‘ ca... all"... RETURNING MATERIALS: MSU Place in book drop to remove this ‘ checkout from your record. FINES UBRAR'ES will be charged if book is returned ‘2— after the date stamped below. NUMERICAL ANALYSES OF SOME NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS by Dongming Wei A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1988 ABSTRACT NUMERICAL ANALYSES OF SOME NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS by Dongming Wei In this work, several problems governed by the so-called " p-harmonic " equation as well as those by the corresponding parabolic equation are studied. More specifically, we are concerned with problems governed by following two equations: -div(|Vulp'2Vu) =f, ut-div(|Vulp'2Vu)=f. In Chapter 1, steady state problems with both homogeneous and nonhomogeneous Dirichlet boundary conditions are studied. Some known results are collected in preparation for the subsequent chapters. Some less known results are given along with their proofs. These latter include a maximum principle and the continuity of the conformal capacity with respect to the parameter p as well as, in a special case, with respect to boundary change. In Chapter 2, we study finite element approximations of these nonlinear elliptic problems. Applying the theory of P. R. Ciarlet, M. H. Schultz and R. S. Varga [3] to the p-harmonic operator equation ( p 2 2), we can show that the finite element solution converges to the true solution as mesh size h tends to zero. We then show that a minimizing sequence of the p-harmonic functional which converges to the discretized solution, can be constructed in the finite element approximation space 811(9). In surport of the theory, numerical experiments are conducted and test problems along with their finite element solutions are presented including finite element solution for the minimizer of the conformal capacity of a ring with a slit in the middle. In Chapter 3, using the method of lines we show that the nonlinear parabolic problem Dongming Wei experiments are conducted and test problems along with their finite element solutions are presented including finite element solution for the minimizer of the conformal capacity of a ring with a slit in the middle. In Chapter 3, using the method of lines we show that the nonlinear parabolic problem governed by the p-harmonic operator has a unique weak solution which is more "classical" than the weak solution obtained by applying the theory of J. Kac'ur, in the sense that it satisfies the equation pointwise with respect to time. A L2 error estimate for the error between the true solution and its semidiscrete approximation is made. We note that the traditional elliptic projection technique fails in this case for it does not provide us with an error estimate for the time derivative of the difference between the projection and the true solution. This difficulty is overcomed by using a time dependent auxillary problem which one may call a " parabolic projection". It is shown that as the mesh size h tends to zero the semidiscretize solution converges to the true solution in the L2 norm. Numerical computation of the semidiscrete solution is also conducted. In Chapter 4, a nonlinear dam problem which is also governed by the p-hannonic equation is studied. The formulation of the so called "free" boundary value problem, which differs from the above is applicable for flows with relatively large seepage velocities for which Darcy's law is no longer applicable. A weak formulation is presented and it is shown that this weak problem has a solution. ACKNOLEDGMENTS First I would like to express my gratitude to Professor David H. Y. Yen, my thesis advisor, for his guidance and patience. Also I would like to thank Professor Glen Anderson for providing me his papers related to my research problems, and Professor Mary J. Winter who took time to read my drafts and give helpful suggestions. Thanks to all my teachers at Michigan State University for their excellent teaching in my graduate studies. I especially thank my wife, Xiaoling, and my parents for their love and constant encouragement. TABLE OF CONTENTS List of tables Introduction Chapterl Some preliminary results §l.l Notations and preliminaries 1.1.1 Notations 1.1.2 Sobolev spaces 1.1.3 Poincaré's inequality and sobolev imbeddings 1.1.4 The p-harmonic functional and the p-harmonic operator §l.2 Boundary value problems 1.2.1 Existence and uniqueness 1.2.2 A property of the homogeneous equation with Dirichlet data 1.2.3 A Maximum principle 1.2.4 Continuity of the conformal capacity as a function of p 1.2.5 Continuity of the minimizer of the conformal capacity of a ring with respect to boundary change Chapter 2 Finite element approximation of the solution to the p-harmonic equation with Dirichlet data §2.l A finite element approximation of the true solution it §2.2 An Approximation of the finite element solution uh §2.3 Numerical experimental results 2.3.1 A test problem iv WQQMMM 13 13 14 17 19 27 29 29 33 39 39 2.3.2 Computation of the minimizer for the conformal capacity of a ring Chapter 3 The nonlinear semi-parabolic equation §3.l An existence and uniqueness result §3.2 A semidiscret approximation of the initial boundary value problem §3.3 Numerical solutions of the semidiscrete problem Chapter 4 A nonlinear dam problem §4.l A nonlinear porous media equation 4.1.1 Physical Background 4.1.2 Clasical formulation of the dam problem 4.1.3 Weak formulation of the dam problem §4.2 Existence of a solution to the dam problem 4.2.1 An approximation problem 4.2.2 An existence theorem Summary Bibliograhpy 4 1 47 47 55 63 7O 7O 7O 71 72 73 74 78 82 84 LIST OF TABLES 1. Error between true solution and finite element solution 2. Numerical solution for the slit problem with p = 3 3. Numerical solution for the slit problem with p = 4 4. Numerical solution for the semidiscrete problem with p = 3, t = 8 5. Numerical solution for the semidiscrete problem with p = 7, t = 8 6. Numerical solution for the semidiscrete problem with p = ‘7, t = 100 41 43 45 68 INTRODUCTION In this work, several problems governed by the so-called " p-harmonic " equation as well as those by the correSponding parabolic equation are studied. More specifically, we are concerned with problems governed by following two equations: -div(qu|p'2Vu)=f, ut-div(qulp'2Vu)=f. In Chapter 1, steady state problems with both homogeneous and nonhomogeneous Dirichlet boundary conditions are studied. Some known results are collected in preparation for the subsequent chapters. Some less known results are given along with their proofs. These latter include a maximum principle and the continuity of the conformal capacity with respect to the parameter p as well as, in a special case, with respect to boundary change. In Chapter 2, we study finite element approximations of these nonlinear elliptic problems. Relatively few results seem to be available in the literature that are devoted to finite element solutions of nonlinear elliptic boundary value problems. And the majority of the available ones are done in Hilbert spaces, such as Hk(Q). For a problem that has a potential, I say, over a Hilbert space H, classical methods such as the method of steepest descent, the Newton method, the secant modulus method and the differential homotopic method, etc. have been successful under rather stringent conditions, e.g. typically assuming that the second Gateaux derivative of the funtional I be positive definite and uniformly bounded 1 from above in the following sense: (1) 2t." h II2 S J"(u, h, h) (2) I J"(u, h, h) | .