˇ0M2r˘0b(M;;˘0)M2+b2(M;;˘0)32x0y0:(4.28)Sincer˘0b(M;;˘0)=r 2(M;;˘0)r 1(˘0);thuswehavejT1jCMZ2ˇ0r 2+r 1x0y0:Thatisfordimensionn=3wehavethedesiredboundwhichisjT1jC(M)C()d:(4.29)108WecontinuetotheuniformboundednessforT1inhigherdimensionsn>3.Weusesphericalcoordinatesindimensionn1togetthefollowingT1=Zj0j<Mf(0)0=Zˇ1=0:::Zˇn3=0Z2ˇn2=0ZM0f(0(r;J(n;r;drd;wheref(0(r;=b0r0bj0j2+b2n2=b(r;;x0)@rb(r;;x0)j0j2+b2(r;;x0)n2;J(n;r;=rn2(sin1)n3:::(sinn4)2(sinn3);and=(1;;n2):WeuseintegrationbypartstocomputethefollowingZM0(br@rb)rn2r2+b2n2dr=ZM0(br@rb)rr2+b232rn3r2+b2n32dr:(4.30)LetA1(r):=A1(r;;x0)=rn3r2+b2n32:Noticethatb(0;;x0)=+ 2(x0) 1(x0)>0andA1(0)=0.Thusbyintegrationbyparts,itfollowsthatZM0(br@rb)rr2+b232A1(r)dr=A1(M)b(M;;x0)pM2+b2(M;;x0)+ZM0bpr2+b2A01(r)dr;(4.31)109whereA01(r)=(n3)0@rn4r2+b2n32rn3(r+b@rb)r2+b2n121A(4.32)WetreatthelastintegralasfollowsZM0bpr2+b2A01(r)dr=(n3)ZM0brn4r2+b2n22brn2r2+b2n2b2@rbrn3r2+b2n2dr=(n3)ZM0brn4r2+b2n22(br@rb)rn2r2+b2n2(b2+r2)@rbrn3r2+b2n2dr=(n3)ZM0(br@rb)rn4r2+b2n22(br@rb)rn2r2+b2n2dr:(4.33)Substituting(4.33)and(4.31)in(4.30),itfollowsthatZM0(br@rb)rn2r2+b2n2dr=1n2A1(M)b(M;;x0)pM2+b2(M;;x0)+n3n2ZM0(br@rb)rn4r2+b2n22dr:(4.34)Repeatingthesameprocessn22timesifnisevenandn32timesifnisodd,weseethatZM0(br@rb)rn2r2+b2n2dr=8>>>><>>>>:Pn32j=1Cj(n)Bj(M)+~Co(n)RM0(br@rb)rr2+b232drifnisodd;Pn22j=1Cj(n)Bj(M)+~Ce(n)RM0(br@rb)r2+b2drifniseven;(4.35)whereBj(M):=Bj(M;;x0)=Aj(M;;x0)b(M;;x0)pM2+b2(M;;x0);(4.36)110Aj(r)=rn(2j+1)r2+b2n(2j+1)2;(4.37)~Co(n)=n32Yj=1n(2j+1)n2j;(4.38)~Ce(n)=n22Yj=1n(2j+1)n2j;(4.39)and~Cj(n)=1n2jj1Yi=1n(2i+1)n2i:(4.40)ToestimateT1itisenoughtoboundBj(M;;x0)Bj(M;;y0)becausetheterminvolv-ingtheintegralRM0(br@rb)rr2+b232dristreatedinthecasen=3andalsotheterminvolvingtheintegralRM0(br@rb)r2+b2dristreatedinthecasen=2[1].NowweboundthetermBj(M;;x0)Bj(M;;y0)Bj(x0)Bj(y0)=Mn(2j+1)b(x0)M2+b2(x0)(n2j2)b(y0)M2+b2(y0)(n2j2);whereb(x0):=b(M;;x0).Usingthemeanvaluetheorem,itfollowsthatBj(x0)Bj(y0)Mn(2j+1)rb(˘0)M2+b2(˘0)(n2j2)(n2j)b2(˘0)rb(˘0)M2+b2(˘0)(n2j+22)x0y0;111forsome˘0liesonthelinesegmentjoiningx0toy0.ThuswehaveBj(x0)Bj(y0)C(n;j;M)1+b2(˘0)rb(˘0)x0y0C(n;j;M)C()d:(4.41)Thereforefromallaboveestimates,wehavethedesiredboundforT1,thatis,jT1jC(n;M)C()d:(4.42)IfweproceedinthesamefashionaswedidforT1andreplacebbya,^bby^aandnoticingthataand^aareindependentof0,wewouldboundT2bythesameboundthatwefoundforT1.Thatis,jT2jC(n;M)C()d:(4.43)ForthelastintegralT3wehaveT3=Zj0j<M 0r 2(0+x0)0r 1(x0)j0j2+b2n20r 2(0+y0)0r 1(y0)j0j2+^b2n2!0WewriteT3asT3=T3;1+T3;2+T3;3+T3;4;whereT3;1=Zd<j0j<M0r 2(0+x0)r 1(x0)r 2(0+y0)+r 1(y0)j0j2+b2n20;112T3;2=Zd<j0j<M0r 2(0+y0)r 1(y0) 1j0j2+b2n21j0j2+^b2n2!0;T3;3=Zj0j<d0r 2(0+x0)r 1(x0)j0j2+b2n20;andT3;4=Zj0j<d0r 1(y0)r 2(0+y0)j0j2+^b2n20:WecaneasilyboundT3;1byT3;1dr 1+r 2Zd<j0j<M0j0j2+b2n20:Forany0<<0,wewouldhaveT3;1d0C()Zd<j0j<M01+(0)j0j2+b2n20:Thusbyusingpolarcoordinates,weobtainthefollowingT3;1Cd0C()ZMdr(0)+n1r2+b2n2dr:ThereforewehavetheboundT3;1C(;0;M)d0C():(4.44)ForT3;2weusethemeanvaluetheoremforsomez2thatliesonthelinesegmentjoining^b2113tob2toobtainthefollowingT3;2CZd<j0j<Mr 2(0+y0)r 1(y0)02+z2n21b^b0b+^bj0j2+b2n2j0j2+^b2n20:Letassumethat^b2<z2<b2.Thenweclearlyseethat0b+^b202+b2:ThereforewehaveT3;2CZd<j0j<Mr 2(0+y0)r 1(y0)b^bj0j2+^b2n2:Usingthebound(5)fromLemma4.4,weobtainthefollowingT3;2CZd<j0j<Mr 2(0+y0)r 1(y0)r 2+r 1d1+j0j2+^b2n20:=T3;2;1+CZd<j0j<Mr 2(0+y0)r 1(y0)r 2(0+y0)+r 1(y0)dj0j2+^b2n20:=T3;2;2:TheintegralT3;2;1followseasily,thatis,T3;2;1C()d:(4.45)114ForthesecondintegralT3;2;2,wehavetheboundT3;2;2CdZd<j0j<Mr 2(0+y0)2+r 1(y0)2j0j2+^b2n20:ThenweapplyLemma4.3forsome0<<1toboundthedenominatorandLemma4.2toboundthenumerator,itfollowsthatT3;2;2CdZd<j0j<M 2(0+y0)21+j0j(1+)n2^b(1)n20+CdZd<j0j<M 1(y0)21+j0j(1+)n2^b(1)n20:Choosing=12ninbothintegralsandnoticingthat^b 2(0+y0)aswellas^b 1(y0),wegetthefollowingboundT3;2;2CdZd<j0j<M 2(0+y0)(1)1+j0jn0+CdZd<j0j<M 1(y0)(1)1+j0jn0:ThusweobtaintheboundT3;2;1C()d:(4.46)ThenthedesiredboundforT3;2followswhen^b2<z2<b2.Thatis,wehaveprovedthatT3;2C()d:(4.47)Whileforthecaseb2<z2<^b2weneedsomemofortheaboveworkinorderto115getthebound(4.47).ForthiscasewewouldhavethatT3;2CZd<j0j<Mr 2(0+y0)r 1(y0)b^bj0j2+b2n2;whereb=+ 2(0+x0) 1(x0).Weseethatthefunctionbdependsonthevariables0andx0whilethefunctionsr 2andr 1dependonthethevariables0andy0sowewillperturbthethefunctionsr 2andr 1inordertomatchtheirvariableswithvariablesofthefunctionb.ForthatwethefollowingdecompositionT3;2~T3;2;1+~T3;2;2+~T3;2;3;where~T3;2;1=CZd<j0j<Mr 2(0+x0)r 1(x0)b^bj0j2+b2n2;~T3;2;2=CZd<j0j<Mr 2(0+y0)r 2(0+x0)b^bj0j2+b2n2;and~T3;2;3=CZd<j0j<Mr 1(x0)r 1(y0)b^bj0j2+b2n2:Evidently,wecanbound~T3;2;1byC()difweusetheearlierworktheforT3;2becausewehavethealternativebound(6)forb^bthatinvolvingx0insteadof(5)thatinvolvingy0.Fortheintegrals~T3;2;2and~T3;2;3wetriviallygetthedesiredboundbyusingtheoldercontinuityoftheauxiliaryfunctionsandthebound(4)forb^b.Thatis,thedesiredbound(4.47)stillvalidwhenb2<z2<^b2.AfterT3;2,wearereadytoboundthelasttwo116integralsT3;3andT3;4inthedecompositionoftheT3term.IfwelookatT3;3=Zj0j<d0r 2(0+x0)r 1(x0)j0j2+b2n20;andT3;4=Zj0j<d0r 1(y0)r 