Homogenization of general quasi-linear Dirichlet problems with quadratic growth in perforated domains

J. Math. Pures Appl., 76. 1997, p. 431-416 HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS WITH QUADRATIC GROWTH IN PERFORATED DOMAINS By Juan CASADO-DfAZ ABSTRACT. - In this paper, we study the homogenization of a Dirichlet problem in perforated domains for an operator which is the perturbation of the Laplace operator by a general nonlinear term with quadratic growth in the gradient. We show that a new term, which does not depend on the gradient, but which is nonlinear, appears in the limit problem. We also give a corrector result. 0. Introduction The goal of the present paper is to study the homogenization problem (‘) (0-l) { - Au’ + H(z,zf, Vu’) = f in ‘D’(V); UE E H,1(W) n L”(W), where Q2’ is a sequence of open sets which are contained in a fixed bounded open set R c RN, f is a function in L”(R) and H(z, s, <) : 0 x R x RN H W is a Caratheodory function which has a quadratic growth in the variable [ and is of classe C2 in the variable (S> 0. The existence of a solution for the problem (0.1) has been established by L. Boccardo, F. Murat and J.P. Puel in [B M P] and its uniqueness by G. Barles and F. Murat in [B M]. From these works, we also deduce that uE is bounded in Hi (0) n L” (0). Therefore, extracting a subsequence, VY converges weakly in Hi(R) and weakly-* in L”(R) to a function U. The questions which we address here is to find the problem satisfied by the function u and a corrector result. It is well known (see [C Ml, [DM Ml], [DM M2], [DM G]) than even for the linear problem (0.2) - Au’ = f in 27’(V), uE E H,l(RE), (‘) Here and in what follows, we consider the functions uE as defined on the whole of 12 by setting rh6 = 0 on s1 \ 62” (see Appendix: Notation). JOURNAL DE MATHBMATIQUES PURES ETAPPLIQU6ES. - 0021-7824/1997/05/$ 7.00 0 Gauthier-Villars 432 J. (‘ASADO-DfAZ the equation satisfied by the function u is not in general -Au = f in D’(0), but is of the type -Au + ~IJ, = f in D’(0), where a positive measure /L vanishing on the sets of zero capacity appears. In the case of equation (O.l), the nonlinear term H(.c, u”, V 11,~) leads us to a more complex equation, in which the new term which appears is no more linear in II,, but is of the form T(x, U)/L for some nonlinear function T(Lc, s) and for the same measure 1~ which appears in the linear case. In [C3], we have studied the particular case where H(:x:. II,‘, v UC) = Au’ - y IV 1~’ I’, X > 0, and have proved that in that case T(x.s) = s. For what concerns the sequence P, we will assume in the present paper that (0.3) 3 ,c E H’(R), f = 0 in 0 \ QE. p - 1 in H1(62) weakly. This implies in particular that the holes 51 \ 0’ are sufficiently small. As proved in [C2], this hypothesis is very close to the hypotheses assumed in [C M] (see also [K M]) to study the homogenization problem (0.2). A typical example is the case Sz” = R \ T", where T’ is the union of balls of radius E* the centers of which are periodically distributed at the edges of a cubic network of size E. Hypothesis (0.3) implies the existence of a subsequence 1~’ which vanishes in 12 \ 12’ and which converges weakly to 1 in H’(R) ( see the precise properties of 7~’ in Section 1, Theorem 1.3) such that the following corrector result holds: If u E Hi (0) n L”(R), the solution u” of (0.2) satisfies (0.4) ‘(f - ,Y#U + 0 in H,‘(0) strongly or equivalently v UC - V ‘/I. - ,uV G -+ 0 in L2 (0)” strongly It is however proved in [C3], by the study of the example H(:c. %I,‘, V 11~) = X U’ - y /V ,llE 1 2, that (0.4) does not hold in general for the quasi-linear problem (0.1). We will nevertheless use ‘IU”U as a test function in the proofs below to estimate V 1~’ when ~1,~ is the solution of the quasi-linear problem (0.1): In some sense, we will compare V 1L and V u + UV ~1’. We follow the general method designed by L. Tartar (see [T]) to study a homogenization problem which consists to test the equation by special test functions. We HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 433 introduce here an original variant of this method, which consists to make a comparison between V 11: and V u + UV wE when uE is the solution of (0. l), u its weak limit and ~1’ the corrector for the linear problem. The important fact is that W~YL is no more a corrector for the nonlinear equation (0.1) (i.e. (0.4) does not hold here) but this comparison will nevertheless to reconstruct the limit equation. Our proof will also make use of nonlinear test functions (as done in [B M P]) and of a change of unknown function (as done in [B M]) to pass from the quasi-linear equation (0.1) to another equivalent quasi-linear equation which satisfies a good “structure condition”. The homogenization of the quasi-linear problem (0.1) could as well be carried out without assuming any hypothesis on the sequence R”. In this case it is sufficient to replace the sequence UI’U by the corrector given in [DM G] (see also [DM Mull, [DM MUM]). One could as well consider the case where -A U” is replaced by a monotone or even pseudomonotone operator --d%~ (~(5, ,uE, V 1~~) acting on w,‘,p(0) (in this case the function H has to have a growth less than I<]“): One has to use in this latest case the corrector results of [DM Mu2]. In view of the technical difficulties which appear in the present paper, and which are mostly due to the use of the technique of change of unknown function which traces back to [B M], we have prefered to limit ourselves to the case where we assume that (0.3) holds. We hope that the reader will be happy of our choice. The method we use in the present paper (i.e. the comparison of V %I,” with V u + UV ~1’) is also sucessful in the study of the homogenization of Dirichlet problems for nonlinear monotone and pseudo-monotne operators of Leray-Lions type. The mehod is presented in [C4] in the simple case where monotone operators defined on W,t,ll(0) are considered and where an hypothesis similar to (0.3) is made on the sequence f12”. The general case of monotone systems without any hypotheses on the sets 6F is treated in [C G]. Note finally that even if the basis of the technique used in [C4] and [C G] is the same as in the present paper, the situation is simpler there since no change of unknown function is necessary when no “quadratic” perturbation occurs. The main results obtained in the present paper can be summarized as follows: Let 62 c Iwav be an open bounded set and let 0’ be a sequence of open sets contained in 62 such that (0.3) holds true. We consider a Caratheodory function H : (2 x R x R” ti US’ such that for almost every z E 62 the function H(x; s, <) is of class C2 in s and [ and has at most a quadratic growth in E. We also assume that