<_ All h "2, where I. and A are positive constants. See for example, Ne‘cas, J. [25]. The first assumption is essential and guarantees a unique solution to the problem. The second assumption which gives rise to a uniform upper bound for the inverse of the Hessian of the functional ( while it plays an important role in Newton's method and the method of differential homotopy ) is often too strong for nonlinear problems and can be relaxed, at least for the functional in this work. Also, working in a Hilbert space has the advantage of having the gradient of the functional being in the Hilbert space itsef, and thus allows one to construct gradient related iterates converging directly to the hue solution. There is a large class of nonlinear elliptic problems which have solutions in Banach spaces such as Wk’p(Q), p >1, p at 2, in which direct approximations using gradient method fail since gradients of these functionals may not belong to Wk’pGI). And assumptions (1) and (2) exclude some of the most common examples of nonlinear elliptic PDEs, for example: (a) The minimum surface problem Find 11 6 111(9), such that it minimizes the functional J(u)=J‘\/l+qul2dx, nIaQ=q> , (be H162). One can show that 1Vhl2 SJ"(u,h,h)Sn[|Vhl2dx \/(1+qu|2)3 for which assumption (1) is not satisfied. (b) The p-harmonic problem Find u e Wk’p(§2), such that u minimizes the functional J(u)=% r[Ivttll’tix.-(f,u), u IaQ= , e wktpm). One can also show that for u e Wk’pm) and h e Wla’pm) o.) allhllpSJ"(u+ 9h,h,h)S(p-1)llu+ ehllp'zllhllzfis es 1, for which assumption (2) is not satisfied. For a class of abstract monotone operator equations in reflexive Banach spaces, the study of the Galerkin or projectional method for approximating the solutions have been studied by P. R. Ciarlet, M. H. Schultz and R. S. Varga [3]. Applying their theory to the p-harmonic operator equation ( p 2 2), we can show that the finite element solution converges to the true solution as mesh size h tends to zero. We then show that a minimizing sequence of the p-harmonic functional which converges to the discretized solution, can be constructed in the finite element approximation space Sh(Q). In minimizing the functional over the finite dimensional 811(9) we do not encounter the difficulty that the gradient of the functional may not be in Sh(Q). The inequalities (it) above guarantee the convergence of the minimizing sequence to the discretized solution. In surport of the theory, numerical experiments are conducted and test problems along with their finite element solutions are presented including finite element solution for the minimizer of the conformal capacity of a ring with a slit in the middle. In Chapter 3, we use the method of lines. The method is interesu'ng from a numerical point 4 of view. We shall show that the nonlinear parabolic problem governed by the p-harmonic operator has a unique weak solution which is more "classical" than the weak solution obtained by applying the theory of J. Kacur, in the sense that it satisfies the equation pointwise with respect to time. A L2 error estimate for the error between the true solution and its semidiscrete approximation is made. We note that the traditional elliptic projection technique fails in this case for it does not provide us with an error estimate for the time derivative of the difference between the projection and the true solution. This difficulty is overcomed by using a time dependent auxillary problem which one may call a " parabolic projection". It is shown that as the mesh size h tends to zero the semidiscretize solution converges to the true solution in the L2 norm. Numerical computation of the semidiscrete solution is also conducted. It is of interest to note that even though we have a global solution for the parabolic problem, and the semidiscrete problem governed by a system of nonlinear ordinary differential equation with prescribed initial data seems to have a global solution,for multidimensional cases, as indicated by numerical computation, it is not justified. In Chapter 4, a nonlinear dam problem which is also governed by the p-harmonic equation is studied. The formulation of the so called "free" boundary value problem, which differs from the above, is applicable for flows with relatively large seepage velocities for which Darcy's law is no longer applicable. A weak formulation is presented and it is shown that this weak problem has a solution. CHAPTER 1 Some Preliminary Results §l.l Notations and preliminaries 1.1.1 Notations a nonnegative integer C a universal constant which in different context may take on different values p a positive real number ( in most cases p > 1) R+ set of all positive real numbers 0 bounded convex domain in Rn (n = 2, in most cases) | l l2 norm in Rn ll llp LPG!) norm 852 boundary of Q x=(xl,x2,m,xn) -------a point in Rn ( , ) symbol of duality or inner product in a Hilbert space —) , —-* symbol denoting convergence and weak convergence respectively A Laplace operater Vu gradient of u div 1: divergence of u V a Banach space V* dual space of a Banach space V II II* norm for V* II II norm for V 1.1.2 Sobolev spaces We shall work with following Sobolev spaces: W‘Pm) = { u I Dau e me), for l a l s k }, with norm defined by _ or _ a p l/p llullk‘p—stkllD NIP-Ems“ n[ID ul dx 1 , n where a = (a1, a2 , . an ), | a I = 2 | on I, where ai's and k are nonegative integers i=1 and Dan is the ath generalized derivative of u. It is well known that 1) The space Wk'p(§2) with this norm is a reflexive Banach space, i.e., any bounded set in Wk‘p(§2) is a relatively compact set in Wk’p(Q); 2) The set C°°(§2)( thus Ck(§2) ) is dense in the space Wk‘p(§2). W 39(9) is defined to be a subspace of Wk‘p(Q) which is the closure of C379) ( thus (35(0)) with respect to the norm defined above. Let S be a surface of class C1 lying in Q. It can be shown using Holder's inequality and Newton-Leibnitz formula that 7 Ilullp SCllull1p foranyueC1(§2). L(S) w’m) For any 11 e Wl‘p(§2), we can choose a sequence { un } in C162) which converges to u in the norm of Wl'p(§2). The inequality llun-umll p SCIlun-um ll 1,p L (S) W (9) tells us that [ un } is again a Cauchy sequence in Lp(S) which shall consequently converge to a function f in Lp(S). It can be easily shown that f is independent of the sequence { un } using the above inequality, and thus it is natural to define the " trace" or the value of u on S to be f . Similarly, for a function u e Wk'p(Q), it is possible to define, on the basis of the above inequality, traces of all derivatives Dau of order | or I S k - 1. It is also known that WE’WQ) = { u eWk'p(Q) I Dau = 0 on 89 in the sense of traces for all | or. I S k-l} and II Vu llp = [ J|Vulpdx ]1/p defines a norm on WS‘WQ) which is equivalent to the norm ll u III p. In most of the the work below we use the space Wla‘pfll), and equip it with the following norm ll u II = II Vu llp throughout. A comprehensive account of Sobolev spaces and their properties has been compiled by Adams [1] and generalized by Maz'ja [24]. ° 1.1.3 Poincaré's inequality and Sobolev imbeddings It is well known that for p 21, ll u ups cu Vu up for any u e wg'Pm) and since Q is bounded, we also have llulquCll u III, for 1 Squ. Thus we obtain m1; Ifl Squ, then llu uqs cu Vu llp,foranyue wg'Pm). In Chapter five we shall need the following lemma. W For any positive integer k , wfipm) -> whim) —> mo) is a chain of imbeddings. For a proof of this lemma see Adams [1], p.45. 