2(0+y0)j0j2+^b2n20:Weseethattheyarebothinthesamestyle.SoweworktoboundT3;3only.Ontheotherhand,theboundforintegralT3;4willfollowinthesamemanner.TobeginboundingT3;3,weaddandsubtractr 2(x0)andusingtheadvantageoftheintegralZ0<d0r 1(x0)r 2(x0)j0j2+a2n20=0;tothebdecompositionT3;3=Zj0j<d0r 2(0+x0)r 2(x0)j0j2+b2n20:=T3;3;1+Zj0j<d0r 2(x0)r 1(x0) 1j0j2+b2n21j0j2+a2n2!0:=T3;3;2:Sincer 2isoldercontinuous,weeasilyboundT3;3;1byC()d.Forthesecondintegral,weusethemeanvaluetheoremforsomez2thatliesonthelinesegmentthatjoininga2tob2toobtainthefollowingT3;3;2CZj0j<d0r 2(x0)r 1(x0)02+z2n21jbajjb+ajj0j2+b2n2j0j2+a2n20117Firstweassumethata2<z2<b2.Thenbyusingthebound0ja+bj202+b2;itfollowsthatT3;3;2CZj0j<dr 2(x0)r 1(x0)jbajj0j2+a2n20:usingthebound(2)fromLemma4.4,wegettheboundT3;3;2CZj0j<dr 2(x0)r 1(x0)r 2(x0)0+r 20+1j0j2+a2n20:ThenweeasilyhavethefollowingT3;3;2C()Zj0j<d01+j0j2+a2n20+CZj0j<dr 2(x0)20j0j2+a2n20+CZj0j<dr 1(x0)20j0j2+a2n20:WeclearlyseetheintegralisboundedbyC()d.Thesecondandthethirdintegralwillbetreatedinthesamemanner.Toboundthesecondintegral,weuseLemma4.3withthechoice=12ntoboundthedenominatorandalsoweuseLemma4.2toboundthenominator.ThenthefollowingfollowsZj0j<dr 2(x0)20j0j2+a2n20CZj0j<d01+n 2(x0)(1)1+0:118ThusthesecondandthethirdintegralareboundedbyC()d.Thereforewhenb2<z2<a2weobtainT3;3C()d:Nowforthecaseb2<z2<a2whichisalmostthesameworkbutwithsomemoWeclearlywouldhavetheboundT3;3;2CZj0j<dr 2(x0)r 1(x0)jbajj0j2+b2n20:usingthebound(3)fromLemma4.4,wegettheboundT3;3;2CZj0j<dr 2(x0)r 1(x0)r 2(0+x0)0+r 20+1j0j2+b2n20:Clearlythetermwith0followstrivially.Sowejustneedtoboundtheintegral~T3;3;2=Zj0j<dr 2(x0)r 1(x0)r 2(0+x0)0j0j2+b2n20:Weperturbbyaddingandsubtractingthefunctionr 2(0+x0)togetthefollowing~T3;3;2Zj0j<dr 2(0+x0)r 2(x0)r 2(0+x0)0j0j2+b2n20+Zj0j<dr 2(0+x0)r 1(x0)r 2(0+x0)0j0j2+b2n20;119thentheboundfollowstrivially.ThusthedesiredboundforT3;3follows.ThatisT3;3C()d:(4.48)Finally,weenclosethissubsectionbycombiningalltheaboveboundstogetthedesiredboundforR3whichisjR3jC(n;;0;M)k˚kC()d0;80<<0:(4.49)Therefor,atthispoint,Theorem(4.1)hasbeenprovenwhen>0.NowweshowthevalidityofTheorem(4.1)underthecase=0.Thatis,weneedtoshow!nq1+r 12J02L(C2);C1))C();8<0;(4.50)whereC()approaches0whenapproaches0.Thebound(4.50)followsifweshowthatfor'2C2),thefollowinglimitholdslim!0J2'(x)=J02'(x);x21\B(R):Infact,sinceJ2isuniformlyboundedinoperatornorm,thenitfollowsthatJ02'(x)2C1)\B(R);120andJ02'C1)C()k'k:For'2C2),06=x21\B(0;R0)theoperatorJ02isbyJ02'(x)=ZRn1k02(y0;x0)˚(y0)dy0;(4.51)wherek02(y0;x0)= 2(y0) 1(x0)r 1(x0)(y0x0)jx0y0j2+ 1(x0) 2(y0)2n2 2(x0) 1(x0)r 1(x0)(y0x0)jx0y0j2+ 1(x0) 2(x0)2n2:Whereasforx=0,theoperatorJ02isbyJ02'(0)=ZRn1 2(y0)y21+:::+y2n1+ 22(y0)n2˚(y0)dy0:(4.52)Afterusingthechangeofvariables0=y0x0in(4.51)andrecallingthat˚hascompactsupportinB0(M),itfollowsthatJ02'(x)=Zj0j<Mk02(0;x0)˚(0+x0)0;121wherea0(x0)= 2(x0) 1(x0),b0(x0;0)= 2(0+x0) 1(x0)andk02(0;x0)=b0(x0;0)0r 1(x0)j0j2+b20(x0;0)n2a0(x0)0r 1(x0)j0j2+a20(x0)n2:Proposition4.5.Let'2C2).Thenforany2<0andx21\B(R0),wehavelim!0J2'(x)=J02'(x):Proof.Forx6=0,itisclearthatb0(x0)>0anda0(x0)>0.Thuslim!0k(0;x0)=b00r 1(x0)j0j2+b20n2a00r 1(x0)j0j2+a20n2;a.e0<M:Alsowenoticethatthekernelfunctionkisuniformlyboundedbecausethedenominatorisboundedawayfromzero.Thereforebyusingthedominatedconvergencetheorem,itfollowsthatlim!0J2'(x)=Zj0j<M0B@b00r 1(x0)j0j2+b20n2a00r 1(x0)j0j2+a20n21CA˚(0+x0)0:Thatis,lim!0J2'(x)=J02'(x);x6=0:(4.53)122Forx=0,wehavethefollowingJ2'(0)=Zj0j<M0B@+ 2(0)j0j2++ 2(0)2n2j0j2+2n21CA˚(0)0:Letf(0)=+ 2(0)j0j2++ 2(0)2n2j0j2+2n2;andf0(0)= 2(0)j0j2+ 2(0)2n2:Bythemeanvaluetheorem,weclearlyseethatf(0)n 2(0)j0jn;whichisintegrableonthedomainf0<Mg.Thusbythedominatedconvergencetheoremweobtainthepointwiseconvergence.Thatis,lim!0J2'(0)=Zj0j<M 2(0)j0j2+ 22(0)n2˚(0)0=J02'(0):1234.4ConvergenceofJ2.InthisSectionwestudytheconvergenceoftheoperatorJ2asanoperatorfromtheBanachspaceC2)totheBanachspaceC1)forany<0,where0istheregularityoftheboundariesjandistheregularityofr j,j=1;2.Beforewestartshowingthepointwiseconvergence,recalltheoftheoperatorJ2,whichisJ2'(x)=Zj0j<Mk2(0;x0)˚(0+x0)0;wherek(0;x0)=b(0;x0)0r 1(x0)j0j2+b2(0;x0)n2a(x0)0r 1(x0)j0j2+a2(x0)n2;b(0;x0)=+ 2(0+x0) 1(x0);b0(0;x0)= 2(0+x0) 1(x0);a(x0)=+ 2(x0) 1(x0);anda0(x0)= 2(x0) 1(x0):Theorem4.6.[1]Fix0<2<0.Thenforany0<0<<0andforall'2C2),wehaveJ2'0!J02'as!0;124where0!meanstheconvergencetakeplaceinC01).Proof.For'2Cand0<,wearguetheconvergenceofJ2bycontradiction.IfthesequenceJ2'doesnotconvergetoJ02'inC001).ThenthereisasequencefJn2'gthatsatisfyingJn2'J0'0>C:(4.54)forsomeC>0.WehaveshownthatfJn2'gisuniformlyboundedinC1)andsinceC1)iscompactlyembeddedinC01),thenthereexistsasubsequencefJnj2'gthatconvergestoJ22C01).FromLemma4.5,wehavepointwiseconvergenceforJ2'2.Thatislim!0J2'(x)=J02'(x);x21\B(R0):Fromtheuniquenessofalimit,wehaveJ2=J02'2andthiswouldcontradict(4.54).ThusJ2'!J02'inC0;0<:125Chapter5UniformoldercontinuityforI2.InthischapterwedemonstratethetheuniformoldercontinuityoftheoperatorI2intheoperatornormL(C2);C1)).WewillshowthattheoperatorI2inanydimensionn2isuniformlyboundedbythequantity121+C()(1+0)whichisexactlythesamequantitythathasbeenfoundintheearlierworkfordimensionn=2[1].However,iftheboundweregreaterthan121+C()(1+0),theinvertibilityoftheoperatorthatin(3.69)mayfail.Wecanseethatclearlyfromtheoftheoperator=0BBBBB@!nr1+r 