for a strictly positive constant X and for almost every :c E 0 we have and that the first and second derivatives in (s, <) of H satisfy reasonable growth conditions (actually the same as the derivatives of ]<12, see (1.2) and (4.3) for the precise hypotheses made on H). Then we have the following homogenization theorem for the quasi-linear problem (0.1) (which easily results from Theorem 5.1, Theorem 6.3 and Remark 6.3): THEOREM 0.1. - There exists a subsequence of F (still denoted by E), a positive bounded Bore1 measure p which vanishes on the sets of zero capacity (11, is the same measure which JOIIRNAI, IX MATH6MATIQUES PURES ET APPLIQI&XS 434 J. CASADO-DfAZ appears in the homogenization of the linear problem (0.2)) and a Carath&odoT @nction T : 0 x R H W, such that for any function f E Lc-(G), the unique solution U’ of (0. I) converges strongly in W,“p(0), 1 5 p < 2, weakly in Hi(a), and weakly-* in L”(0) to a function u which is the unique solution of the problem: (0.5) { - Au + T(rc: U)LL + H(z, u, D U) = f in D’(0), u E H;(0) n L”(0). For p-almost every x E R, the function T(:E, .) is increasing, satisfies T(z, 0) = 0, and is locally Hiilder continuous, i.e. satis$es IT@, sl) - T(x, s2)1 5 C(s)lsl - &, where s = max{(sll, 1~21) and where C : [0, +oo) H [0, +oo) and X : [0,+x1) H [l: +zo) are increasing. We also have the following corrector result (this result is stated in Theorem 7.1): THEOREM 2. - Let E be the subsequence extracted in Theorem 0.1. Define for s E R and r1 E N the function s:, as the solution of (0.6) C - As:& + ns: + H(z: sz, D SE) = KY in D’(flE), s:, E H;(v) n L”(F) and set P,“(s) = V sFL. Consider on the other hand a step function y(x) = C:f, six&, (XL.‘), where the Qi are closed subsets of R which satisfy p(Q, n Q:;) = 0 for % # j and where the si are real numbers. Let Q and t be dejined by: Q = U Qi, 1; = max{w{ll G NIL=), II Y lb(~)). i=l Then (0.7) I IV UE - v u - P,(X) y(x))12 dx 1 T-5 : where 6 : [0, +m) H [0, foe) and 1 : [O,+oo) H [ 1, -tc0) are increasing functions which do not depend neither on the sets Qi, nor on the function y and nor on the right hand side f of (0. I). The above result provides an approximation of V IL’ in L2(fl)N. Indeed when Y(x) = c:“=, %xQ,b) IS a step function which is defined on closed sets Qi with p(Q, n Qj) = 0 and when y is close to ‘1~ in L1( 0, dp) (it is possible to construct such test functions, see Remark 7.4), then Vu + P:(z, y(z)) is close to V 2~~ in LZ(R)N. Formally the idea is to replace V ZL& by V SE, where s is the value of U(X) at the (frozen) point .z. This replacement is nevertheless impossible, since the function V s’, = I’,“(,, s) is not TOME 76 - 1997 - No 5 HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 435 a continuous function with respect to s and this leads us to the use of an approximation by a step function y. The idea for the introduction of s: is the following: For s E Iw given (which will be u(z~) for a given x0), we would like to find some fs E L”(R) such that the solution sE of (0.8) { - As’ + H(z, sE, V s”) = fs in 27’(fl’) SE E H,1(V) n L”(R’) has the property that sE tends to s. This is impossible for several reasons: The first one is that s does not belong to IIt (a), since s # 0 on da. This could be solved by replacing s by scp(z) with cp E D(R), but a new difficulty appears: Passing to the limit in (0.8) would give, according