1.1.4 The p-harmonic functional and the p-harmonic operator germinal; The functional J: Vc_; wl'Pto) —-> R defined by 1 p J(u) =3 qul dx-(f,u ), where f e V* and V is a subspace of Wl'p(Q), is called the p-harmonic functional on V. One can easily calculate the first and second Gateaux-derivatives J ' and J": J'(u,h) = JIVulp'2(Vu, Vh)dx-(f,h), J"( u, h , k) = J! Vu IP'2( Vh, Vk )dx + (Mri I Vu tP'4(Vu. Vh)( Vu. Vk)dx. We apply Holder's inequalities to the above expressions and obtain following inequalities: p-l lJ'(u,h) ISIIuII llhll + ||f|l*||h||, p-2 lJ"( u, h , k) l S (p-1)ll u II II h II II k ll. Thus, by these inequalities and Remark 1.3 of J. Ce’a [9], we have Lemma 1,3 For p 2 2 , J is twice G-differentiable and hence I has both a gradient G(u) and a Hessian H(u) at u e V. It is well known that the p-harmonic functional is a strictly convex,coercive and weakly lower semicontinuous functional ( see P. G. Ciarlet [4], pp. 314-316 ), and its Euler equation 1‘Iqulp'2(Vu,Vv)dx=(f,v),forany vs V. 10 is the variational equivalent of the well-known p—harmonic equation - div(|Vulp'2Vu)=f in the theory of quasi-conformal mapping. W The operator A: v c: whim) -> v* defined by (Au,v) = {II Vu IP'2( Vu, Vv )dx is called the p-harmonic operator. Remark: The above operator is well defined for V = W362) , and V = Wl'p(Q), for we have the following inequalities: -l |A(u,v)|SJ|Vulp'1|Vvldx S [ Jl Vu lp dx](p )/p.[ le Idx ]. i.e., |A(u, v) I 5 II u llp‘lllvll 5 II It llp'lll v um. Definin’gn1,3 A : K -) V* is call a monotone operator from a convex closed set K of a Banach space V, if (Au-Av,u-v)20,foranyu,veK and a strict monotone operatpr if farther we have 11 (Au-Av,u-v)=0ifandonlyifu=v. W The above A is " hemicontinuous " if A( (1-t)u + tv ) converges weakly to Au as t—iO, for any u, v GK. 9911321135; The p-harmonic operator A is hemicontinuous. Proof: Let f(x,t) = I V((1-t)u+tv) IP'2V((l-t)u+tv), and g(x) = ( IVu l+lel )1"1 . Clearly lim f(x,t) = I Vu lp'IVu. Using Helder's inequality, one has t—)0 ‘1 gm ‘1" 5[J( I W '+| Vv I )de 1(p'IWIQI1’Ps IQI( llu|l+ ||v||)p'1. Applying Lebesque's dominated convergence theorem to f(x,t), we get lim Af(x,t)V§dx = J I Vu IP'1(Vu,V§)dx. t->O The following lemma is proved by R. Glowinski and A. Marroco [17]. Lemma 1,4 There exist constants or > O and B > 0 , such that (a) For p 2 2 , otII u-v IIPs ( Au — Av ) (u - v), 12 llAu-Avll*SB(llull+l|vll)p'2- llu-vll. (b) For 2 > p >1, aIIu-vll25(llull+llv|l)2'p-(Au-Av)(u-v), II Au - Av ||* 5 [3 II It -v||p-1. for any u, v e V c: Wl’p(Q). Corollary For p >1, A is strictly monotone and continuous. Lem—mLLj For p 2 2 and for any u, v e Wl'p(§2), there exists a real number 9 6 [0,1] such that on h IIp s (H(u + an, h, h ) s (p-l) II 11 + an Ilp’zll h II2 where h = u - v, and 6 depends only on u, v. Proof: For any n, v e Wl‘p(Q), by Taylor's formula there exists a 9 6 [0,1] satisfying J'(u,u-v)=J'(v,u-v)+J"(u+6(u-v),u-v,u-v). Using this and lemma 1.4 (a), we get aIIu-vllpS(Au-Av)(u-v)=J'(u,u-v)-J'(v,u-v) 13 =J"(u+9(u-v),u-v,u-v) S(p-1) II u + e( u- v )llp’zll u - v ”2, i.e., all h IIp s (H(u + 9v, h, h) s (p-l) II u + eh Ilp'2|| h ”2. §l.2 Boundary value problems 1.2.1 Existence and Uniqueness of some boundary value problems Let g 6 C(89), such that g has an extension, say (D, which belongs to Wl'pm). Then (D I 39 = g in the trace sense. Let I be the p-harmonic functional. Define wé'1’(r2)={ ue Wl’p(Q)| uIaQ= g I We consider the following two problems: Probleml. Find u e Wé’pal), such that u minimizes J( u ) over Wé’pdl). Problem2. Find w e wgpm) , such that w minimizes J( w + ) over wg'Pm). 11195212:le In Probleml and Problem 2 above, we have 1) J(u) has a unique minimizer u in Wé’pal) and J(w+ (D ) has a unique minimizer w in wol'Pm); 2) u = w+ (I) . 14 Proof: Let j(w) = J(w+ (I) ). Then it can be verified that i) j is strictly convex; ii) j is weakly lower semicontinuous; iii) j is one time G-differentiable; iv) j is coercive i.e., 1imj(w) = +oo as llwll —-) +oo; since it is known that J has the above properties. Now, an application of Theorem 1.2 and Theorem 1.3 in J. C6a [8], pp.62-63, implies that j has a unique minimizer w in wg'Pto) which is the unique solution of its Euler equation j'(w,v) = 0 for any v e Wé‘pm). Let u e wgtpm), then v = u - o e w01:P(Q). Thus u = v + , and we have J(u) = J( v + o ) 2 J(w + o ), where w is the unique minimizer of j in whim). This says that w + <1) is a minimizer of J in Wé’pal). Suppose that I1] is another minimizer of J in wfi'Ptfz) and u1= w1 + o. Then J(w1+) .<. J( w + (D ), i.e., j(wl) S j(w). Hence w1 = w since j has only one minimizer in Wé‘p(§2). Comma The following variational problems are equivalent 1) Find w cwohpt‘o), such that j'(w,v) = 0, for any v e wg'Pto). 2) Find u e ngrP(Q) , such that J'(u,v) = O, for any v e Wé'p(§2). 1.2.2 Some properties of the homogeneous equation with Dirichlet data 15 For f =0, J(u) = §Jl Vu lp dx. In order to minimize J over ng’p(§2) it suffices to minimize the integral J I Vu lpdx over ng’p(Q). Define rpm) =inf IVu Ip dx, where K = wglpm). (1.1) K We call I‘p(Q) the conformal capacity of Q. W Any minimization sequence of (1. 1) converges to the exact solution. Proof: Let {un} be a minimizing sequence, i.e., a sequence of functions in ng’p(§2) satisfying no): lim J IVunlpdx, n->+oo where for simplicity we have omitted the subscript p in I‘(Q). By Clarkson's inequality ( Adams [1]: Sobolev spaces p.37) one has for p 2 2 V -V V V Al "”2 unlpdx+Jl “m; unlpdx Sill Vum lpdx + 121' Vun lpdx, (1.2) 16 for10, there exists an integer N > 0 , such that whenever n,m .>. N, one has r[IVumIpdxs 1m) + e , Aqunlpde no) + 2, Using (1.2 ) and (1.3 ) we get r{IVum - Vun lp Vzllm + VUn lpdx) 2 dx) sZ-(rto) + e) +1-(I‘(Q)+8) A I SI‘(Q)+ 8 - I‘(Q)= 8 ,for p22. V -V / - [Al —2—"“‘ unlpdx)]l(p-1)S[I‘(Q)+ 811/021) _ [ Al Vum '5 Vun Ide]l/®-1))S[F(Q) + e ]l/(p-l)-[r( Q)]1/(p-1) for 1+oo. This and the Poincare's inequality imply that { un} is a Cauchy sequence in wli’to) and hence lim AIVunlpdx =Jqulpdx, n—H—oo i.e., rm): s‘II Vu lpdx. The uniqueness results from the strict convexity of J. 1.2.3 A maximum principle Let u be the weak solution to div(|Vulp'2Vu)=0,inQ. ulaQ=. Then J l Vu lp'2( Vu, Vv)dx =0 for any v e Wé'p(§2) 18 and especially, JIVu IP'2( Vu, Vv)dx = o for any v c c0162). (1.4) Theorem 1.3 u attains its maximum and minimum values on the boundary an for n > p . Proof: Suppose, on the contrary, supnu > supanu = supan). Let uo = supaQCD. Then there exist a constant C > 0, and a subregion no c: Q , such that v = u - (uo +C) > 0, in (20, andv =Oonaflo. In fact, we can choose (20 = { x l v(x) > 0 }. 