1(x0)2J2+I2!nr1+r 2(x0)2J1+I11CCCCCA;where=k+12(k1)andkistheconductivityinthesubdomains,and0hasbeenchosentosatisfy12<1+02<jj.Thusfortheoperatortobeinvertible,itthattheoperatorsbeboundedbyaquantitystrictlylessthanjj.Theinvertibilityandtheconvergenceoftheoperatorwillbefullyillustratedinthenextchapter.TheproofoftheuniformoldercontinuityofI2issimilartobuteasierthanwhatwehaveseenforJ2inChapter4.Westateandprovethemaintheoremofthischapter,Theorem5.1.Givenany0<<0andany0<0,theoperatorI2isacontinuous126linearoperatorfromC2)toC1),<0.Moreover,wehave!nq1+r 12I2L(C2);C1))121+C()(1+0);whereC()convergesto0asconvergesto0,uniformlyin.Remark5.2.Weusethe3.2toshowtheuniformboundtotheoperatorI2.Thatis,for'2C2)andx;y21\B(R0),weshowthatmax I2'1;supjxj;jy0I2'(x)I2'(y)jxyj!121+C()(1+0);whereI2'1=supjx0I2'(x):Let'2C2),x21\B(R0).RecalltheofI2I2'(x)=ZRn1+ 2(x0) 1(x0)r 1(x0)(y0x0)jx0y0j2++ 2(x0) 1(x0)2n2˚(y0)dy0;where˚(y0)=E'(y0)q1+r 2(y0)2hascompactsupportinB0(R0)andboundedbythenormk˚k1+C()(1+0)k'k:(5.1)Forx=(x0;xn)21\B(0;R0),wedecomposetheoperatorI2asI2I1,whereI1'(x)=ZRn1r 1(x0)(y0x0)jx0y0j2++ 2(x0) 1(x0)2n2˚(y0)dy0;(5.2)127andI2'(x)=ZRn1+ 2(x0) 1(x0)jx0y0j2++ 2(x0) 1(x0)2n2˚(y0)dy0:(5.3)BeforewestartprovingtheTheorem,recallfromLemma3.6thatZRn1dz0jz0j2+1n2=!n2:(5.4)WewilluseitfrequentlyduringtheproofoftheTheorem.5.1L1-normforI2,>0.InthissectionweelaboratetheuniformboundforI,when>0.WediscusstheuniformboundsforitstermsI1andI2separately.Tobeginthediscussionoftheuniformboundedness,weassumethatx=(x0;xn)21\B(0;R0).5.1.1L1-normforI1,>0.Since˚hascompactsupportinB0(R0),weseethatI1'(x)=ZRn1r 1(x0)(y0x0)jx0y0j2++ 2(x0) 1(x0)2n2˚(y0)dy0=ZB0(R0)r 1(x0)(y0x0)jx0y0j2++ 2(x0) 1(x0)2n2˚(y0)dy0:128Let0=y0x0,thenwehaveI1'(x)=Zj0j<Mr 1(x0)0j0j2+a2n2˚(0+x0)0;(5.5)wherea:=a(x0)=+ 2(x0) 1(x0):SinceZj0j<Mr 1(x0)0j0j2+a2n2˚(x0)0=0;thenclearlyweseethatI1'(x)=Zj0j<Mr 1(x0)0j0j2+a2n2˚(0+x0)˚(x0)0:ThuswehavetheboundjI1'(x)jr 11k˚kZj0j<M01+j0j2+a2n20:Sincea=+ 2(x0) 1(x0)>0,weobtainjI1'(x)jr 1k˚kZj0j<M01+n0C(M;)C()k˚k:(5.6)129Thereforefrom(5.1)and(5.6),wehavetheboundjI1'(x)jC()k'k:(5.7)5.1.2L1-normforI2,>0.FromtheofI2,wehaveI2'(x)=ZRn1aj0j2+a2n2˚(0+x0)0:Byusingthechangeofvariable0=a$,itfollowsthatjI2'(x)j=ZRn111+j$j2n2˚(a$+x0)d$k˚k1ZRn111+j$j2n2d$=!n2k˚k1:(5.8)Againfrom(5.1)and(5.8),weseetheuniformboundjI2'(x)j!n21+C()(1+0)k'k:(5.9)Finally,wecombine(5.7)and(5.9)togetthedesireduniformboundforII'1!n21+C()(1+0)k'k:(5.10)1305.2olderestimateforI2,>0.Fortheoldercontinuity,wechoosex=(x0;xn)andx=(x0;x0n)21\B(0;R0).AsbeforeweuseaforthecombinationoftheC1functions 1and 2actingatthepointx0withtheperturbationconstant,thatis,a:=a(x0)=+ 2(x0) 1(x0);andsimilarlyfora,butatthepointx0,thatis,a:=a(x0)=+ 2(x0) 1(x0):InordertogetthedesiredboundforI,weusethesamedecompositionforthetermsI1andI2thatpresentedindimensionn=2[1]withsometechnicalmo5.2.1olderestimateforI1,>0.FromtheofI1,wehaveI1'(x)=Zj0j<Mr 1(x0)0j0j2+a2n2˚(0+x0)0;andI1'(x)=Zj0j<Mr 1(x0)0j0j2+a2n2˚(0+x0)0:(5.11)Let}=x0x0and0=01+}in(5.11),itfollowsthat131I1'(x)=Z01+}<Mr 1(x0)(01+})01+}2+a2n2˚(01+}+x0)01:(5.12)Since˚hascompactsupport,wehavethefollowingI1'(x)=ZRn1r 1(x0)(01+})01+}2+a2n2˚(01+x0)01:(5.13)Letting0=01in(5.13),itfollowsthatI1'(x)=ZRn1r 1(x0)(0+})j0+}j2+a2n2˚(0+x0)0:(5.14)NowweformthefollowingdecompositionI1'(x)I1'(x)=ZRn1r 1(x0)r 1(x0)0j0j2+a2n2˚(0+x0)0+ZRn10B@r 1(x0)0j0j2+a2n2r 1(x0)(0+})j0+}j2+a2n21CA˚(0+x0)0:=A+B:(5.15)ForAwehaveA=ZRn1r 1(x0)r 1(x0)0j0j2+a2n2˚(0+x0)0:132SinceZRn1r 1(x0)r 1(x0)0j0j2+a2n2˚(x0)0=0;thenthefollowingboundfollowsjAjr 1j}jZRn10j0j2+a2n2˚(0+x0)˚(x0)0r 1j}jk˚kZj0j<M01+j0j2+a2n20C(M;)C()k˚kj}j:Usingolderestimate(5.1)for˚,thedesiredboundforAfollowsjAjC()k'kj}j:(5.16)WeproceedtodtheboundforB,whereB=ZRn10B@r 1(x0)0j0j2+a2n2r 1(x0)(0+})j0+}j2+a2n21CA˚(0+x0)0133WedecomposeBintofourtermsasfollowsB=Zj0j<4j}jr 1(x0)0j0j2+a2n2˚(0+x0)˚(x0)0Zj0j<4j}jr 1(x0)(0+})j0+}j2+a2n2˚(0+x0)˚(x0)0Zj0j<4j}jr 1(x0)(0+})j0+}j2+a2n2˚(x0)˚(x0)0+Zj0j>4j}j0B@r 1(x0)0j0j2+a2n2r 1(x0)(0+})j0+}j2+a2n21CA˚(0+x0)˚(x0)0:ThuswehavethedecompositionB:=B1+B2+B3+B4;whereBiisthecorrespondingintegralintheabovedecomposition,i=1;2;3;4.WebeginbyestimatingtheintegralB1,thatis,B1=Zj0j<4j}jr 1(x0)0j0j2+a2n2˚(0+x0)˚(x0)0:Usingtheoldercontinuityof˚,weeasilyhavethefollowingboundjB1jr 1k˚kZj0j<4j}j01+n0C()k'kj}j:(5.17)134SimilarlywecangettheappropriateboundforB2.Toshowthat,wehaveB2=Zj0j<4j}jr 1(x0)(0+})j0+}j2+a2n2˚(0+x0)˚(x0)0:oldercontinuityof˚allowustohavetheboundsjB2jr 1k˚kZj0j<4j}j0+}j0+}j2+a2n20+}0C()k˚kZj0j<4j}j0+}1+n0C()k˚kZj0j<5j}j01+n0:Using(5.1)andintegrating01+nover0<5j}j,weobtainthedesiredboundjBjC()k'kj}j:(5.18)ForB3,wesubstitute0=0}inordertowriteB3inthefollowingformB3=Zj0}j<4j}jr 1(x0)0j0j2+a2n2˚(x0)˚(x0)0:SinceZj0j<12j}jr 1(x0)0j0j2+a2n2˚(x0)˚(x0)0=0;135thenwehaveB3=Zfj0}j<4j}jnj0j<j}j2gr 1(x0)0j0j2+a2n2˚(x0)˚(x0)0:WeclearlycanboundB3asfollowsjB3jr 1k˚kj}jZfj0}j<4j}jnj0j<j}j2g01n0r 1k˚kj}jZj}j2<j0j<5j}j01n0:Therefore,jB3jC()k'kj}j:(5.19)ForthelastintegralB4,wehaveB4=Zj0j>4j}j0B@r 1(x0)0j0j2+a2n2r 