to Theorem 0.1, -A(scp) -t T(z, scp)dp + H(z, scp, sV ‘p) = fs in D’(R) and in general fs does not belong to L”(R). For this last reason, we introduce a new parameter n. and the penalization n(sz - s) in (0.6); passing to the limit in (0.6) for n fixed implies that stL tends to sn in Hi(R) weak, with: C - As,, + T(x, G)P + H(x:, s,, V s,) = ns in W(n), s, E H;(a) II L”(R) and it can be proved that s, tends to s when n tends to infinity. Contents 1. Some preliminary results about quasilinear problems with quadratic growth and homogenization in perforated domains ...................... 1.1. Quasilinear problems with quadratic growth .................. 1.2 Homogenization in perforated domains ..................... 2. Estimates on the gradients of the solutions and first results on the homogenization problem. ................................ 2.1. Strong Wi’p(fi) 0, < 2) convergence ...................... 2.2. First estimates on V uE .............................. 2.3. Structure of the limit of (0.1) ........................... 3. Comparison of the gradients of two sequences of solutions ........... 4. Dependence of the function E with respect0 to u ................ 5. Construction of the function 7’ ........................... 6. The homogenization result and a property of the function T .......... 6.1. A first homogenization result ........................... 6.2. The uniqueness of the limit problem ....................... 6.3. The homogenization result ............................ 7. Corrector ....................................... Appendix: Notation ................................... A. 1. Standard notation ................................. A.2. Specific notation .................................. References ........................................ 436 436 442 444 445 446 449 451 457 462 465 465 466 469 470 473 473 474 475 JOURNAL DE MATH6MATtQUES PURES ET APPLlQUBES 436 J. CASADO-DiAZ 1. Some preliminary results about quasi-linear problems with quadratic growth and homogenization in perforated domains 1.1. Quasilinear problems with quadratic growth We first recall some results about the existence and uniqueness of the solution of the problem (1.1) i - Au + H(x, 7~. Vu) = f in D’(O), u, E H,1(0) n L”(0). where 0 is an open set contained in 0. Following the ideas of [B M P] and [B M], we will obtain for this problem some estimates which will be useful in the homogenization of (0.1). Let us assume that the CarathCodory function H : 12 x R x RX H R satisfies the following hypotheses: i) For almost every z E R and every < E R”, the function H(z, .: <) is continuously differentiable and there exists a constant X > 0 such that 0.2) ii) There exist two increasing functions ~u(~ and II : [0, +a) H [0, +30), such that REMARK 1.1. - It is enough to assume that ‘11~ and u are just bounded on the bounded sets of [0, +co). We then obtain increasing functions by defining 2jg(s) = ~up(~<~<,~ ,~~,(t) and G(s) = SUI)~<~<,~ t)(t). L. Boccardo, F. Murat and J.P. Puel proved in [B M P] (see also [Cl]) the following existence result for (1.1). THEOREM. - Assume that 8 is an open set, 8 c f2, and that H satisJies (1.2) and (1.3) and that f E L”(O). Then there existsa solution u of(l.1) such that 11 u lIH;(o) and 11 u IIL-co, are bounded by constants which depend only on X, 710, 71, 11 f l)I,-(i2) and the measure qf@. In fact, we have (1.4 11 u IIL-(o,L awl+ II f III?