520 is open, since v is continuous ( le’to) being compactly embeded in c762), 0 s y < 1- §- ) . Obviously, v e wg'l’too), and since C(l)(50)is dense in wg'l’too), we can choose a sequence { C“ } c: C3630) converging to v. Since C3650) c Cam) , and by (1.4 ), one gets nil Vu lp'2( Vu, VCn )dx = 0 , for any n. 0 Now, IaLI Vu Ip'2( Vu, Vu )dx - ”II Vu Ip'2( Vu, VCn )dx I 0 IA; Vu IP’2( Vu, Vu - V§n )dx I s (ZIOI Vu Ip'IIVu - thI dx 19 s[ (LI Vu lpdx ](p'l)/p-[ {1)qu — Vt, Ide ]”p implies that lim I Vu lp'2( Vu, VCn )dx = IV ulp'2( Vu, Vu )dx . n—-)+oo ' Hence, 4; Vu lpdx = 0 which again implies that Vu =0, and v = constant in 90, i.e., u = uo +C, v = 0 in $20, a contradiction! We conclude that supau = supanu. Similarly, we can show that infa u = inf a 52 u. 1.2.4 Continuity of the conformal capacity as a function of p m J(u) is a continuous function of u. Proof: This result follows from the identities below: i)For00, 20 {[1 mp. Iglpldx s 2pM1"l [ “[I f— glpdx]1/p. W I‘p(§2) is a continuous function of the parameter p. Proof: Let up be the unique minimizer, i.e., up satisfies rp(o)=J IVup Ide. Let q > 1 be fixed. We consider the continuity of I‘p(§2) at q in the two cases: Case 1) p—)q+ II‘p(Q)-I‘q(Q)I=IJIVuplp- Aquqlql s IAIVuplpdx - r{IquII’dear IAIqulpdx- JIqulqul Suppose that Al qulp dx< +oo , forp close to q. JIquldeZianIVulp dx = r[IVupIp dx. 21 -2 _dx _ ‘ Let r—q(>1)anddm-n ,wherelQl-thevolumeoffl. then A I Vup Ip dx = An Vup Ip)’dx = [ Au Vup Ip)’dm] IQI. Let f(y) = y'. One can easily show that f is convex in (o, +oo ) since f(y) = y"1 is increasing in (o, +00 ). By Jenson's Inequality, one gets J(l VupIq)’dm] IO! 2[ {II Vup I‘l dm 1’Ir2I 2 IQI"‘[ A! Vuplqu ]’ 2 IQI‘“[ A I qu I‘l dx 1’, since {I I Vup lq dx 2 AI qu qux . Therefore, one has 5jIquII’dx 2 r{IVupIF’dx 2I§2I"1[ JIqulqu ]' which implies li_rn IquIde2 1i Ier'l[Jquqlqu]r P-9q+ p q+ =JIqulq dx. 22 Let 8 > 0 be sufficiently small; choose p such that q < p < q - 8 , by using Holder's inequality, one has JIquIde s[ JIquIq'e dx ]q"5IoI‘5"l which implies that li—m r{IquIPs [‘jIVtrqI‘l'E dx ]l/(q'8)IoI8/q. p->q+ Letting 8 —-) 0 , one obtains __ 1 lim Iqulps[ JIqulqu]/q. p-+q+ Therefore, we have shown that . 1 11m Iqu|p=[ r{quqlq dx ]/q. 13—) (1* Case 2) p—) q ' We have u e Wl’p(Q) c Wl’q(§2), since q > p and IQI < +oo. By definition, 23 JIVup Ip dx S {II qulp dx which implies and 1 - [Jl Vup IPJVP dx SI 4|qu Iq dx] ”In“q mp. Thus if; IVuplp dxs f{IquIq dx, p->q' __ 1 lim[ IVupIp11/p de[A|quIq or)“l p-> Q‘ and the set { [ A I Vup |p dx]1/p} is bounded from above by a constant, M say. Now we wish to show that um IVuplp dx2 r[IquIq dx. P—>Q' 24 One can easily show that lim [an] a = [ [in an ]‘IL for an> 0, 0 < or < +oo. n—>oo n -> oo Let 81 > 0 and choose pk such that q - 81< pk < q , pk < pk“, k=1, 2, 3, . We have Jqupqu'el dx s[ r[IVukapk dx](q-81)/pklnl(pk.q+81). Thus ".— q-el hm I Vupk I dx Pk ‘9 C1' 5 lim— {[ IVukapkdx]l/pk}q'81IoI81/q} Pk-Xl' 5{ Hr; [ IVukapk dx ]1/pk}q'81t2I81/q} P159 ‘1’ '8 s Mq llle/q. As a result, { Vupk} is a bounded set in the Banach space (Lq.€1(Q))n. Since “PIE (D e Wé'q'e'm). JIupk- (DI‘I‘Eldx s r[Iwupk- 82 , we can again choose a subsequence of { upk1}’ say, [ upkzl which converges weakly to u in Wl’q'82(fl). Let 81 > 82 > 83 > . We then have a nest of sequences { upkll D [ upkz} 3 { “Pk3} ~--, such that for each fixed integer n 2 1, [ “Pkn} converges weakly to u in Wl’q‘£“(fl). Choose the diagonal sequence { upkk}, then it converges weakly to u in wl.q-en(Q) for each n. And for each 8“ > 0, k > n, JIVu Iq-endx s 1i_m Jqupkqu-en Pkg-9 Q" . - (p -q+8n)/p s llm [ JIVupkklpkk dx](q an”may "k kk Pkg-“1' 26 s [( Lint Jqupkklpkk dx)1/p klr](“""")ll2I en/q. Pick-“1' Letting 8n -) 0 we have Al Vu qux S [ lim ( Jqupkklpkk dx)1/pkk]q S rIIqulqu. Pick—"1' Thus, JIVu qux = r[I qu lqu and we have u = uq. It then follows that AIququxS 11.111. [AIVupkklpkkdx] Dirk-9 (1’ s lint [JIVupkklpkk dx] 5 Aquqlqu, pin-Kr lim IVupkklpkk dx= Al qu qux. Pkk-9 (1' We have thus proved that for every sequence { pk } which converges to q from below, there exists a subsequence { pkk} such that 27 lim JIVupkklpkk dx= rIIququx Pkg-9 ‘1’ and hence lim r{IVuplp dx= JIqulqu. P“) q' By using Clarkson's inequality and Theorem 1.4 we have Corollary, The minimizer of the conformal capacity is continuous with respect to p in the following scense: lim JIVup -qu Ip dx=0 p-HI Note: For p > q, one has rIIquI‘ldx s JIVupqux s [J(IVupI‘lf/qu ]qm( IQI ) (P ' ‘1)- Taking limts, by Theorem 1.4, it gives lim Jqupqux =9qu qux. p -* q Using this together with Clarkson's inequality, one attains 28 lim JIVup- qu qux =0. P '9 9"” 1.2.5 Continuity of the minimizer of the conformal capacity of a ring with respect to boundary change Let 1 S k S n-l, Qt be the ring consisting of the open unit ball in Rn minus the closed concentric k dimensional ball of radius t. Let u(t) be the minimizer corresponding to rptoo. m5 u(t) is a continuous function of t. I 9 I e t - t. I Proof: Let t > t > o and K = w; ”((29, K =w; Pain). Thenfiz-‘Mc K . Applying Clarkson's inequality to u(t) and u(t') , using the fact that I‘p(Qo is a continuous function of t ( G. D. Anderson [2], p.30), we can show that lim I u(t) - u(t')lpdx = 0. t'-9 t Chapter 2 Finite element approximation of the solution to the nonhomogeneous p-harmonic equation with Dirichlet data §2.1 A finite element approximation of the true solution u Let Sh(Q) be a conformal finite element approximation subspace of Wfififl), dim 811(9) = k; let also (I) e Wl’p(§2) , p 2 2 . It is known from Chapter 1 that the following two problems are equivalent: (1) Find u e “,1.me such that J Wu IP'2( Vu, Vv )dx = ( f, v) for anyv e wkpto) andu=)II"2(Vw+,Vv)dx = ( f, v) for any v e woltpto). 29 30 Define (Tw,v)=J |V(w+),u-v) =(A(u+) -A(v+) , (u+)) 2 or H u - v HP. 2) T is bounded, i.e., T maps bounded subsets of wg'Pm) into bounded subsets of (WI'PIQ)>*. Proof: Let u e wg'l’m) and II 11 II 5 M, where M > 0. Then I(Tu,v)I=| rIIV(u+)Ip'2(Vu+,Vv)dx — f(v) II 31 s J I V(u+cr>)IP“IVvIdx + II f Il*|l v II 1)/ s [A IV(u+)IP](p' pllvll+|lfl|*||v|l = (II u+ II"1 + II f II*) 5 (MP1 + II f Il*). Now, let w e Wé’pdl) be the solution of problem 2). We have (Tw,v)=0 foranyve wg'l’m). Let whe 811(0) be the solution of (T wh ,v )= 0 for any v e Sh(Q). The monotonicity of T implies that on wh-O IIPS(Twh-T0,wh-0)SIIT(0) II* II thI, o II wh II 1’" 5 II T(0) |l*. 32 Thus, ”WM IT(0) ||*)1/P'1 Let v be an arbetrary element in 811(9) . Then ( Twh- Tw ,wh -v ) = 0, since wh - v e thy’pal). This and the monotonicity of T yields allwh-w IIpSI(Twh-Tw,wh-w)| =I(Twh-Tw,w-v)ISIITwh-Twll*IIw-vll. (2.1) The boundedness of T and that of the set I Wh I let us conclude that there exists a positive constant K > 0, such that II T wh- Tw II* S K is independent of 811(9) and allwh—wllpSKIIw-vll foranyve W396». (2.2) ( 2.1 ) and ( 2.2 ) together then yield the inequality 33 allwh-wllpSKinfUIw-vll: vs 81.62)}. Let uh: Wh + 1'1th and u = w + o , where 1],, is the finite element interpolation of f over 811(9) and u is the true solution of problem 1). Then we have alluh-ullps dunno-(pup + Kinf[ lIw-vII: ve shm) I. The above inequality and a classical finite element interpolation theorem (see P. R Ciarlet [3]) lead to the following theorem: W1 Assume that the true solution It Wl’p(§2) of problem 1) belongs to W2’p(§2). Then IIu-uhllpS[(ClhluI2’p )p+ KCZthlzp], where C1 , C2 and K are constants. §2.2 An approximation of the finite element solution uh In this section we will construct a sequence of functions in 811(9) which converges to the finite element solution “h . Let { Ni}. be the global finite element basis for Sh(Q). We consider 1: 19m 34 "‘1 m Sh(§2)={vl v=2xiNi, (x1,x2,...,xm)eR} i=1 by denoting functions v= zlxiNi and W: ZyiNi rn 811(9) by X and Y respectively, i=1 X, Y in Rm. A functional j : Rm -—> R is defined by J(X)=J(v+]'[h)=-p-1JIV(V)+I'Ih , w, w) and shown that IIwI|=[§£|2YiVNII wdx]/pSIYI[A(2|VNII)p1.dx]fP i=1 LetC= [A(§;IVI~I,I)"1.dx]/p By Lemma1.3we have -1 |j'(X)Y I .