1(x0)(0+})j0+}j2+a2n21CA˚(0+x0)˚(x0)0:Weworkwithsomemanipulationsinordertohavethedesiredbound.Westartbyassuming13601=0+}inthesecondintegrandtodecomposeB4asfollowsB4=Z3j}j<j0j<4j}jr 1(x0)0j0j2+a2n2˚(0+x0)˚(x0)0+Zj0j>3j}jr 1(x0)0j0j2+a2n2˚(0+x0)˚(x0)0Z01}>4j}jr 1(x0)01012+a2n2˚(01}+x0)˚(x0)01:=C1+C2;(5.20)whereC1=Z3j}j<j0j<4j}jr 1(x0)0j0j2+a2n2˚(0+x0)˚(x0)0andC2=Zj0j>3j}jr 1(x0)0j0j2+a2n2˚(0+x0)˚(x0)0Z01}>4j}jr 1(x0)01012+a2n2˚(01}+x0)˚(x0)01:WeeasilycanboundC1byC1r 1k˚kZ3j}j<j0j<4j}j01n+0C()k'kj}j:(5.21)137ForC2,recallthat}=x0x0thenweseethatC2=Zj0j>3j}jr 1(x0)0j0j2+a2n2˚(0+x0)˚(x0)0Z01}>4j}jr 1(x0)01012+a2n2˚(01+x0)˚(x0)01:WerewriteC2inthefollowingformC2=Zj0j>3j}jr 1(x0)0j0j2+a2n2˚(0+x0)˚(x0)0Z01}>4j}jr 1(x0)01012+a2n2˚(01+x0)˚(x0)01+Z01}>4j}jr 1(x0)01012+a2n2˚(01+x0)˚(01+x0)01:=D1+D2;(5.22)whereD1=Zj0j>3j}jr 1(x0)0j0j2+a2n2˚(0+x0)˚(x0)0Z01}>4j}jr 1(x0)01012+a2n2˚(01+x0)˚(x0)01:andD2=Z01}>4j}jr 1(x0)01012+a2n2˚(01+x0)˚(01+x0)01;138WestartwiththeeasiesttermwhichisD2andafterutilizingthat˚iscompactlysupportedinB0(M),clearlywecanseetheboundsjD2jZ3j}j<01<Mr 1(x0)01012+a2n2˚(01+x0)˚(01+x0)01k˚kj}jZ3j}j<01<M 1(x0)1+01012+a2n201(byusinglemma4.2)k˚kj}jZ3j}j<01<M 1(x0)1+0101(1+)n2a(1)n201(byusinglemma4.3)k˚kj}jZ3j}j<01<M 1(x0)(1)2(1+)011n+201bychoosing=1nC()k˚kj}jZ3j}j<01<M011+2n01C()k˚kj}j:ThereforeweendwiththeboundjD2jC()k'kj}j:(5.23)ToestimateD1werewrite0=01intheintegralforD1togetthefollowingD1=Z01>3j}jr 1(x0)01012+a2n2˚(01+x0)˚(x0)01Z01}>4j}jr 1(x0)01012+a2n2˚(01+x0)˚(x0)01:139NowwerewriteD1inthefollowingformD1=Z01>3j}j0B@r 1(x0)01012+a2n2r 1(x0)01012+a2n21CA˚(01+x0)˚(x0)01+Z01>3j}jr 1(x0)01012+a2n2˚(01+x0)˚(x0)01Z01}>4j}jr 1(x0)01012+a2n2˚(01+x0)˚(x0)01:LetusouttheboundforthelasttwointegralsinD1anddenotethembyD1;2.Thatis,D1;2=Z01>3j}jr 1(x0)01012+a2n2˚(01+x0)˚(x0)01Z01}>4j}jr 1(x0)01012+a2n2˚(01+x0)˚(x0)01:ThenweeasilyseethatD1;2Z3j}j<01<5j}jr 1(x0)01012+a2n2˚(01+x0)˚(x0)01r 1k˚kZ3j}j<01<5j}j011+n01:ThuswehavethefollowingboundD1;2C()k˚kj}j:(5.24)140FortheintegralinD1weneedtousethemeanvaluetheoreminordertogetthedesiredbound.LetusdenoteitbyD1;1anduse0insteadof01tosimplifythewriting,thatisD1;1=Zj0j>3j}j0B@r 1(x0)0j0j2+a2n2r 1(x0)0j0j2+a2n21CA˚(0+x0)˚(x0)0:Inanaiveway,wecanhavetheboundD1;1r 1k˚kZj0j>3j}j01+02+a2n202+a2n2j0j2+a2n2j0j2+a2n2:(5.25)Usingthemeanvaluetheoremforsomez2thatliesbetweena2anda2,wehavethefollowingD1;1C(n)r 1k˚kZj0j>3j}j01+02+z2n21a2a2j0j2+a2n2j0j2+a2n2:(5.26)Withoutlossofgeneralitywemayassumez2a2andsameworkwillholdwhenz2<a2.Therefore,wehavetheboundD1;1C(n)C()k˚kZj0j>3j}j01+jaajja+ajj0j2+a2j0j2+a2n2:(5.27)Sincea2<a2,wecaneasilyhavetheinequalityja+ajj0j2+a2j0j2+a2n21j0j1+n:(5.28)141Bysubstitutinginequality(5.28)in(5.27),wehavetheboundD1;1C()k˚kZj0j>3j}j0njaaj0:(5.29)Sincea=+ 2(x0) 1(x0)anda=+ 2(x0) 1(x0).Thenweeasilyhavetheboundjaajj}jr 2+j}jr 1ThuswecanseethatD1;1C()k˚kj}jZj0j>3j}j0n0C()k˚kj}jZ1r=3j}jr2drC()k'kj}j:(5.30)Combining(5.24)and(5.30),theboundforD1follows,thatis,jD1jC()k'kj}j:(5.31)Whilecombining(5.23)and(5.31)givestheboundforC2,thatis,jC2jC()k'kj}j:(5.32)142ThereforeB4isboundedby(5.21)and(5.32),thatis,jB4jC()k'kj}j:(5.33)ForBtermwecombine(5.17),(5.18),(5.19)and(5.33)togettheboundjBjC()k'kj}j:(5.34)Finally,toletthisendwecombine(5.16)and(5.34)togetjI1'(x)I1'(x)jC()k'kjxxj:(5.35)5.2.2olderestimateforI2,>0.Wefollowthesameideathatpresentedfordimensionn=2in[1]tondtheoldercontinuityforI2.FromtheofI2wehaveI2'(x)=ZRn1aj0j2+a2n2˚(0+x0)0:Weusechangeofvariable0=a$togetI2'(x)=ZRn111+j$j2n2˚(a$+x0)d$:143WeformI2'(x)I2'(x)=ZRn111+j$j2n2˚(a$+x0)˚(a$+x0)d$:Byusingtheoldercontinuityof˚,wehavethefollowingboundjI2'(x)I2'(x)jk˚kZRn111+j$j2n2a$+x0a$x0d$k˚kZRn111+j$j2n2x0x0x0x0jx0x0j+(aa)$jx0x0jd$k˚kx0x0ZRn111+j$j2n21+j$jjaajjx0x0jd$:Weboundjaajjx0x0jC 2 11Ck 2k10+k 1k10:(5.36)Substituting(5.36)intheaboveboundforI2,itfollowsthatjI2'(x)I2'(x)jk˚kx0x0ZRn111+j$j2n21+C()j$jd$:(5.37)Noticethat1+C()j$j1+j$j2n2!11+j$j2n2as!0:144Thusbyusingthedominatedconvergencetheorem,weseethatZRn11+C()j$j1+j$j2n2d$!ZRn1d$1+j$j2n2=!n2as!0:ThereforewehavejI2'(x)I2'(x)j!n2+C()k˚kx0x0:(5.38)Involvingthebound(5.1),wegetthedesiredolderboundforI2jI2'(x)I2'(x)j12!n+C()(1+0)k'kx0x0:(5.39)Combining(5.35)and(5.39),wehaveI2'(x0)I2'(x0)12!n+C()(1+0)k'kx0x0:(5.40)ThustheolderestimateforI2follows,thatis,I2'C1)12!n+C()(1+0)k'k;(5.41)andthetheoremisprovedwhen>0.Forthecase=0,itfollowsifweprovethatfor'2C2),thefollowinglimitholdslim!0I2'(x)=I02'(x);x21\B(R);145andthiswillbetheaimofthenextSection.5.3ConvergenceofI2.TheoperatorI2thatwedealwithinthischapterisnicerthantheoperatorJ2becauseitisproducedfromthenormalderivativeofthesinglelayerpotentialthatontheconstanthypersurface.WhereastheoperatorJ2isthebetweenthenormalderivativeofthesinglelayerpotentialthatontheapproximatesurface2andtheoperatorI2.WehaveshownintheprevioussectionsthatI2iscontinuouslinearoperatorfromC2)toC1)forany<0,aswellasitslimitoperatorI02isagaincontinuouslinearoperatorfromC2)toC1)forany<0.Beforeproceedingtostudytheconvergence,itisappropriatetowritedownthedeoftheoperatorI2inordertovisualizethesingularityofitskernelfunction.For'2C2)andx21\B(R0)werewrotetheoperatorI2inthefollowingformI2'(x)=I2'(x)I1'(x);wherefor>0orx6=0theoperatorsI1andI1arebyI1'(x)=Zj0j<Mr 