(0, x The estimate for II u IIH; co) is more complicated and will not be given explicitly. (It is easily deduced from the following Lemma by taking cp = 1 and T = 0 in (1.7). LEMMA 1. - (see [B M P]) Assume that 0 is an open set, 0 c St, and that H satisfies (1.2) and (1.3) and that f E L”(8). C onsider a constant n/r > 0 and a function u E H1(0) n L”(O) such that (I ‘U IIL-cc+j< n/r, and dejine f E H-l(@) + L1(0) by: (1.5) -Au-t H(x,u, Vv) = .f in D’(O). TOME 76 - 1997 - No 5 HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 437 Let h E Cl(R) be the function dejined by h(s) = 2&003)” which depends only on M. This function satisfies (1.6) h(0) = 0, h'(s) - 2w(M)lh(s)l > 1, V’s E W. Then, for any T and ‘p such thaz 7- E Hl(0) n L”(O), ‘p E H1(0) n L”(O), p 2 0, (U - ~)cp E H,1(0); we have (’ ) (h(u-r)lp+2P,(M) J ’ IOr121h(zmjq 0 Proof. - Taking h(a - T)(P E I$(@) n L”(O) as test function in (1 S), we obtain s VuVh(u-r)cp+ s h(u - r) vu v p + =@(f, h(u - Q&j. ’ .I H(z, u, v u)h(u - r)(p 0 Since V /L(IL - r) = h’(u - r) V (IL - r) and (1.3) we have: (1.8) - .I’ pVrVh(u-r)- h(u-r)VuVp 0 .I 0 + boo 1; IQ - +P + 4W J, W21hb - 91~. Using the inequality IVu12 5 2lV (r~ - r)12 + 21Vr12 in (1.8) and carrying the term 2v(M) so JV (r~ - r)121h(u - r)lp to the right-hand side of (1.8) and using (1.6), we deduce (1.7). n We will use stronger hypotheses about H to obtain a uniqueness result for Problem (1.1). Specifically let us assume that for almost every 1c E R the function H(z, ., .) is continuously differentiable and that there exists an increasing function Q : [O, +cc) H [0, +cc) such that (fW,o) E L”(fl), (‘) Here and in what follows, (f,v)~ denotes the duality pairing between H-l(8) + L’(6) and Hi ((3) n LK (0) (see Appendix: Notation). JOURNAL DE MATH~MATIQUES PURES ET APPLIQ&ES 438 .I. CASADO-DfAZ REMARK 1.2. - Assumption (1.9) implies the existence of increasing functions 110: li : [O> +cQ) H [O, + co such that the function H satisfies inequality (1.3). ) The following lemma results from a computation which is due to G. Barles and F. Murat (see proof of Theorem II.1 in [B MI). LEMMA 1.2. - Assume that 0 is an open set, 0 c 0, and that H satisfies (1.2) and (1.9). Let u E HI(O) n L”(O) and f E H-l(@) + L1(0) which satisfy (1.10) { - Au + H(z, U, Vu) = f in D’(S), ~1 E H1(0) n L”(0): 11 u I[L~(~>)< M, for some M > 0. Define for A > 0 and K > 0 the functions $ and 19 = li,-l by: (1.11) (1.12) Qs such that s < ilog( Then there exists two constants A, K > 0, which are increasing with respect to M such that the function 6 = 19(u) satisjies (1.13) - A& + B(.2.,ii, Vii) = f in D’(O), ii E HI(@) n L”(O), where f^ is defined by f f = 1//‘(G) and where the function B : 0 x W x RN H R is a Carathkodory function which satisjies a property similar to (1.9): For almost every :c E R the function B(z, . , .) is continuously differentiable and there exists an increasing function /3 : [0, +oo) H [0, +CG) such that pLo,o) 6 L”(Q), (1.14) I I~(x,i,[)l <Ll(ldl)(l+li12). a.e.n:EQ v(S,i)EWXR”, (~(~,6_i)(<II(l~~l)(l+(CI). a.e.zE@ V(s^,i)EWxRN. Finally, there exists also a constant n > 0, which depends on A and K, and thus on M, and which is increasing with respect to M such that TOME76- 1997-No5 HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 439 Proof. - The results of Lemma 1.2 are proved in [B M] (proof of Theorem 