<_IJ'(v+1'Ih,w) IS H v +I'IhCDIIp II w II + II fII*|I w II -1 s C( II v + I'Ihllp + II f II*) I Y I, (2.3) I (j'(X),Y,Y) I s I J"( v + I'Ihd), w, w) I S(p-1)II v + I'IhCD IIp‘ZII w II II w II 35 2 2 2 2 -2 2 SC (p-1)IIv+r1hII‘* IYI 5C (p-1)(Ilvll+llI'[hll)p IYI s C2 (p-1)( CI x I + III'Ihd) II)!” I Y I2. (2,4) We now establish the following theorem: Theorem 2,; Let j: Rm—) R be the functional defined previously, and let X* denote the true solution uh . Then, for each Xk e Rm, there exists a real function gk (p) which attains a ( negative ) minimum value for some pke R+. Let Xk+1= Xk - pkj'(Xk), or choose 11“ > 0 in such a way that j(Xk - nkj'(Xk)) is minimal, and let Xk+1= Xk - nkj'(Xk). Then there exists a C > 0, such that I X* - Khan-1) s c -j 0 and p1‘ > 0 such that gk(Pk)= inf gk(P)=-Mlt >-°° peR+ Set Xk+1= Xk - pkj'(Xk) or choose 11“ > 0 in such a way that j(Xk - nkj'(Xk))is minimal, and let Xk+1= Xk - nkj'(Xk). Then in either case, we have 37 jIXk+1)-j(x1< )5 lJ"(X" >I2gt< pk) = - Ij'(Xk)|2 Mk and it follows that j(xk) - j(xk+1) 2 Ij'(xk )I2 Mk, 10‘") 21'0““). (2.6 ) Claim: { Il vkll } is bounded. In fact, by (2.6) j(xk) Sj(x0) for k = 1, 2, 3, ---, hp 'kIde A kadx 5 a? vOIpdx A vadx which yields i.e., A] mm s pr! fv'kdx + AI VOIde -pJ vadx Sp IItIIq II VkIIp + {II volpdx my! vadx 3 p CIIfIIq II W“ + p llfllq( Ill'IhCDII + II nhcbup) +AI volpdx m“! fvodx, 38 i.e., II VI: It? s p CIIflIq II kaI + p IlfIlq( III'IhCDII + II nhoIIp) +JI votpdx -p A fvodx. From this last inequality one can deduce that there is a constant N > 0, such that llvkll s N fork =1, 2, 3, , and hence C2( 1) 2 p-1 1,2 2 gk(p).<_-p+—-§'—[N+pC(N +I|f|l*)] p. The function on the right hand side obviously has a minimum, say - M, M >0, which is bounded away from - co. Now -Mk= inf gk(P)S-M reR+ leads to Mk 2 M, and then ltxk)-l(x1<+1> .>. lJ'(X“)|2Mk .>. j'(Xk)|2 M. (2.7) Let x denote the finite element solution uh , then using j'(x)(x - xk) = 0, we have j'(XkX X - xk) = j'(XkXx - X") - j'(X)(X - Xk) 39 = 1'07"in - 9k) - J'( \7 XII-‘7“) = ( A(§/")- AW ) )('V - 9k) 2otIIv-kaIp2ot*I x - kuP, where a* > 0. I j'(xk) I 2 ot*I x- xk IIH which together with (2.7 ) yields j(xk) -j(xk+1) 2 (a*)2MI x- xk I29“). and this leads to the assertion of the theorem. §2.3 Numerical experimental results The following computations are conducted on a VAX VMS computer. 2.3.1 A test problem For p = 4 and Q = U ( the unit disck ), the following p-harmonic problem - div( IVulp'ZVu ) = 32( x2 + yz), u lag = 1 40 has an exact solution u = 2 - (x2 + y2). Numerical experiments using a 6 quadratic finite element mesh shows that the maximum error between the true solution u and the approximates of the true solution uh of the approximate problem, for a mesh size h = 0.5, can be as low as 102, while using a 24 quadratic finite element mesh with a mesh size h = 0.125, it can be as small as 104. Further refining the mesh size does not seem to improve this error significantly as indicated by the computations using a 96 quadratic finite element mesh with mesh size h=0.0625. These numerical experiments show that maximum errors always occur at the point (0, 0) where l Vu I = 0 and the equation is degenerate. The approximations are relatively good around a neighborhood of EU and get worse and worse towards the center (0, 0), as indicated by the picture below and Table 1. I\ \/ Note: The interior surface designates the computer generated finite element solution which approximates ti}, and the exterior designates the the true solution 11 . 41 Table 1. Error between true solution and finite element solution Mesh Size 11 It of Interior Nodes Max | 0n " ug—ll Max | u- uhl 0.50000000 ,5 ICE-03 4.165992183-02 ICE-07 2.215548533-02 I 03-05 8.757286333-04 72 °' ‘ 2500000 I 02-09 875542548 3-04 I 03-05 2.406089992-03 0.06250000 288 102-07 1.465887342-04 I 013- I 0 1.46 I 772982-04 2.3.2 Computation of the minimizer for the conformal capacity of a ring For P = 4, (2 = U ( the unit circle), 89 = 8U U {(0,0)}, the following p - harmonic problem - div( quIp'ZVu ) = 0, u IBU = l, u(0,0) = 0 3 has an exact solution u = ‘I x2 + y2. Let P, Q be as above. Let an = 8U U B(t), where B(t) is the slit { (x, 0) : - 5-5 x s 5- I. Consider the following problem - div( IVulp'2Vu ) = 0, uIaU = 1, u '3“) = O 42 The solution to this problem is continuous with respect to t (Theorem 1.5). Thus, for small t, we use the solution to the first problem as our initial guess for the computation of the solution to the second problem. By the Corollary following Theorem 1.4, the solution to the above problem is also dependent continuously upon the parameter p, which suggests the use of the solution for the problem with p = 4 as initial guess and then by incrementing or decrementing to reach other values of p. One may use the exact solution for the problem with p =2 as initial guess as well. The figure below is a 24 quadratic finite element mesh with interior nodes 1 - 69, boundary nodes 70 - 89. On the slit there are three nodes 87, 88, 89. The length of the slit t is 0.5. :zétfls o :50 907: '1'2 6 9 17 21878988 42 2054 626665 The numerical solution to the above problem using this mesh is given below: Table 2 shows the solution for the problem with p = 3 at the first 38 nodes ( Refer to Figure 2.2), Table 3 is the case when p = 4. 43 Table 2. Numerical solution for the slit problom with p = 3 NODE# X-COORDINATE Y-COORDINATE SOLUTION U 1 -0.87500000 0.00000000 0.90504586 2 -0.75000000 0.00000000 0.80218206 3 -O.72145482 0.14350629 0.79083332 4 -0.69290965 0.28701257 0.80448275 5 -0.80839459 0.33484800 0.90610410 6 -0.62500000 0.00000000 0.69037377 7 -0.57742471 0.23917715 0.69464293 8 -0.61161987 0.40867133 0.79527366 9 -0.50000000 0.00000000 0.56480939 10 -0.48096988 0.09567086 0.55716340 ll -0.46l93977 0.19134172 0.57308536 12 -O.42895978 0.29366076 0.59698820 13 -0.39597980 0.39597980 0.64476603 l4 -0.46315494 0.46315494 0.7 30327 1 8 15 -0.53033009 0.53033009 0.80971644 16 -0.61871843 0.61871843 0.90854929 l7 -0.37500000 0.00000000 0.41970348 18 -0.31935823 0.18405921 0.427568 32 19 -0.29366076 0.42895978 0.61047472 Table 2. (cont'd) 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 -0.40867133 -0.25000000 -0.21338835 -0. 17677670 -0.18405921 -0.19134172 -0.23917715 -0.28701257 -0.33484800 -0.08838835 -0.09567086 -0.14350629 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.61161987 0.00000000 0.08838835 0. 17677670 0.31935823 0.46193977 0.5774247 1 0.69290965 0.80839459 0.17677670 0.48096988 0.72145482 0.08838835 0. 