In this paper we prove an existence result for Leray-Lions quasilinear elliptic operator with discontinuous coefficients. The idea of the proof is based on compactness results for the sequences of solutions to regularized problems obtained via the Compensated Compactness, Young measures, and Set-Valued Analysis tools

Gwiazda, Piotr

Zatorska-Goldstein, Anna

English

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Weierstraÿ-Institutfür Angewandte Analysis und Stochastikim Forschungsverbund Berlin e.V.Preprint ISSN 0946  8633An existence result for the LerayLions typeoperators with discontinuous coecientsPiotr Gwiazda1, Anna ZatorskaGoldstein2submitted: 31st January 20031 Institute of Applied Mathematics and Mechanics,Warsaw UniversityBanacha 2PL 00-913 Warsaw, PolandE-Mail: pgwiazda@hydra.mimuw.edu.pl2 Institute of Mathematics, Polish Academy of Scienceniadeckich 8PL 00-956 Warsaw 10, Skr. poczt. 21, PolandE-Mail: azator@mimuw.edu.plNo. 811Berlin 2003WIAS2000 Mathematics Subject Classication. 49J45, 35J65.Key words and phrases. nonlinear elliptic equations, discontinuous operator, compensatedcompactness, Young measures, set-valued analysis.Edited byWeierstraÿ-Institut für Angewandte Analysis und Stochastik (WIAS)Mohrenstraÿe 39D  10117 BerlinGermanyFax: + 49 30 2044975E-Mail: preprint@wias-berlin.deWorld Wide Web: http://www.wias-berlin.de/AbstractIn this paper we prove an existence result for Leray-Lions quasilinear el-liptic operator with discontinuous coecients. The idea of the proof is basedon compactness results for sequences of solutions to regularized problems ob-tained via the Compensated Compactness, Young measures, and SetValuedAnalysis tools.1 IntroductionRecall the Leray-Lions operator in the formdivA(x; u;ru) +B(x; u;ru)where functions A : RRm ! Rm and B : RRm ! R satisfy the classicalassumptions: the growth conditions:jA(x; s; )j  C1(k(x) + jsjp 1 + jjp 1);jB(x; s; )j  C2(k(x) + jsjp 1 + jjp 1); the monotonicity condition:(A(x; s; 1)  A(x; s; 2))  (1   2) > 0for all 1 6= 2; the coercivity conditions:A(x; s; )    c1jjp;B(x; s; )  s  c2jsjp:for almost all x in an open subset   Rm and for all s 2 R,  2 Rm . The constantsC1, C2, c1, c2 are strictly positive, and k 2 Lp0(). The other classical assumptionis that A and B are Carathéodory functions (i.e. measurable with respect to therst variable and continuous with respect to the second and third). Under thesehypotheses the operator is a bounded continuous pseudomonotone operator of Leray-Lions type from W 1;p0 () into its dual W 1;p0(). (see [3], [5], [12], [10], [13]).1In this paper we consider the equationdivA(x;ru) +B(x; u;ru) = fand we relax the assumption on the continuity of A in the second and B in the secondand third variable. Instead, we assume that the functions A :   Rm  ! Rm andB :  R  Rm  ! R are Lebesgue measurable and such that:1. there exist nitely many open sets Ul such thati) Rm =Skl=1 Ul and Ui \ Ul = ; for i 6= l,ii) AjUl(x; ) is continuous for l = 1 : : : k,iii) there exists continuous extention Al(x; ) of AjUl(x; ) onto Ul.2. divA(x;ru) = f   B(x; u;ru) 2 W 1;p0(),3. B(x; ; ) is Borel measurable and convolution B with a standard Friedrich'smollier '" gives a smooth function.4. jA(x; )j  C1(k(x) + jjp 1);5. (A(x; 1)  A(x; 2))  (1   2) > 0 for all 1 6= 2;6. A(x; )    c1jjp.7. jB(x; s; )j  C2(k(x) + jsjp 1 + jjp 1),The assumption B(x; s; )  s  c2jsjp is not necessary if we assume that  is abounded set and one can use a Poincaré inequality. The above assumptions arevalid throughout the paper and we will not write them in formulations of theorems.Given functions A and B, we extend their graphs (i.e. we close the graphs andmake a set of values in each point convex; this procedure give an unique extension)to obtain multifunctions A and B. Our main result is the followingTheorem 1.1. Assume  to be an open set with jj < +1, and f 