11.1). Let us recall the main points of this proof. When K > eAM, the domain of definition of ZY covers [-M, M] . When u is a solution of (1.1) the change of unknown function li = ?Y ( U) implies that ii is a solution of (I. 13) where It is then easy to prove that (1.14) holds true. In order to prove (I. 15), it is sufficient to follow the proof of [B M]. One first fixed A sufficiently large, A = Ao(M) (with A,(M) = KZ -t 1 in the notation of [B MI). Then for K large enough (more precisely, K > Ko(M, X) where A is the constant which appears in (1.2)) one obtains, when n is large enough (n 1 no(M7 X)) (1.16) which is the desired result. REMARK 1.3. - As in Remark 1.2, the inequalities in (1.14) imply the existence of two increasing functions Co, 6 : [0, + ) cc H [0, +co) such that the function B satisfies an inequality similar to (1.3). The following result provides similar estimates to those of Lemma 1.1, which will be used later for the homogenization of problem (0.1). The ideas used in the proof follow from [B M]. LEMMA 1.3. - Assume that 0 is an open set, 0 c R and let B : R x R x IRK H R be a Carath&odory function which satisfies properties (1.14) and (1.15). Consider G,,, 5 E Hyq n L”(R), with 11 0 [IL-(O)< M, 11 6 [(L-(G)< M, and f,jj E H-l(0) + L1(0) which satisjfy (1.17) - Ail+ B(z,&,V&) = f^ in D’(O), {- A8 + B(x, 6, V 6) = lj in D’(0). Set S(s) = IsIn-’ s. Then, for any function cp E H1 (0) n L”(O), cp > 0, we have the following estimates: i) Let G = li - 2. Assume that 29 E Hi(O), we have: (1.18) 1 1 z Q .I S’(;)IV&12p+ S[ (B(z,i&Vii) - B(z>6,Vli))S(Lj) + b(&)~V&/2 p 0 2 I = (f - ij, S(ij)(p)Q - I S(G) v (2) v $9. .Q Moreover (B(x,i&Vii) - B(~,~,VG))S(LJ) + ;S’(&)lVljl” 1 > 0 a.e. in 0. JOURNAL DE MATHBMATIQUES PURES ET APPLlQUtES 446 J. CASADO-DiAZ Using (2.1) and the fact that H(z, uE’ , V k) is bounded in L1 (fl) independently of E’, we conclude to the existence of a constant C > 0 such that J IV ZE’12X {/zq<p} 5 0, + CP - .I v (w”‘u) v T&“). R R By (P6) the last term converges to zero. Since for E’ small enough we have A6 c {X E R : /,z”‘(x)/ < p}, we deduce that lim sup J IV &I2 < cp. c'-0 .-In Since p is arbitrary, this implies that (2.4) Writing 51) VZE' lllf2(Q) A+ 11 rw II&) pp: we obtain that for any p such that 1 < p < 2, 2” converges strongly to zero in W,‘>“(R). Since U/IL converges strongly to u in W,‘*p(0) by (P4), we have that uE’ converges strongly to u in WllP(fl). Finally, since the above reasoning holds not only for uE but for any subsequence of uE, Theorem 2.1 is proved. n 2.2. First estimates on V Q The following lemma extends in some sense the resuts of Corollary 1.2 to the case where w’u is replaced by G, the solution of (0.1). LEMMA 2.1. - Assume that H satisjies (1.2) and (1.9). Consider f E, f, uE and u which satisfy (2.3), a function cp E H1 (n)n L” (a), p > 0 and a sequence ofjiinctions @ such that { cp E H1(R) n L”(R), cp 2 0. y’i” E H1(S1) n L”(0), $’ 2 0, II ti” IIH’(R) + II @ lb(n)< (7. Then there exists a sequence of functions pE which satis$es (2.5) pE E HA(W) n L”(W), pe 2 0 pE - 0 in H;(0) weak, (I ,oE I/L-(n)< CM such that I J IV (u" - u)pj!J'p 5 CM IV WE12?pp (2.6) R J + (/l:w~l~~)~(~,v~~l~~~~)~+o~. TOME 76 - 1997 - No 5 HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 447 Proof. - Applying estimate (1.7) to equation (2.2) with 0 = fltE, u = u”, T = w’u, cp = $“cp and setting