1(x0)0j0j2+a2(x0)n2˚(0+x0)0;I2'(x)=ZRn1a(x0)j0j2+a2(x0)n2˚(0+x0)0;wherea:=a(x0)=+ 2(x0) 1(x0):146Whileforx=0and=0,wehavethatI01'(0)=0;andI02'(0)=˚(0)!n2:Proposition5.3.Let'2C2).Thenforany<0wehavetheuniformconvergencelim!0I2'I02'1=0;whereisthesmoothfunctiondin(3.11)whichthefollowingproperties01,equalsto1intheballB(0),supportedintheballB(R0)andkrk10.Proof.For'2C2)and06=x21\B(R0),weformthefollowingI1'(x)I01'(x)=Zj0j<M0B@0r 1(x0)j0j2+a2(x0)n20r 1(x0)j0j2+a20(x0)n21CA˚(0+x0)0:WetakeadvantageoftheintegralZj0j<M0B@0r 1(x0)j0j2+a2(x0)n20r 1(x0)j0j2+a20(x0)n21CA˚(x0)0=0;togainsomeregularityfromtheoldercontinuityofthefunction˚.Thatis,weformthe147followingI1'(x)I01'(x)=Zj0j<M0B@0r 1(x0)j0j2+a2n20r 1(x0)j0j2+a20n21CA˚(0+x0)˚(x0)0:Itisaworthwhiletosplitthedomainoftheintegralf0<Mgintotwosubdomainsf0<gandf<0<Mgandthenweproceedtoarguetheuniformconvergenceoneachsubdomainseparately.Forthat,werewriteI1'(x)I01'(x)=I1;1+I1;2;whereI1;1=Zj0j0B@0r 1(x0)j0j2+a2n20r 1(x0)j0j2+a20n21CA˚(0+x0)˚(x0)0;andI1;2=Z<j0j<M0B@0r 1(x0)j0j2+a2n20r 1(x0)j0j2+a20n21CA˚(0+x0)˚(x0)0:WeeasilycanboundI1;1bythefollowingI1;1r 11k˚kZj0j01+n0:148ThuswehaveI1;1Cr 11k˚k:(5.42)Thereforeweclearlyseetheuniformconvergencefromthebound(5.42)overthesubdomainf0<g.AgainforthesecondintegralI1;2wetriviallyobtainthefollowingboundI1;2k˚kZ<j0j<Mr 1(x0)01+1j0j2+a2n21j0j2+a20n20:Noticingthat,0a0(x0)<a(x0)anda(x0)a0(x0)=.Thenbyapplythemeanvaluetheoremforsomez2thatsatisfyinga20(x0)<z2<a2(x0),wehavethefollowingI1;2Ck˚kZ<j0j<Mr 1(x0)00a(x0)+a0(x0)j0j2+a20(x0)n2j0j2+a2(x0)0since0a(x0)+a0(x0)202+a2(x0);weconcludethatI1;2Ck˚kZ<j0j<Mr 1(x0)0j0j2+a20(x0)n20:ByusingLemma4.2toboundthefunctionr 1(x0),wehaveI1;2Ck˚kZ<j0j<M 1(x0)1+0j0j2+ 2(x0) 1(x0)2n20:149AlsobyusingLemma4.3toboundthedenominator,wegetthefollowingI1;2Ck˚kZ<j0j<M 1(x0)1+0j0jn2(1+) 1(x0)n2(1)0:Bychoosing=12n(1+),itfollowsthatI1;2Ck˚k(1)Z<j0j<M0n+1+0:ThenweclearlyseethatI1;2Ck˚kZ<j0j<M01n+1+0:ThereforeweobtainthefollowingboundI1;2Ck˚k:(5.43)Thustheuniformconvergencefollowsonthesubdomainf<0<Mg.AftertheconvergenceofI1.WecontinuetoshowtheuniformconvergenceoftheoperatorI2.For06=x21\B(0;R0),wehaveI2'(x)I02'(x)=ZRn1 a(x0)˚(0+x0)j0j2+a2(x0)n2a0(x0)˚(0+x0)j0j2+a20(x0)n2!0:Fortheintegrandweusethechangeofvariables0=a(x0)$,whileforthesecond150integrandweusethechangeofvariables0=a0(x0)$,thenitfollowsthatI2'(x)I02'(x)=ZRn111+j$j2n2˚(a(x0)$+x0)˚(a0(x0)$+x0)d$Since˚isoldercontinuous,thenweeasilyobtainthefollowingboundI2'(x)I02'(x)k˚kZRn1j$j1+j$j2n2d$:ObservethatZRn1j$j1+j$j2n2d$=Zj$1j$j1+j$j2n2d$+Zj$j>1j$j1+j$j2n2d$;Zj$1j$j1+j$j2n2d$ZRn111+j$j2n2d$=!n2:andZj$j>1j$j1+j$j2n2d$Zj$j>1j$jnd$C1:ThereforewehaveI2'(x)I02'(x)Ck˚k:Thatis,I2convergesuniformlytoI02whenx6=0.Finally,weshowtheuniformconver-151gencewhenx=0.InthiscasewehaveI2'(0)I02'(0)=ZRn1 ˚(0)j0j2+2n2˚(0)1+j0j2n2!0:AfterchangingthevariablesweobtainthefollowingI2'(0)I02'(0)=ZRn1˚($)˚(0)j$j2+1n2d$ThenweclearlyseetheboundI2'(0)I02'(0)Ck˚kZRn1j$j1+j$j2n2Ck˚k:Thereforethetheoremfollows.ThefollowingTheoremshowstheconvergenceofI2'inC01)forany0<where'2C2)anditcanbeprovedinthesamemannerasTheorem4.6.Theorem5.4.[1]Fix0<2<0.Thenforany0<0<<0andforall'2C2),wehaveI2'0!I02'as!0:152Chapter6Theconvergenceandthemainresults.InthisChapterwewillshowtheoperatorsT1existforany00andareuniformlyboundedintheoperatornorm.WehaveshowninSection3.1theoperatorsTareinvertiblefor>0,butithasprovenin[1]thattheoperatorsTdonotconvergeinnormtotheirlimitingoperatorT0.FromelementaryfunctionalanalysisweknowthatIftheoperatorsTweretoconvergeinnormtoT0andif(T0)1exists,thenweimmediatelywouldgettheuniformboundednessfor(T)1inoperatornorm.Therefore,accordingtotheabsenceofthenormconvergence,weneedtostudyfurtherpropertiesfortheoperatorsTinordertohavetheuniformboundednessfortheirinverses.In[1]forthedimensionn=2,theyhaveusedthenotionofcollectivelycompactoperatorstogettheinvertibilityofT0andtheuniformboundednessoftheoperatorsT1.Wewillmimictheideathatpresentedin[1]toobtaintheuniformboundednessfor(T)1inanydimensionn2.6.1PreliminariesLetX,YandZberealorcomplexBanachspacesandletBbetheclosedunitballinX,thatis,B=fx2X:kxk1g:153DenotebyL(X;Y)theBanachspaceofboundedlinearoperatorsT:X!Ywiththeusualoperatornorm,kTk=supx2BkTxk:ItisknownfromelementaryfunctionalanalysisthatasubsetX0ˆXisrelativelycompactX0issequentiallycompactX0istotallybounded.Recallthatasetistotallyboundedifforany>0,thereexistsacoverby-balls.InthisChapter,wewillusekTnTk!0todenoteforconvergenceinnormandTn!Ttodenoteforpointwiseconvergence(strongconvergence),thatisTnx!Tx8x2X:AnoperatorK2L(X;Y)iscalledcompactthesetKBisrelativelycompact.Theconceptofcompactoperatorsplaysanimportantroleinsolvingequationsoftheform(IK)x=y:(6.1)Inappliedmathematics,equation(6.1)isknownasFredholmalternative,thatis,(6.1)has154uniquesolutionthehomogeneousequation(IK)x=0;(6.2)hasonlythetrivialsolutionx=0.Insuchacase,theFredholmoperatorIK:X!Yhasaboundedinverse(IK)1.Thepracticalsolutionofequation(6.1)oftendependsontheapproximationoperators.Thatis,toasolutionof(6.1),itoftenrequirestoasolutionofthecorrespondingapproximateequation(IKn)xn=y;(6.3)whereKnarecompactoperatorsandkKnKk!0.Then(IK)1existsforsomeNandallnNthereexistuniformlybounded(IKn)1.Insuchacase,(IKn)1(IK)1!0:Wenoticethisschemerequiresconvergenceinnorm.Insomecases,convergenceinnormcannothold.Indeed,itispossibletohavepointwiseconvergenceonly.Therefore,weseekanalternativeschemethatrequirespointwiseconvergenceinsteadofthenormconvergence.Suchaschemeisindeedavailable:ForasequenceofcollectivelycompactoperatorsKn,suchthatKn!K(whereKisacompactoperator).Then,(IK)1existsifandonlyifforsomenNtheoperators(IKn)1existandareboundeduniformly,insuchacase(IKn)1!