17677670 0.33838835 0.50000000 0.62500000 0.75000000 0.87500000 0.80162344 0.22841150 0.23564667 0.29371180 0.45646261 0.60040672 0.71261770 0.81498220 0.91099626 0.26187623 0.59148427 0.80594057 0.13924920 0.23260342 0.43594700 0.60515736 0.71602673 0.81694234 0.91 193450 45 Table 3. Numerical solution for the slit problem with p = 4 NODE# X-COORDINATE Y-COORDINATE SOLUTION U 1 -0.87500000 0.00000000 0.88960447 2 -0.75000000 0.00000000 0.77312628 3 -0.72145482 0.14350629 0.76042371 4 -0.69290965 0.28701257 0.77616685 5 -0.80839459 0.33484800 0.891035 34 6 -0.62500000 0.00000000 0.64984193 7 -0.57742471 0.23917715 0.65507319 8 -0.61161987 0.40867133 0.76607838 9 -0.50000000 0.00000000 0.51656542 10 -0.48096988 0.09567086 0.50881131 1 l -0.46193977 0.19134172 0.52550728 12 -0.42895978 0.29366076 0.55238978 13 -0.39597980 0.39597980 0.60321682 l4 -0.46315494 0.46315494 0.695 261 84 15 -0.53033009 0.53033009 0.78300313 16 -0.6187 1843 0.61871843 089430492 17 -0.37500000 0.00000000 0.36898650 18 -0.3 1935 823 0.18405921 0.3 8003707 Table 3. ( Cont'd ) 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 -0.29366076 -0.40867133 -0.25000000 -0.21338835 -0.17677670 -0. 18405921 -0.19134172 -0.23917715 -0.28701257 -0.33484800 -0.08838835 -0.09567086 -0.14350629 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 46 0.42895978 0.61161987 0.00000000 0.08838835 0.17677670 0.31935823 0.4619397? 0.5774247 1 0.69290965 0.80839459 0.17677670 0.48096988 0.72145482 0.08838835 0. 17677670 0.33838835 0.50000000 0.62500000 0.75000000 0.87500000 0.56883128 0.77438942 0.19022453 0.19733620 0.254777 84 0.41286403 0.55874870 0.677 85474 0.79003372 0.89765573 0.22739322 0.55095480 0.78001258 0.11888648 0.20290609 0.39408723 0.56473057 0.68216396 0.79267491 0.89894493 Chapter 3 The nonlinear semi-parabolic equation §3.l An existence and uniqueness result Let A : wll’m) —+ (wl:P(o))* be the operator defined by Au(v) = (II Vu Ip‘2(Vu, Vv)dx. Throughout this chapter, we shall assume that Q is a bounded convex domain in R'1 with smooth boundary 652, and p _>. 2. Lemma 3,1 For any g e Wl’pm), the following problem (Au, v) = (g, v), for any ve wétpm) u lag = no lag has a unique solution u e Wl’p(§2), where uo e Wl’p(§2), and ( - , - ) is understood either as the usual inner product in L262) or as the duality for a pair in wgp(o)x (Wé’pm) )*. Proof: Since 9 is a bounded set we have (W':"<£2»* : WWII) :> wl’Ptnl. 47 48 where q = 25% . The lemma then follows from the Corollary of Theorem1.1. Consider following nonlinear evolution equation %%+Au=f, xco,te(0,TI, (3.1) MM) = 110(X). KS 89 . t6 (0. T]. (3.2) u(x,0) = uo(x), x 6 9, (3.3) where no 6 wltpm), f e (w‘:P(o))*. Definio'on3,1 We say that u is a solution of (3.1), (3.2), (3.3) if u(t.x> e L°°<0.T;w1"’ 0 as t'—) t. . 1 Thus, w e C(0,T; L2(§2)). Let u*(t)= Jw(t)dt + no. Using Fubini's theorem, dun 42.9. t t (un(t)-U*(t).V)= (i J (( -d-t--W).V))dtdx= (I A“ d, -W).V))dxdt. Thus lim ( un(t) - w(t), v) =0 for any v 6 W336», uniformly over [0,T], and we have u(t) = u*(t), iii—f: w(t) e L°°(0,T; ng’mmLzm», i.e., u e L1:°°(0,T;wg'I’(o)), (glil , v) + (Au,v) = (f,v) , for any v e “1343(9). This completes the proof of the theorem. §3.2 A semidiscrete approximation of the initial boundary value problem. Let Sh(Q) g Wl’p(§2) be a finite element space, and l'Ih : Wl'p(Q) —> Sh(Q) be defined by 11 Hhu = Zli(u)Ni. Uh is known as the finite element interpolation operator, I N; }i=1,n are ' 1 1: the global basis functions for Sh(Q) and [ 11(u) h: 1," corresponds to the global degrees of freedom ( see J. T Oden & G. F. Carey [27], volume iv, pp. 63-64 ). Obviously (l'Ihu )t = I'Ihut, and a classical theorem on global interpolation error estimates in the finite element theory leads immediately to Lemma, 3.2 Suppose that { T h }h is a regular family of triangulation of Q. We then have 56 a) pr>2andk=0.1,then II u(t) - 111,110) "(mo 5 Ch” I “(9 l1.p.Q II w(t) ' fibula) "lepfl S Chl'kl 1110) l1.15.9 b) pr>1andk=0,1,then II u(t) - Hhu(t) "m, s Ch“ I u(t) Igpn II u,(t) - Hhudt) "kph S Chz'kI I110) I2.p.o for any u(t), ut(t) e Wl’p(Q), where C is a constant independent of u, t and h and I'Ih is the finite element interpolation operator corresponding to I T h }h- 1'1 Let U(x,t) =OZXi(t)Ni be the semidiscrete Galerkin approximation solution to the 1: following problem dU (71?,V)+(AU,V)=(f,V),00 and a function u (x,t) e Ll’”(0,T; Wl’p(§2)) such that llu-Ullgsllu- I'Ihull2 + I r 1 exp (-JyIIE(t)IIzp'2dt){IIE(0)112+ 11%:1112expp JyIIE(s)II§'2dsdt I. T where e(t) = u (t)- I'Ihu(t), and Iim III def,” 112d: =0. h —> 0 o I t Mag Ilu-UIIZSIIu- mun2 + I|U(0)-I'Ihu(0) 112+f11 distinzdt , O and therefore lim IIu-UII2=0. h—)0 Proof: We introduce an auxillary problem: Find u e Wl’p(§2) satisfying (%,v)+(AI'Ihu,v)=(f,v), 0 =(f v) (Al—1W”-(dB—“3M:1 = (%%.V) + (AU.V>-0, { y(0) > 0 has solution: 1 t Y(t)SCXP(- JYY(T)q'ldT){ y(0)+Ja(T)CXP(- JYY(S)q'1dS )dT }- T (3. 17) 60 t . . . . d q __ q-l Proof. Multiplying both Sides of 3%4' y y - 01(t) by exp( Jyth) dr), one has I I 1: exp( Jwtzlq'ldt>§,1+ exp( JW(t)q’ldr) YYqS attlexpt JW(‘t)q'ldt). i.e., . t I $124 exp( Jyy(t)q'1dt)y] s o(t)exp( Jyyctfl'ldt). Integrating the last inequality over [0,t] we then have t t T exp( JW(t)q'ldt)y(t) - M) s Iattlexpt - jyy(s)q'1ds)dr. 0 0 Thus, I t T W) Sexp( - JMfiq'ldtM y(0) +Ja<0exp< - Iwq"ds )dt }. 0 Now we return to the proof of the theorem. Applying Lemma 3.9 to (3.17), we get 61 t I I II E 1125““. OIYIIEUNEZ (mm B (0) "2+ J de('c) 1: 2 I—dt—iI2 exp( - OJyIIE(s)II‘.: ds)dt}. Hence, IIu-UIIZSIIu-I'IhuII2+IIHhu-Ull2S IIu-I'IhulI2-1- t r I exp( - JwE(t)II;2dt ){ IIB(0)I|2 +J119%(:—)112 exp ( - J'yIIE(s)II’:2ds)dt}. 0 it Let us recall that the semidiscrete Galerkin approximation solution U(x,t) =£Xi(t)Ni i=1 satisfies (3.14) and (3.15) which can be rewritten as Bx(t) = F(X(t)), (3.18) X(0) = To , (3.19) where B is the nxn matrix with entries given by bi, j = ( Ni , N j ), and F is a n-vector given by . Fk(x(t)) = (f. N0 - (A(ixi(t)Ni). N0. and 70 = B-IC. Ci = ( uo ,Ni ). k = 1. n. 1: Since B is positive definite and F is continuous, this nonlinear system of ordinary differential equations has a unique solution, at least locally. By using Lemma 1.4 (a), we can show that for any X, Y in R", |F(X)-F(Y)ISC(|XI+IY|)p'2IX-YI. where 62 p-1 “ C=B(MaxIINill)[:IlVNIP/2] ,IVNI=21VN,I. 1SiSn 1:] Thus IF(X) I 5 C1 (IX IP'I+ 1) ,CI = Max(C, Max I(f, Ni)l). lSiSn Remark: Although we can prove the existence of global solution to the problem (3.1), (3.2), (3.3) in the sense of definition3.1, and its semidiscrete approximation problem (3.14), (3.15), which is a system of nolinear ordinary differential equations with prescribed initial data, seems to have a global solution. we however only have following local existence result. Consider following one dimensional initial value problem: v(t) - Cl( v(t)I*1+1). V(O) = V0 , vo= I Vol. For p = 3, this problem has solution: tan'lv = C1t+tan'1v0. . . . . rt In order that this problem have a posrtlve solution, one must have 0 S Cu + tan'lvo S 2- , i.e., 0 S t S él-(gh tan'lvo). By Corollary 6.3 of J. Hale [19], p.32, we conclude that for p =3, the solution to the initial value problem (3.18), (3.19) at least exists on the interval 1 1: _1 [0. 51'(f- tan V0)]- 63 §3.3 Numerical solution to the semidiscrete problem Let the initial boundary condition no in (3.2), (3.3) be x2 + y2. Let the domain (2 be the unit circle. Firstly, finite element method is used to compute and assemble A and F in (3.18), (3.19), and then a Runge-kutta method of order four is applied to 5(0) =B-1F(X0». (3.20) X(0) = 70 . (3.21) Numerical solutions to (3.20), (3.21) for some values of p and t are given in following tables, using a space mesh size of 0(10'1) and a time step size of 10'5 with a tolerance of 103. In table 3.1 p=3, t = 8 and in table 3.2 p=7, t = 8, while in table 3.3, p = 3, t = 100. 