ZE = uE - w%, - J h(f) v 28 v (+p) + CM IqqI$“(P + CM IV (w”4l”lw)leP. w J w J w Let us now estimate the various terms of the right-hand side of (2.7): By (2.1) we have (fE,h(~E)~Ep)RE = 0, since h(x’) - 0 in H1(R) weak. Applying (P6) to 21’ = uh(zE)@~ which belongs to Hi(R’) and tends weakly to zero in Hi(R), and using 1) u IjLW(ol< M, we obtain for the second term = 0, - .I’ ,tJEpuV wE V h(S) R - 0, - - s V wE V (t/Apuh(z”)) + @&(ZE)V WE V u cl s R + I* puh( x’) V wE V I/? + R .I $“uh(z”) V wE V cp R = J cpuh(zE) V wE V 4” + 0, R ~~~(s,lV-E,z~)‘(~lV~E121,~E,lz,)~+~,. For what concerns the third term of the right-hand side of (2.7) we write - lc h(x’) VuE V (@v) = - /, h(2) V (u’ - u) V ($‘cp) + 0, = -/ h(z”)cpV(u’-u)V$‘+O, R The fourth term is CM s Ih(sz’)lu’,‘p = 0,. RE For the fifth term, we have: C M .I Iv W4121h(~‘)l~“cp < CM s [IV wE12u2 + Iw’~~~VU(~] Ih(.z’)l@p RE n = CM .I JV wEJ2u2Jh(zE)J@p + 0, < CM )V’W~)~$~~ + 0,. R s R JOURNAL DE MATHeMATlQUES PURES ET APPLIQU6ES 448 J. CASADo-DiAZ Taking into account the estimates obtained for the right-hand side of (2.7) we have proved that: (2.8) Using (1.30) with $’ = $‘cp and the fact that I/ 7~ IjL=;(L2)< M, we have Taking in (2.9) $’ = 1 (which is licit), we obtain I’ IV (If - 11,)12y 5 C,$I . s1 I pw’l*y + 0,. . S? which substituted in (2.9), implies (2.6) with fjE = Cnl )/b(~‘)/~. n COROLLARY 2.1. - Assume that H satisjes (1.2) and (1.9). Consider f’, f, 11,~ and IL which sati.~j$ (2.31, as well as @: 4~ E H,’ (0) f? L” (n), with $’ > 0, $” bounded in L” (12) and $’ converging strongly in HA(R) to 41. Then Pro@ - Take y = 1 in (2.6) and observe that sa IV w”I*$” --+ .I;2 Q&L by (P6) with ,(,’ = r~I,jz*yy andu= p= 1. n COROLLARY 2.2. - Assume that H satisjes (1.2) and (1.9). Consider f’, f, 1~’ and 11, which satisfy (2.3) and let Y/I’ and 9 be such that y E H’(O) n L”(O), p > 0, ,I)’ E H’ (12) n L”(Q), Y/f > 0, f - 0 in Hi (62) n L”(O) weak- * . Then we have (2.12) Proqf. - Inequality (2.11) follows from (2.6), (1.31) and I( pE ((Lm(61)i C.M, while inequality (2.12) is deduced from (2.11) and from IVu’12 5 2~V(~r”-u)~*+2~V7~~*. I TOME 76 - 1997 -No 5 HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 449 2.3. Structure of the limit of (0.1) The estimates above obtained allow us to give a first result about the structure of the problem obtained by passing to the limit in (0,l). THEOREM 2.2. - Assume that H satisjes (1.2) and (1.9). Consider f”, f, ~~~ and II, which satisfy (2.3). Then, there exists a function E E L”(R, dp) with 11 E I(~~(n,,l,,)< CIll, such that II, is a solution of the problem: (2.13) - AU + Ep + H(z, 9~: Vu) = f in D’(O)! II, E H;(O) n L”(O), or equivalently (“): The ,function E is dejined by: i.e., since H(n:,u”.VuE)uf is bounded in L’(R), Ep - up + H( 2, IL; V U) = iii; H(z. IL’, V uE)wE in Mb(O) weak- * . Proof. - For cp E D(0) we use (as for the linear case, see [C M]) TLI”~ E H@) n L”(F) as test function in (2.2). We obtain (note that the integrals can be written on the whole of 62) Using (P6) and (2.1) we deduce that, as E + 0 Since H(z. ~b’. V ‘u~)?u~ is bounded in L’(62), we deduce from (2.16) that: H( z, If, v uE)71JE - v = f + Au - up, in Mb(O) weak- * . (“) Recall that H,’ (62) f’ 15” (0) c L%(R. rllc) (see Appendix: Notation) JOURNAL DE MATH6MATlQUES PURES ET APPLIQU&S 450 J. CASADO-DiAZ On the other hand, using (1.9) and 0 < W” < 1, we