2 W 1;p0().Then there exist a function u 2 W 1;p0 () and measurable selectionseA(x) 2 A(x;ru(x)),eB(x) 2 B(x; u;ru(x)) such thatZeA(x)r'(x)dx + ZeB(x)'(x)dx = Zf(x)'(x)dxfor all ' 2 C10 .Malý and Ziemer [7] consider a regularity theory for quasilinear elliptic operatorswith discontinuous coecients. They assume that A(; s; ), B(; s; ) are Lebesguemeasurable and A(x; ; ), B(x; ; ) are Borel measurable. However, to solve theexistence problem, they need to assume that A and B are Carathéodory functions2(see p. 162, and p. 253). It is natural to ask if the assumption can be relaxed. Itwas one of motivations to this work.The other motivation comes from physical applications: let consider the free energyfunction of the formE(x; u;ru)with standard polynomial growth and coercivity condition, strictly convex in thelast variable but which is only dierentiable in the sens of Clarke with respect tothe second and third variable. Then Euler-Lagrange equations are of Leray-Lionstype but with discontinuous coecients. Similar motivation one can nd in thehemivariational inequalities (see [11], [9]).The main idea of the proof is based on the application of Young measures in anonstandard setting, forced by the discontinuity of A, combined with the bitingdivcurl lemma from Zhang [14]. We really need the biting version, as the termA(x;ru)  ru is only L1 under our growth conditions.Steps of the proof are as follows:1. We regularize functions A and B by convolution with the function '" and putthe term B" on the left side of the equation.2. Using Biting div-curl lemma, and characterization by Young measures we provea compactness result for a sequence of regularized solutions in W 1;q0 () for allq < p.3. At the end we use Convegence Theorem from the theory of the Set-ValuedAnalysis to obtain an existence of weak solutions.Remark 1. Using similar methods as in [3] and [5] one can show the strong con-vergence of gradients in the space Lp().Remark 2. In [6] there are considered generalizations of the pLaplacian in theformdivf(ajrujp 2 + bjrujq 2)rugwhere a; b > 0 and p; q 2 (1;+1). Our arguments, combined with some moregeneral characterization of Young measures (see [4]) allow us to consider the casep 2 [1;+1), q 2 (1;+1).2 ToolsIn the following theorems the notation is slightly dierent from that of [14], [1] and[8]. We use it to facilitate the reading ot the proof of Theorem 1.1.3Theorem 2.1 (Biting divcurl lemma). Let  be an open set of Rm and (zj),(Aj) be sequences in Lp(;Rm) and Lp0(;Rm) respectively, such thatzj * z in Lp(;Rm)Aj * eA in Lp0(;Rm)curlzj is bounded in Lp(;Rm)divAj is compact in W 1;p0())Then there exist subsequences, still denoted (zj), (Aj) and a nonincreasing sequenceof measurable sets (Ek), Ek   with jEkj ! 0, such thatzj Aj * z  eA in L1(nEk) for k = 1; 2; : : ::The next theorem is a slight modication of the Fundamental Theorem on Youngmeasures (see e.g. [8] Th.3.1).Theorem 2.2. Assume   Rm to be an open set of a nite measure and letzj :   ! Rm be a sequence of measurable functions. Denote j;lx= (Æzj(x)  'j)jUl.Then there exists a subsequence still denoted by zj and a family of weak-* measurablemaps l :  *M(Ul) such that1. x =Pllx 0, andkxkM(Rm) =XlZUldlx 1for a.e. x 2 ;2. j;l* l in L1w(;M(Ul));3. If for some p > 0 there exist constant C such that for all j:Rjzjjp  C, thenkxkM(Rm) = 1;4. If 3. holds and a set E   is measurable, F 2 L1(; C0(Ul)), and for everyl the family fRUlF ()dj;lx()g is relatively weakly compact in L1(E), thenZUlF ()dj;lx()*ZUlF ()dlx() in L1(E);5. If 4. holds and F 2 L1(; C0(Rm) thenZRmF ()dÆzj(x)()*ZRmF ()dx() in L1(E);i.e. x is a usual Young measure generated by a sequence zj(x);46. If x = Æz(x) a.e. in  then zj(x)! z(x) a.e. in .The idea of the proof is based on the fact that L1w(;M(Ul)) is the dual spaceto L1(;C0(Ul)) and the proof is similar to that in [8], pp. 3134. Passing fromthe functions zj which take values in Rm to maps which take values in the spaceof measures in Rm , we take instead of Æzj(x) measures of the form Æzj(x)  'j, andconsider them on the sets Ul where the function A is continuous. The assertion 5.comes from the fact: fPlj;l   Æzjg* 0 in the space L1w(;M(Ul)).The following theorem is the special case of Convergence Theorem which is wellknown in SetValued Analysis, (see e.g. [1]).Theorem 2.3. Let   Rm be an open set with nite Lebesgue measure, A :Rm  ! 