54 Table 4. Numerical solution for the semidiscrete problem with p =3, t = 8 X—COORDINATE Y-COORDINATE SOLUTION U -0.87500000 0.00000000 0.76463747 -0.75000000 0.00000000 0.56188232 -0.72145480 0.14350629 0.54057568 -0.69290966 0.28701258 0.56187630 -0.80839461 0.33484802 0.76463431 -0.62500000 0.00000000 0.3 8997555 -0.57742471 0.23917715 0.38998687 -0.61161989 0.40867132 0.54060304 -0.50000000 0.00000000 0.249635 80 -0.48096988 0.09567086 0.24004136 -0.46193975 0.19134171 0.24963513 -0.42895979 0.2936607 6 0.26979628 -0.39597979 0.39597979 0.31332624 -0.46315494 0.46315494 0.42833182 -0.53033006 0.53033006 0.56191176 -0.61871845 0.61871845 0.76462603 -0.37500000 0.00000000 0.1402831 1 -0.31935823 0.18405920 0.13554138 -0.29366076 0.42895979 0.26979664 Table 4. ( Cont'd ) -0.40867132 -0.25000000 -0.21338835 -0. 17677669 -0. 18405920 -0.19134171 -0.239177 15 -0.28701258 -0.33484802 -0. 12500000 -0.08838835 -0.09567086 -0.14350629 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.61161989 0.00000000 0.08838835 0.17677669 0.31935823 0.46193975 0.5774247 1 0.69290966 0.80839461 0.00000000 0.17677669 0.48096988 0.72145480 0.00000000 0.08838835 0.17677669 0.33838835 0.50000000 0.62500000 0.54060304 0.06227459 0.05314219 0.0622961 l 0.13554141 0.24963425 0.38998622 0.56187677 0.76463443 0.01551264 0.03889032 0.2400391 1 0.54057479 0.00002941 0.00773707 0.0310951 1 0.11419977 0.24962461 0.38997543 66 Table 5.Numerical solution for the semidiscrete problem with p = 7, t = 8 X-COORDINATE Y-COORDINATE SOLUTION U -0.87500000 0.00000000 0.76541054 -0.75000000 0.00000000 0.56246543 -0.72145480 0.14350629 0.54107714 -0.69290966 0.28701258 0.56246454 -0.80839461 0.33484802 0.76541030 -0.62500000 0.00000000 0.39057919 -0.57742471 0.23917715 0.39058045 -0.61161989 0.40867132 0.54108149 -0.50000000 0.00000000 0.25000602 -0.48096988 0.09567086 0.24048448 -0.46193975 0.19134171 0.25000498 -0.42895979 0.29366076 0.27024037 -0.39597979 0.39597979 0.31360778 -0.46315494 . 0.46315494 0.42896938 -0.53033006 0.53033006 0.56247145 -0.61871845 0.61871845 0.76540917 -0.37500000 0.00000000 0.14062278 -0.31935823 0.18405920 0.13586640 -0.2936607 6 0.42895979 0.27024046 Table 5. ( Cont'd) -0.40867132 -0.25000000 -0.21338835 -0.17677669 -0.18405920 -0.19134171 -0.23917715 -0.28701258 -0.33484802 -0. 12500000 -0.08838835 -0.09567086 -0.14350629 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.61 161989 0.00000000 0.08838835 0.17677669 0.3 1935 823 0.46193975 0.57742471 0.69290966 0.80839461 0.00000000 0.17677669 0.48096988 0.72145480 0.00000000 0.08838835 0.17677669 0.33838835 0.50000000 0.62500000 0.54108149 0.06250143 0.05334897 0.06250264 0. 1 35 86637 0.25000492 0.39058039 0.56246454 0.76541030 0.01562470 0.03906449 0.24048401 0.54107702 0.00000226 0.00781223 0.03125156 0.1 1450540 0.25000456 0.39057925 68 Table 6. Numerical solution for the semidiscrete problem with p =3, t = 100 X-COORDINATE Y-COORDINATE SOLUTION U -0.87500000 0.00000000 0.74913752 -0.75000000 0.00000000 0.55976617 -0.72145480 0.14350629 0.54036266 -0.69290966 0.28701258 0.55805087 -0.80839461 0.33484802 0.74900985 -0.62500000 0.00000000 0.38026512 -0.57742471 0.23917715 0.38231325 -0.61161989 0.40867132 0.54388785 -0.50000000 0.00000000 0.25064573 -0.48096988 0.09567086 0.236395 85 -0.46193975 0.19134171 0.24970487 -0.42895979 0.29366076 0.26798773 -0.39597979 0.39597979 0.328497 68 -0.46315494 0.46315494 0.41558808 -0.53033006 0.53033006 0.564307 69 -0.61871845 0.61871845 0.74743307 -0.37500000 0.00000000 0.13669094 -0.31935823 0.18405920 0.13302509 -0.293 66076 0.42895979 0.26801610 Table 6. ( Cont'd) -0.40867132 -0.25000000 -0.21338835 -0.17677669 -0.18405920 -0.19134171 -0.23917715 -0.28701258 -0.33484802 -0. 12500000 -0.08838835 -0.09567086 -0. 143 50629 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.61161989 0.00000000 0.08838835 0. 17677669 0.31935823 0.46193975 0.5774247 1 0.69290966 0.80839461 0.00000000 0. 17677669 0.48096988 0.72145480 0.00000000 0.08838835 0. 17677669 0.33838835 0.50000000 0.62500000 0.543 89095 0.06083285 0.05148761 0.06123241 0.13300553 0.24964806 0.38226894 0.55808139 0.74901742 0.01429211 0.03752954 0.23611800 0.54025221 0.00112834 0.00702544 0.03025743 0.11115801 0.24973403 0.38037065 Chapter 4 A Nonlinear Dam Problem §4.l A nonlinear porous media equation 4.1.1 Physical background In describing flows in an area adjacent to a pumping well in a coarse grained aquifer and flow through rockfill dams and banks where the traditional Darcy's Law fails to hold, Missbach [32] postulated the so called " exponential law": — Vh = C IVIm'lv , m > o, (4.1) where V = the average seepage velocity of the flow, (If: the coefficient of permeability, and h = the piezometlic head For an incompressible fluid, we have the continuity equation: div V = 0. (4.2) Assume that C = 1. Eliminating V from (4,1) and (4,2) yields -2 div(IVhIp Vh)=0,p=la+1. (4,3) When m=1, (4,1) reduces to Darcy‘s Law: 70 71 V=th,k=z1-:- and (4,3) reduces to the Laplace equation: A h = 0. 2 Note: When dealing with a flow through a porous medium, the velocity head |I_\2’lgl_ can usually be neglected compared with the piezometric head h, even when the velocities are large enough to cause the problem to be nonlinear [32]. In Missbach's postulate, the exponent m lies between 1 and 2, i.e., 1 S m S 2. Here we generalize it to m > 0, irrespective of the physical considerations. 4.1.2 Clasical formulation of the dam problem Consider a flow through a rockfill dam with a lower impervious boundary F1 as shown in the picture, where A is the wet region, 8* is the so called " free boundary" ,and 13,1 is the upper fluid contacted boundary F33 the lower fluid contacted boundary, F2 the dry boundary of the dam, and finally (2 the interior of the dam. 2 Q: / \I/ 5* I‘ A 3,1 w \r3,2 ’1‘ V "l‘ 72 The problem is to find a pair ( h , A ) satisfying 2 div ( IVhIp' Vh ) = 0 in A, (4.4) h = hi on I733 , i=l,2, (4.5) h = y on S*, (4.6) gig-=0 onS’I‘UI‘l. (4.7) Remark: the boundary conditon (4.6) is due to the fact that on 8* the fluid is pressure free. In fact h = pressure + y, and thus h 2 y , A ={ (x,y) : h(x,y) > y }, Ac = { (x,y) : h(x,y) = y (pressure free region) }. In the rest of this chapter, we shall always assume that p > 1, also all the equalities and inequalities hold in the sense of "almost every where". 