(IK)1:155Letusintroducethenotionofcollectivelycompactoperatorswhichisageneralizationofthenotionofcompactoperators.Collectivelycompactoperatorsareveryimportanttoolsforsolvingintegralequationsofthesecondkindandhavebeenstudiedinseveralpapers[4,5,6].Weuse[3,5]tointroducetheofcollectivelycompactandsomebasicresultsthatareneededforourwork.6.1.[3]AsetKˆL(X;Y)iscalledcollectivelycompactifthesetKB=fKx:K2K;x2Bgisrelativelycompact.Wenoticefromtheofcollectivelycompactness,everyoperatorincollectivelycompactsetiscompact.Example6.2.[3]LetX=`2.Kn2L(X);n=1;2;,byKnx=(xn;0;0;):LetK=fKng.ThenKBisboundedanddim(KX)=1.ThusKiscollectivelycompact.ThefollowingLemmashowsthatpointwiselimitofcollectivelycompactoperatorsiscompact.Lemma6.3.[3]LetK;Kn2L(X);n=1;2;,besuchthatK=fKn;n=1;2;gisacollectivelycompactsetandKnconvergespointwisetotheoperatorK.ThenKiscompact.156Proof.KBˆfKnx:n1;x2Bg=fKngB:ThusKiscompact.Lemma6.4.[5]LetKˆL(X;Y)beacollectivelycompactsetandMˆL(Z;X)beaboundedset.ThenthesetKMiscollectivelycompact.Proof.LetBbetheclosedunitballinXandB0betheclosedunitballinZ.SinceMisboundedthenthereexistr>0besuchthatkMk<rforallM2M.ThusKMBˆrKB0:SinceKiscollectivelycompact,thenKMBiscompact.ThereforeweseethatKMiscollectivelycompact.Remark6.5.[3]Acollectivelycompactsetisaboundedsetofcompactoperators,buttheconverseisfalseaswewillseeinthefollowingexample.Example6.6.[3]LetX=`2withtheorthonormalbasisf':2IgandletPbetheorthogonalprojectionontotheone-dimensionalsubspacespannedby'.ThenPx=(x;')';kPk=1;andPiscompactsincedimP`2=1.ThusK=fP;2Igisaboundedsetofcompact157operatorsinL(`2).However,Kisnotcollectivelycompactbecause'=P'2KB;82I;and''=p2for6=:ThereforeKBisnottotallybounded.ThefollowingLemmaisveryusefulanditplaysafundamentalroleinourworkbecauseitjustrequirespointwiseconvergence.Lemma6.7.[3]LetXbeaBanachspaceandK,KnbeboundedlinearoperatorsonXforn=1;2;:::.AssumethatfKngiscollectivelycompactandKiscompactandKn!K.Then,(IK)1existsifandonlyifforsomenNtheoperators(IKn)1existandareboundeduniformly,inwhichcase(IKn)1!(IK)1.6.2CollectiveCompactnessandconvergenceofK2.AfterintroducingtheconceptofcollectivelycompactnessinthepreviousSection,weusetheconcepttoshowthesetK=fK2;0<<0giscollectivelycompact,wheretheoperatorsK2areinSection3.3.Theorem6.8.Letbedwith0<2<0.TheoperatorsK2:C2)!C1),0<<0,formacollectivelycompactfamilyofoperators.Proof.For'2C2)andx21\B(R0),recalltheoftheoperatorK2from158Section3whichisK2'(x)=1!nZ2(xyen)(x)jxyenjn'(y)d˙(y)+1!nZ2(xyen)(x)jxyenjnE'P(y)d˙(y);where(x)=1q1+r 1(x0)20B@ 1(x0)11CA;Pistheprojectionmapfrom2ontoRn1andEistheextensionoperatoronC2)intoC(Rn1)thatdein(3.13).Now,whenjxj<2weclearlyseethatfory0<thetwointegrandsintheaboveexpressioncoincide.ThuswehavethefollowingK2'(x)=1!nZf2\jy0jg(xyen)(x)jxyenjn'(y)d˙(y)+1!nZf2\jy0jg(xyen)(x)jxyenjnE'P(y)d˙(y);andmoreexplicitlywiththehelpoftheauxiliaryfunctionswerewritetheoperatorK2inthefollowingformK2'(x)=1!nZf2\jy0jg(xyen)(x)jxyenjn'(y)d˙(y):=K2;1'(x)+1!nq1+r 1(x0)2Zjy0j(y0x0)r 1(x0)+ 1(x0) 2(y0)jx0y0j2+( 1(x0) 2(y0))2n=2˚(y0)dy0:=K2;2'(x):Thenweproceedtodiscusstheregularityandthecompactnessofeachtermthatinvolvedintheaboveexpressionseparately.First,letusassumethatH1andH2bethekernel159functionsthatassociatedwiththeoperatorsK2;1andK2;2respectively.Thatis,H1(x;y):=(xyen)(x)jxyenjn;(6.4)andH2(x;y):=1q1+r 1(x0)2(y0x0)r 1(x0)+ 1(x0) 2(y0)jx0y0j2+( 1(x0) 2(y0))2n=2:(6.5)FortheoperatorsK2;1'(x)=1!nZf2\jy0jgH1(x;y)'(y)d˙(y);jxj<2:WeobviouslyseethekernelfunctionH1(x;y)is0oldercontinuousforany0<01withrespecttoxandsmoothwithrespecttoyaswellasH1isuniformlyboundedindependentlyof.ThusthecompactnessoftheoperatorsK2;1follows.Similarly,fortheoperatorsK2;2'(x)=1!nZjy0jH2(x;y)˚(y)dy0;jxj<2:ThekernelfunctionH2(x;y)is0oldercontinuousforany0<01withrespecttoxandyalsoH2isuniformlyboundedindependentlyof.Therefore,theoperatorsK2;2arecompact.ThusthecompactnessandtheregularityoftheoperatorsK2followwhenjxj<2.Thenwecontinuetodiscussthepropertiesoftheoperatorswhenjxj2.Inthis160casewehaveK2'(x)=1!nZ2H1(x;y)'(y)d˙(y)+1!nZRn1H2(x;y)˚(y0)dy0;(6.6)whereH1(x;y)andH2(x;y)arein(6.4)and(6.5)respectively.Again,clearlyweseethatthekernelfunctionH1(x;y)is0oldercontinuousforanyx21\B(R0)nB(2)andy22eventhough=0aswellasH1(x;y)isuniformlyboundedindependentlyof.Thusthecompactnessandtheregularityoftherstintegralin(6.6)follow.Forthesecondintegralin(6.6),weseethatthedenominatorofthekernelfunctionH2(x;y)nevervanisheseventhough=0becausetheauxiliaryfunction 