2Rmbe a nontrivial setvalued map, with a closed graph, and such thatfor any  2 Dom(A) = Rm , a set A() is bounded and convex. Let us considermeasurable functions zj :   ! Rm and Aj : Rm  ! Rm single-valued functionsuch thati) supy2Rmdist[(y; Aj(y)); Graph(A)]  ! 0, as j ! 0;ii) the sequence fzjg converges almost everywhere to a function z;iii) Aj(zj) 2 L1() and the sequence fAj(zj)g is convergent weakly in L1() to afunction eA 2 L1().Then for almost every x 2  we have eA(x) 2 A(z(x)).The following fact is well known in setvalued analysis (see [1]):Lemma 2.4. Let A(x; ) is a maximal monotone extention of the function A(x; )satisfying the monotonicity condition 5. If eA1 2 A(x; 1) and eA2 2 A(x; 2) then( eA1   eA2)  (1   2) > 0for all 1 6= 2.Notation 1. Throughout the paper by '" we will denote the standard Friedrich'smollier with respect to the variable  i.e. for a xed radial (i.e. depending onlyof jj), nonnegative function ' 2 C10 (Bm) withR'(y)dy = 1, we have '"() :=" m'(="). The symbol e'" will stand for the molire with respect to the variabless and .3 Proof of the main resultProof. Let A"(x; ) = ('"A)(x; ) and B"(x; s; ) = (e'"B)(x; s; ). These are welldened, continuous regularizations of the functions A and B. The growth, coercivity5and monotonicity conditions are preserved under the regularization. Thus, thereexists a weak solution u" to the problemdivA"(x;ru") +B"(x; u";ru") = f;that is a function u" 2 W 1;p0 () such thatZA"(x;ru"(x))r'(x)dx +ZB"(x; u"(x);ru"(x))'(x)dx ==Zf(x)'(x)dxfor all ' 2 C10 .The energy estimates and conditions on A and B yield ku"kW1;p0 C, where theconstant C does not depend on ". Thus ru" and A"(;ru") are bounded sequencesin Lp(;Rm) and Lp0(;Rm) respectively, and up to subsequences we may assumeu"  ! u in Lp(;Rm)u"  ! u almost everywhereru" * ru in Lp(;Rm)andA"(;ru")* eA in Lp0(;Rm)B"(; u";ru")* eB in Lp0(;Rm):In order to show that eA is a measurable selection from A and u is the desired weaksolution we use theorem 2.3 and thus we need to show that ru"  ! ru almosteverywhere.For almost every x 2  set "x= Æfru"(x)g. Then of courseA"(x;ru"(x))  ru"(x) =ZRmA"(x; )  d"x():Since A"(x; ) = (A  '")(x; ), one can express the last integral asZRm(A  '")(x; )  d"x() ==ZRm(A  '")(x; )     ((A  Id())  '")(x; )d"x()+ZRm((A  Id())  '")(x; )d"x():(3.1)The rst term of RHS can be written asZRmZRmA(x; )(   )'"(   )dd"x();6and thenjZRmZRmA(x; )(   )'"(   )dd"x()j ZRmZRmjA(x; )jj   jj'"(   )jdd"x():Using the fact that '" has a support in a ball of a radius ", and the growth conditionsof A we can estimate the last expression byC"ZRm(jj+ ")p 1d"x()and a convergence to zero follows from the uniformly boudedness of the pth momentof the measure "x.The second term of RHS of (3.1) can be represent asZRmA(x; )  d"x();where the measure "x= "x'" is absolutely continuous with respect to the Lebesguemeasure. Then ZRmA"(x; )  d"x() = C"+ZRmA(x; )  d"x():Let ";lxbe a restriction of the measure "xto the set Ul and the function Al be acontinuous extension of AjUl onto Ul.Theorem 2.1 impliesA"(x;ru"(x))  ru"(x) * eA(x)  ru(x) in L1(nEk) (3.2)and thus the sequenceRRmA(x; )d"x() is relatively weakly compact in L1(nEk).Then Theorem 2.2 yieldsA"(x;ru"(x))  ru"(x) *XlZUlAl(x; )  dlx() (3.3)in L1(nEk), where the Young measure l is a weak* limit of ";l in L1w(;M(Ul))and x =Pllxis a probability measure for almost every x 2 .Therefore combining (3.2) and (3.3) we obtainZnEkeA(x)  ru(x) (x)dx ==XlZnEkZUlAl(x; )  dlx() (x)dx: (3.4)7for every  2 L1(nEk)Let now  be a nonnegative function from L1(nEk). It follows from the mono-tonicity condition on A that the termXlZnEknZUlAl(x; )  A(x;ZRmdx())  ZRmdx()dlx()o (x)dx (3.5)is nonnegative. We will show that in fact it is equal to zero.After expanding the expression under the inner integral, using the formula (3.4) andthe fact x is a probability measure, we obtain that (3.5) is equal toZnEkeA(x)  ru(x) (x)dx ZnEkXlZUlAl(x; )dlx()ZRmdx() (x)dx ZnEkA(x;ZRmdx())ZRm[  ZRmdx()]dlx() (x)dx: (3.6)The last term is equal to zero becauseZRm[  ZRmdx()]dlx() = 0:We can represent functions A"(;ru") and ru" also by the Dirac measures "xandtheorem 2.2 gives eA(x) =XlZUlAl(x; )dlx()andru(x) =ZRmdx():That however implies (3.6) is equal to zero.Thus, for every lZUlhAl(x; )  A(x;ZRmdx())ih  ZRmdx()idlx() = 0and from Lemma 2.4 it follows thatx = Æfru(x)g;and therefore ru"  ! ru almost everywhere.8Theorem 2.3 now gives eA is a selection from A, eB is a selection from B andZeA(x)r'(x)dx + ZeB(x)'(x)dx = Zf(x)'(x)dxfor all ' 2 C10 . Moreover,eA and eB are measurable as weak limits of measurablefunctions A"(x;ru") and B"(x; u";ru").Acknowledgment: One of the authors (P.G.) appreciates the grant of WeierstrassInstitute for Applied Analysis and Stochastics, Berlin. We would thanks also: DrAgnieszka Kªamajska, Dr Paweª Strzelecki, Professor Krzysztof Cheªmi«ski and Pro-fessor Dr Krzysztof Wilma«ski, for some comments and remarks to this paper.References[1] Aubin J.P., Frankowska H., Set-valued analysis, Birkhäuser, Boston BaselBerlin, 1990.[2] Ball J. M., A version of the fundamental theorem for Young measures, in:PDE's and Continuum Models of Phase Transitions, M. Rascle, D. Serre andM. Slemrod (eds.), Lecture Notes in Phys. 334 Springer, Berlin 1989, 207-215[3] Boccardo L., Murat F., Puel J.P., Existence of bounded solutions for non linearelliptic unilateral problems, Ann. Mat. Pura Appl., IV Ser., 152 (1988), 183196.[4] Kaªamajska A., Between the classical theorem of Young and convergence the-orem in setvalued analysis. To appeare.[5] Leray J., Lions J.L., Quellques resultats de Visik sur les problemes elliptiquesnon lineaires par les methodes de Minty-Browder. Bull. Soc. Math. France, 93(1965), 97-107.[6] Málek J., Ne£as J., Rokyta M., and R uºi£ka M., Weak and measure-valuedsolutions to evolutionary PDEs, Chapman & Hall, London, Weinheim, 1996[7] Malý J., Ziemer W.P., Fine regularity of solutions of elliptic partial dierentialequations, AMS 1997.[8] Müller S., Variational models for microstructure and phase transitions, in: Cal-culus of variations and geometric evolution problems, S. Hildebrandt, M. Struwe(eds.), Lecture Notes in Math. 1713 SpringerVerlag Berlin Heidelberg 1999.9[9] Naniewicz Z., Panagiotopoulos P. D., Mathematical Theory of HemivariationalInequalities and Applications, Marcel Dekker, Inc., New YorkBaselHong Kong1995.[10] Ne£as J., Introduction to the theory of Nonlinear elliptic equations, Wiley,Chichester, New York, Brisbane, Toronto, Singapore, 1986.[11] Migórski S., Ochal A., Optimal Control of Parabolic Hemivariational Inequali-ties, J. Global Optim., 17 (2000), 285300.[12] Showalter R. E., Monotone operators in Banach space and nonlinear partialdierential equations, AMS 1997.[13] Zeidler E., Nonlinear Functional Analysis and its Applications, Vol.2, SpringerVerlag Heidelberg, 1993.[14] Zhang K., Biting theorems for Jacobians and their applications, Ann. Inst.Henri Poincare - Analyse non linearie 7 no. 4 (1990) p. 345-365.10

An existence result for the Leray-Lions type operators with discontinuous coefficients

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An existence result for the Leray-Lions type operators with discontinuous coefficients

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