4.1.3 Weak formulation of the dam problem For g e C1(r'2), using Green's formula and (4.4) and (4.7) we have p-2 JIVhI (Vh ,Vlg) p~28h -Jdiv(IVhIp'2Vh)§+ )[IVhI 3—1: 8 2 -2 IIVhIVng 3" +£1Vh1p'33 3": S*U r1 =r£IVhI"23—§, 3" 73 where I‘ 3 = I73,1U r12- Thus it follows that -2 _ JIVhID (Vh ,Vg) = 0 for any § 5 C1(Q ), g = 0 on [‘3‘ (4.8) Since h = y in AC, we can rewrite (4.8) as Jthlp'ZWh, vg) = [IVyIp'2(Vy, vg) A for any fie C162), §= 0 on F3, or JIVhIPQWh, V5,) = Jxfiy for any i eCI(§2 ), §= 0 on F3, where x is defined by x (x,y) = 1 for (x,y) e A, 0 for (x,y) 5 AC. Now, the weak problem (P) is to find a pair (h, x ) e Wl’p(§2)xL°°($2), satisfying (1) rJ’IVhlp.2(Vh ,Vé) = A x éy for any § er’p(§2), § = 0 on F2 U F3, (ii) h2yin£2,h=hion1"3,i(i=1,2),andh=yonF2, (iii) 0S x SlinS2, and x =lonAC,whereA°={(x,y):h(x,y)=y}. §4.2 Existence of a solution to the weak problem 74 4.2.1 An approximation problem For a > o , we introduce a functional Hg defined on Lp(£2) by 0 if v- y > c Hc(v)- 3%! if czv-yzo 1 H v-y<0 a subset of wl-I’(o) defined by K= { ue Wl’p(S2)Iu=yonI‘2,andu=hionI‘3,i} and a subspace of Wl’p(S2) defined by v: ( ue Wl’p(£2)lu=00n rzur3 I. Consider the following approximation problem (Pg) : Find he 6 K , such that p2 JIthI (th .V§) = JH£(h€)§Y for any §er’p(S2), §= 0 on F2 U F3 . The following theorem is well known ( see Ful’nlt [14], for example): (4.9) 75 W Let K be a nonempty closed convex subset of B, a reflexive Banach space. Let f e V'I'and A : K—) B* be a continuous monotone map. Then, if A is coercive on K, there exists a solution u of the variational inequality ue K, (Au,v-u)2(f,v-u) foranyve K. Morover, if A is suictly monotone, the solution to this variational inequality is unique. Qorollag Let w e Lp(£2). Then the following problem has a unique solution: Find us 6 K such that (Aug,v-ug)2(f,v-ug) foranyveK, (4.10) where A is the p-harmonic operator, (f , é) = Hg(W)§y , fie K. Proof: By Corollary1.4, A is a continuous, strictly monotone and coercive Operater on K. Using Poincare's inequality which is valid for functions in Wl’p(§2) vanishing on part of 852, we can show that f e V*. The result now follows from Theorem 4.1. Iheoremiz For every 8 > 0 , there exist a solution he to the approximation problem (Pg). The set { he I e > 0 ) is bounded in both the LP(Q) and the wI'I’m) norms. Proof: Let us consider the mapping Fe which for w e Lp(S2) associates ug = Fg (w) the solution of the problem stated in Corollary 4.1. Let V be a function in Wl’p(§2), which agrees with us on F3 . Substituting §= 113 - \I’ into (4.9), we then have 76 -2 J IVllglp (V1.18 ,Vug - W) = A He(W)(lle - 111))“ (4.11) The following inequalities are adirect result of Lemma 1.4: Forp22, otIIug-tu IlpS(Aug -A\II )(ue-w). For2>p>1, allug-w||2S(IIu8II+|I\III|)2'p.(Aug-Aw)(u8-\[I). Using (4.9) we have, for p 2 2, 2 an ue - w H" s A He(W)(ue- my - AW" (th .Vtue - ‘11)) 5 (II (he - 1|!)y 1 + IjIVanp'lIVmg - w -1 $1521 Hug-111M + IIVwIIp ||V(u8-\V)|l which gives -1 Otllug-III III"1 5 1121 + IIVIIIIIP . (4.12) 77 Similarly, for 1 < p < 2, we also have otIIue-ty 115(11 us II+IIurII )2'1’ (191 + IIVur 11‘”). (4.13) We now show, using (4.12) and (4.13), that there exists a constant C(Q, w) independent of ug, such that IlugllSC(§2,\II). This follows by considering the two cases for p below: -1 1 1 For p 2 2, we choose C( (2, \II) = [ 01'1(I§2I + IIVIV llp ) ] /(p- ). For 2>p >1,if|l ug II SMax { II V II, 1}, let C( (2, y) =Max { II III II, 1}; otherwise (4.13) implies that -1 2-P p'1 IIueII-IIwIISa (IlugII+II\IIII) (IS2I+I|V\|III ), 1 || ue IIgIIuIII+0t'1(2II 118 II )2'1’ (191 + 11th up. ) IIug IIP'IS —-"-“AL—+ ot-1(IQI + IIVurIIp “1.13” 2 ‘+P 1) 2- -l s llwllp'l + 2 p ot-1(IQI + 11th IIp ), since II 118 II > II 111 ll . We can choose 78 C( £2,111): [ 11 v II “+22? ot-1(If21+ 1va 11‘“) 11/ (M). So, for r large enough, F8 maps B(0,r), the ball of center 0 and radius r in Lp(£2), into itself. The boundedness of us in Wl'p(£2), and the compactness of the identity mapping from Wl’p(S2) implies that F3 is completely continuous. By Shauder's fixed point theorem, F8 has a fixed point he in B(0,r) n wltpm). 4.2.2 An existence theorem 111m} There exists a pair (1!, X )6 Wl’p(9)> 0 we have a function he in Wl’p(S2) which is a solution to problem ( Pg ). The set { he I 8 > 0 } is a bounded set in the reflexive Banach spaces Lp(S2) and Wl’p(§2). We can choose a sebquence 8“ —> 0 such that he —e h in Wl’p(S2) and he —> h in Lp(§2) for some h, also Hg(hg) —- x for some x in Lp(£2). Taking limit in (4.11), we have p-2 JIVhI (Vh ,vg) = {I x a, for any g e wlI’m), g = 0 on 1‘2 U 13. (4.14) Since the convex sets K={ ve Wl’p(52)| v=yonr2,andv=h:onr3,i,i=l,2} 79 and K'={ ve LI’(o) I 0SvS1inS2} are weakly closed in Wl’p(f2) and Lp(§2) respectively (being closed and convex), we conclude that (h. x) e KxK' ; w1:P(o)x me) . On the set 52' = { (x,y) l h > y in S2 }, we have almost everywhere ( after extraction of a subsequence) Hg(hg) —> 0. Thus by Lebesque's Theorem of dominated convergence, Heme) —> 0 in Lp(S2'). From the fact that Hg(hg) —= x and by the uniqueness of the limit, we can deduce that x = Oin 9'. Similarly, we can show that x = 1 in the set I (x,y)lh oo lead to 41%|th ,V(h - y) )= J(h - y)y = £(h - y)ny = 0, (4.16) 8 i.e., 80 -2 JIVhlp= JIVhIp hy. (4.17) 2 2 On the other hand, IVhIpz thIP- by in Q, and thus thIp= IVhIp. by which implies that either h = constant or h = y in Q, a contradiction to h < y in Q. Case2)8QnI‘1= Q Similar to (4.16), using Green's formula, we have Jtvmwwh .V(h - y) >= 41h - y): = £01 - y)ny = I(h - y)ny. a aQflrl i.e., -2 c{IVhI‘L JIVhIp hy - Ih. (4.18) aQflF 1 Claim: h 2 0 proof: Let 01 = I (x,y) I h < 0 ]. If 8010 F1= O, we are in case 1) and there is nothing to be proved. Suppose that 8010 F1: Q . Then 81 Jthlp-th ,V§)= Jéy for any g e Wl’p(01), § = 0 on 801/1‘1. l 1 In particular, h = 0 on 801 IF 1, and h satisfies O[IVhI"'2(Vh ,Vtg) -.- OR, for any g c w1:I’(ol), g = 0 on aol m. l 1 Let 0'; be the reflection of 01 about the x-axis and h' the refection of h about x-axis, then by symmetry, we have 2 III e IIVhH" (Vh*,V§) -_- Ia, for any g e wI.I’(01),§ = 0 on 801. 01 C)1 ' By the Corollary to Theorem4.l this problem has a unique solution h* = 0. Hence h 2 0, and (4.18) implies that h = 0 on 80ln F1, and so -2 JthIp = JIVhIp hy which again leads to either h = 0 or b = y in 01, a contradiction. SUMMARY In this work, several problems governed by the so-called " p-harmonic " equation as well as those by the corresponding parabolic equation are studied. More specifically, we are concerned with problems governed by following two equations: -div(|Vulp'2Vu)=f, ut-div(|Vu|p‘2Vu)=f. Steady state problems with both homogeneous and nonhomogeneous Dirichlet boundary conditions are studied. Some less known results are given along with their proofs. These include a maximum principle and the continuity of the conformal capacity with respect to the parameter p as well as, in a special case, with respect to boundary change. Applying the theory of P. R. Ciarlet, M. H. Schultz and R. S. Varga [3] , finite element approximations of these nonlinear elliptic problems are studied. It is shown that the finite element solution converges to the true solution as mesh size h tends to zero and that a minimizing sequence of the p-harmonic functional which converges to the discretized solution, can be constructed in the finite element approximation space Sh(§2). Numerical experiments are conducted and test problems along with their finite element solutions are presented including finite element solution for the minimizer of the conformal capacity of a ring with a slit in the middle. Using the method of lines we show that the nonlinear parabolic problem governed by the p- harrnonic operator has a unique weak solution which is more "classical" than the weak solution obtained by applying the theory of J. Kaeur, in the sense that it satisfies the equation pointwise with respect to time. A L2 error estimate for the error between the true 82 83 solution and its semidiscrete approximation is made. 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