1alwaysnegativewhenx0>0thereforetheregularityofH2andtheuniformboundednessfollow.Thereforeweobtaintheregularityandtheuniformboundednessaswellasthecompactnessofthesecondintegralin(6.6).FromtheaboveobservationweconcludethattheoperatorsK2arecompactoperatorsfromC2)intoC01)aswellastheyareuniformlyboundedindependentlyof.ThusthefamilyF=fK2;0<<0gisuniformlyboundedinoperatornormfromC2)intoC01).Since,theembeddingC01),!C1);<0;iscompact,thenweseethatthefamilyFiscollectivelycompact.Asaresultfromtheabovetheorem,wehavethepointwiseconvergenceoftheoperators161K2.Thatis,Corollary6.9.Let0<<0,and0<2<0.Thenforall'2C2),wehaveK2'!K02';inC1);asapproaches0.Proof.ThedenominatorsofthekernelfunctionsassociatedtotheoperatorsK2areboundedawayfrom0eventhough=0aswellasthekernelfunctionhasCregularity.Thusbyusingthedominatedconvergencetheoremwecanpassthelimitinsidetheintegralsign.6.3PointwiseConvergenceofT.ThisSectiondealswiththepointwiseconvergenceoftheoperatorTthatinSection3.4bythefollowing,for>0,TasanoperatorinLC1)C2)andhastheformT=+C;where=0BBBBB@!nr1+r 12J2+I2!nr1+r 22J1+I11CCCCCA;andC=0B@K1K2K1K21CA+(1)0B@0L2L101CA:162For=0,TisbyT0=0+C0;where0=0BBBBB@!nr1+r 12J02+I02!nr1+r 22J01+I011CCCCCA;andC0=0B@K1K02K01K21CA+(1)0B@0L02L0101CA:AsaconsequenceofTheorems4.6,5.4andCorollary6.9,weobtainthefollowing,Corollary6.10.Fix0<2<0.Thenforany0<0<andforall('1;'2)2C1)C2),wehave0B@'1'21CA!00B@'1'21CA;inC01)C02);C0B@'1'21CA!C00B@'1'21CA;inC1)C2);163andT0B@'1'21CA!T00B@'1'21CA;inC01)C02):Furthermore,theoperatorsCareuniformlyboundedinLC1)C2).Proof.BytheboundedprincipletheoremtheuniformboundednessoftheoperatorsCfollows.Theorem6.11.For=k+12(k1)and0<0<.Thereexists0>0suchthattheoperators,00,areinvertiblewithinversesthatsatisfying800;1Cjj121+C()(1+0);(6.7)uniformlywithrespecttointheoperatornormLC1)C2)andC()approaches0asapproaches0.Furthermore,forany('1;'2)2C1)C2),wehave10B@'1'21CA!100B@'1'21CAinC01)C02);0<:Proof.FromTheorems(4.1)and(5.1),wehaveshownthatforany00and0<thefollowingboundholds!nq1+r 12J2+I2L(C2);C1))121+C()(1+0);164andinthesimilarmannerwehavethebound!nq1+r 12J1+I1L(C1);C2))121+C()(1+0):Recallthatfromequation(3.10),0hasbeenchosentosatisfytheinequality1+02<jj:Thuswemaychoose>0tlysmallsuchthat121+C()(1+0)<jj:Thereforeareinvertibleandsatisfythebound(6.7).FromCorollary(6.10),itfollowsthatfor('1;'2)2C1)C2)wehavethepointwiseconvergence10B@'1'21CA!100B@'1'21CAinC01)C02);0<:InthefollowingTheoremwewillusethenotionofcollectivelycompactoperatorsthatintroducedinSection6.1.Theorem6.12.For0<<1,thefollowinghold1.TheoperatorsfC1garecollectivelycompactonC1)C2),2.C1!C010pointwiseinLC1)C2)asapproaches0.165Consequently,fC010giscompactoperatoronC1)C2).Proof.ThecollectivelycompactnessofthesetfC1gfollowseasilyfromtheuniformbound(6.7),Theorem6.8,thecompactnessoftheoperatorsKi;i=1;2andthenLemma6.4.Forthepointwiseconvergence,let'2C1)C2).SincetheoperatorsCarecollectivelycompactonC1)C2)and110)'areuniformlyboundedinC1)C2),thenforanysubsequenceCn1n10)'thereexistsfurthersubsequenceCnj1nj10)'whichconvergestosomefunctionL2C1)C2).FromTheorem6.11,wehave1'!10'inC01)C02);0<;andalsowehavefromCorollary6.10thatCareuniformlyboundedinL(C2)C1)).Therefore,Cnj1nj10)'!0inC01)C02):Byuniquenessofthelimit,weobtainthatL0.Thatis,Cnj1nj10)'!0inC1)C2):ThereforethesequenceC110)'convergesto0inC1)C2).Now,weformthefollowingC1C010'=C110'+CC010':166ThusweobtainthatC1convergespointwisetoC010inC1)C2).SinceC0isacompactoperatorand0isinvertible,thenT0=0+C0isaFredholmoperator.ThusbythehelpofTheorem6.12wehaveallingredientstoshowtheinvertibilityofthelimitingoperatorT0.Theorem6.13.TheoperatorT0:C1)C2)!C1)C2)isinvertible.Proof.IttoshowT0isinjective.Let('1;'2)2C1)C2)besuchthatT00B@'1'21CA=0:(6.8)FromCorollary6.10,wehavethatL2'2!L02'2inC01)for0<asapproaches0.ThereforeweobtainZ1L02'2d˙=lim!0Z1L2'2d˙=lim!0Z1@S2'2(xen)d˙(x)SinceS2'2(xen)isharmonicinD1,thenwehaveZ1L02'2d˙=0:ThusweobtainthefollowingZ1(K1)'1+L02'2d˙(x)=Z1(K1)'1d˙(x):167SinceK1istheL2adjointofK1,thenwehavethefollowingZ1(K1)'1d˙(x)=Z11K1(1)'1d˙(x):FromTheorem2.9,wehaveK1(1)=12,thusweobtainZ1(K1)'1+L02'2d˙(x)=(12)Z1'1d˙(x):(6.9)AsimilarresultholdforK2'2+L01'1on2,thatisZ2(K2)'2+L01'1d˙=(12)Z2'2d˙:(6.10)Fromtheassumption(6.8),wehave(K1)'1+L02'2=0on1;and(K2)'2+L01'1=0on2:Thereforbyusing(6.9)and(6.10),itfollowsthat(12)Z1'1d˙=(12)Z2'2d˙=0:Sincejj>12,wehaveZ1'1d˙=Z2'2d˙=0:(6.11)168Now,considerthefunctionw=S1'1+S2'2onRn;(6.12)whereSiisthesinglelayerpotentialoni,i=1;2:thatis,Si'i(x)=Zix;y)'i(y)d˙(y);i=1;2:andx;y)isthefundamentalsolutiontotheLaplaceequation.Fromthesmoothnessofthesinglelayerpotentials,weseethatwispiecewiseharmonic,thatis,wisharmonicinsideandoutsidetheinclusionD1[D2.Byusing(6.11),weeasilyconcludethatw(x)=O(jxj1n)asjxj!1:(6.13)SincewisharmonicoutsideD1[D2,thenbyusing(6.13)weobtainrw(x)=O(jxjn)asjxj!1:(6.14)Furthermore,fromLemma2.18,wehavetheboundsrSj'j0Dj+rSj'j0RnnDj)C'j0<;j=1;2:(6.15)Nowweshowthatwislocallyaweaksolutionto8>>><>>>:div(akrw)=0;inRnnf0g;ak=1+(k1)˜(D1[D2):(6.16)169LetR>>1andobservethatZB(R)akrwrdx=kZD1r(S1'1(x)+S2'2(x))rdx+kZD2r(S1'1(x)+S2'2(x))rdx+ZB(R)n(D1[D2)r(S1'1(x)+S2'2(x))rdx:Usingthedivergencetheorem,itfollowsZB(R)akrwrdx=kZ1@S1'1(x)+@S2'2(x)d˙(x)+kZ2@S1'1(x)+@S2'2(x)d˙(x)Z1@+S1'1(x)+@S2'2(x)d˙(x)Z2@S1'1(x)+@+S2'2(x)d˙(x):Utilizingthejumpconditionsforthesinglelayerpotential(2.23),wehaveZB(R)akrwrdx=(1k)Z1 (K1)'1(x)@S2'2(x)!d˙(x)+(1k)Z2 (K2)'2(x)@S1'1(x)!d˙(x):Byapplying(6.8),itfollowsthatZB(R)akrwrdx=0:Thuswisalocalsolutionto(6.16).Next,weshoww0onRn.Letr>>1and=w˜,170where˜2C10(Rn)besuchthat˜1inB(r1)and˜0outsideB(r).ThenwehaveZB(r)akrwrw˜dx=0:ThatisZB(r)akjrwj2˜dx+ZB(r)nB(r1)akrwr˜wdx=0:(6.17)Forthesecondintegralintherighthandsideof(6.17),weusethedecayconditions(6.13)and(6.14)toobtainthefollowingZB(r)nB(r1)akrwr˜wdxCrn+1!0asr!1:(6.18)Thereforewhenrapproaches1in(6.17),weconcludethefollowingZRnakjrwj2dx=0:(6.19)ThuswisaconstantinRn.Thenbyusingthedecaycondition(6.13),wehavew0inRn.Consequently,involvingthejumpconditionsforthesinglelayerpotential(2.23),wehavethefollowing'i(x)=@@+w@@w=0;x2inf0g;i=1;2:Thenbyusingthecontinuityof'iat0,itfollowsthat'i0inRn.ThusT0isinjective.171SinceT0and0areinvertibleandT0=I+C0100:Thenweseethat0T01=I+C0101:Thatis,I+C0101exists.FromTheorem6.12wehaveI+C1!I+C010inC1)C2):ThusfromLemma(6.7)wehavethatI+C11existforistlysmallandtheyareuniformlyboundedwithrespecttointheoperatornorm.Therefore,weobtainthepointwiseconvergenceI+C11!I+C0101:(6.20)SinceT=I+C1,thenbyusingthebound(6.7)weconcludethat(T)1=1I+C11areuniformlynormboundedandsatisfy(T)1!(T0)1inC01)C02);0<;as!0.ThuswehaveprovedthefollowingTheorem.172Theorem6.14.For=k+12(k1)and<0.Thereexists0>0suchthattheopera-torsT,00,areinvertiblewithinversesthatareboundedindependentlyofinL(C1)C1)),<0.Moreover,theoperators(T)1convergepointwiseto(T0)1asapproaches0inL(C01)C01))forany0<<0.6.4Themainresults.Themainresultsaresimilartothecaseofdimensionn=2.Thesolutionofproblem(2.5)hastherepresentation(2.58)intermsofthesolutions'1;'2to(2.59)andtheharmonicfunctionHfrom(2.49).Asimilarrelationshipholdsbetweenthesolutionu0toproblem(2.4)withtouchinginclusionsandthesolutions('01;'02)toT00B@'01'021CA=0B@@H0j1@H0j21CA;(6.21)whereH0istheharmonicfunctionfrom(2.48).Thisistheassertionofthefollowingtheorem.Theorem6.15.[1]Thesolutionu0,to(2.4),maybewrittenu0(x)=S1'01(x)+S2'02(x)+H0(x);x2;(6.22)whereH0isharmonicinside,anddby(2.48),andthepair('01;'02)2C1)C2)istheuniquesolutionto(6.21).Proof.SineH0isharmonicinsideandsince1and2areC1+0,theright-handsideof(6.21)liesinC1)C2)forany0.Byusingtheorem(6.14),theintegralequation(6.21)thereforhasauniquesolution('01;'02)2C1)C2),forany0.173UtilizingLemma(2.16),weseethat@Hji!@H0ji;inCi)as!0:Soweinferfromtheorem(6.14)that0B@'1'11CA0B@'01'011CA=T10B@@Hj1@Hj21CAT010B@@H0j1@H0j21CA=T1"0B@@Hj1@Hj21CA0B@@H0j1@H0j21CA#+"T1T01#0B@@H0j1@H0j21CA!0inC01)C02);0<0<:Thisconvergenceof'iimmediatelyimpliesthatS1'1(x+2en)!S1'01(x);andS2'2(x2en)!S2'02(x);(6.23)uniformlyoncompactsubdomainsofn1[2)as!0.Considernowthesolutiontoproblem(2.5)u(x)=S1'1(x+2en)+S2'2(x2en)+H(x):FromLemma2.16,weknowthatH!H0uniformlyoncompactsubdomainsofand174ifwecombinethiswith(6.23),itfollowsthatu(x)=S1'1(x+2en)+S2'2(x2en)+H(x)!S1'01(x)+S2'02(x)+H0(x);uniformlyoncompactsubdomainsofn1[2as!0.Sincewealsoknowthatu!u0inH1itfollowsfromuniquenessofthelimitthatu0=S1'01+S2'02+H0;(6.24)oncompactsubdomainsofn1[2.Butbothsidesof(6.24)arecontinuousfunctionsinwegetthatu0(x)=S1'01(x)+S2'02(x)+H0(x);x2:Theorem6.16.[1]Let~>0and0<<0.Thesolutionto(2.5)kukC1~nD1[D2)+kukC1(D1)+kukC1(D2)CkgkL2(@;(6.25)whereCisindependentofandg.Proof.Recallthatuhastherepresentationu(x)=S1'1(x+2en)+S2'2(x2en)+H(x);where'1;'2solves(2.59)inC01)C02),forany<0<0.Fromequation175(2.62),wehaveSi'iC1(Di)+Si'iC1(~nDi)C'iC0i);i=1;2:Duetotheorem(6.14)andthefactthat'1;'2solves(2.59),wehave'1C01)+'2C02)CkHkC10~):ApplyingLemma(2.16),wehavethatkHkC10~)CkgkL2(@:176BIBLIOGRAPHY177BIBLIOGRAPHY[1]H.Ammari,E.Bonnetier,F.Triki,andM.Vogelius.Ellipticestimatesincompositemediawithsmoothinclusions:anintegralequationapproach.Ann.Scient.Ec.Norm.Sup.,48(2):453{495,2015.[2]H.AmmariandH.Kang.Reconstractionofsmallinhomogeneitiesfromboundarymea-surement.Springer,2004.[3]P.Anselone.Collectivelycompactoperatorapproximationtheoryandapplicationstoinetgraleqautions.Prentice-Hall,Inc.,1971.[4]P.AnseloneandR.Moore.Approximationsolutionsofintegralandoperatorequations.J.Math.Anal.Appl.,(9):268{277,1964.[5]P.AnseloneandT.Palmer.Collectivelycompactsetsoflinearoperators.J.Math.,25(3):417{422,1968.[6]P.AnseloneandT.Palmer.Spectralanalysisofcollectivelycompact,stronglyconver-gentoperatorsequences.J.Math.,(25):423{431,1968.[7]A.Bensoussan.Asymptoticanalysisforperiodicstructures.AMSChelsea,1978.[8]E.BonnetierandM.Vogelius.Anellipticregularityresultforacompositemidiumwithtochingersofcircularcross-section.SiamJ.Math.Anal.,31(3):651{677,2000.[9]G.CittiandF.Ferrari.Asharpregularityresultofsolutionsofatransmissionproblem.AmeriMath.Society,140(2):615{620,2012.[10]D.ColtonandR.Kress.Integralequationmethodinscatteringtheory.PureandAppliedMath.,JohnWiley&Sons,Inc.,1983.[11]E.DiBendetto.Partialentialequations,secondedition.BirkhauserBoston,2010.[12]Y.L.E.BaoandB.Yin.Gradientestimatesfortheperfectaconductivityproblem.Arch.RationalMech.Anal.,193:195{226,2009.[13]Y.L.E.BaoandB.Yin.Gradientestimatesfortheperfectandinsulatedconductivityproblemswithmultipleinclustions.Comm.Part.Eqs.,35(35):1982{2006,2010.178[14]D.GilbargandN.Tru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