have for every function cp E D(Q), IJ R [H(z, UE) v uE)wEp - H(3:. u, v u)wEp] <(;1(/(1+~v~urJZ+~8ul2)la--u/I~/+~(1+~vu~I+~V~~)~v(uE-u)ll~l). . I? .R Using that uE - u converges almost everywhere to zero, that I] uE - u /IL-(o)< 2&f, that (2.17) IV UE12 < 2p (u’ - u)l2 + 2lV uj2, pu’I < IV (u” - u,)I + IV211 and that (2.18) IV (uE - u)I - 0 in L’(Q), which is deduced from Theorem 2.1, we get: I.1 R [H(Z, UE, v uE)wEp - H(z, u> Vll)w&;o] / 5 CM I’ IV (u” - “))2l’pI + oc. .n Passing to the limit in this expression and using (2.10) with $F = Iv], we deduce By the Radon-Nikodym’s theorem, we deduce that there exists a function E’ E L”(R, dp) with ]I E’ ljL-(o.dti)I C&f such that which implies that E defined by E = E’ -t u satisfies This proves (2.13) and (2.15). The equivalence between (2.13) and (2.14) follows from a result of J. Deny ([D], see also [Z]) which implies that: I when p E M:(a), E E L”(R, dp), z E H,‘(a) n L”(O); (2.19) then (E~,,z)~ = .! Ezd,u. n n TOME76-1997-N’s HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 451 3. Comparison of the gradients of two sequences of solutions This section is devoted to the proof of the following Lemma, which shows that when uE and w’ are the solutions of two problems (0.1) with right-hand sides f and g, which weakly converge to u and ‘u, then 11 uE - 21’ - u + 71 IIH~CRj can be estimated by II u - u IILl(n,dP). LEMMA 3.1. - Assume that H satisfies (1.2) and (1.9). Consider f E, f, uE and u, and gE, g, vE and v which respectively satisfy (2.3). Define: rE=uE-7f-u+tl. Then, for any function cp E H1(fI) n L”(a), cp > 0, we have (6) W) / I R ~~~l~~~~~(i,,vw~l~~)l-~(~lcw~l~lu-v,~~~)~+u~~ Proof. - It will be performed in eight steps. STEP 1. - In view of Lemma 1.2, there exist two functions $ and 9 = 4-l given by (1 .l 1) and (1.12) such that denoting (3.2) we have (3.3) - AGE + B(z, ?Y, V 6’) = f^E in D’(R’), - A? + B(z, G’, V G”) = 6” in D’(RE), where the function B satisfies (1.14) and (1.15). STEP 2. We have (7) (3.4) Iii - 61 5 CNIIu - VI, (3.5) { IVii”l 5 C,lVu~I piq 5 C~pVEl) (3.6) { p(Q’-ii)I<CIMp(UE-l(l)~+O~* IV (6’ - $)I < c,p (TIE - w)I + Of2) (3.7) pw12 5 c,(pq2 + 1 v (v’ - v)121ii’ - VYI”) + ofl, (6) Here and in what follows XM denotes a generic constant with X&f 2 0, which can change from a line to another and which is increasing with respect to M (see Appendix: Notation) (7) Here and in what follows 0: denotes a sequence which converges to zero in X and which can change from a line to another (see Appendix: Notation) JOURNAL DE MATHhMATIQUES PURES ET APPLIQU6ES 452 J. CASADO-DiAZ where analogously to r’, +’ denotes Proof. - Inequality (3.4) is clear since rY is locally Lipschitz-continuous. The equality V GE = rY’( 71,‘) V ~1,~ implies IV IjE 1 < CA1 JV ~~~ 1 and analogously, we have lVC’I 5 CJ~pffl. The first inequality of (3.6) (the second one is similar) is deduced from v (6' - 6) = 19'(d) v lLC - f/Y'(u) v 'VI, = ?Y'(?f) v ( lLE - ,?A) + (7Y'(1LE) - lY'(lL)) v 1I, = lY'(lLZ) v (d - IL) + (f In order to prove (3.7), we write which implies (3.7). STEP 3. - Define cc -- = IL ,il’ - ,ru’(G - ‘il). Then, for any function cp E H1 (0) n L” ((I), cp > 0, we have Proof. - Write IV +“I* I, 4p (ii” - ii)l’ + 4p (3 - ,??)I” + 4p (ui’fi - *ii)l” + 4lV (w’il - ;,)I’ and then apply (1.30) with ‘4’1’ = ‘p, (3.6) and (2.6) with @ = 1. STEP 4. - For any function cp E H1 (0) n L”(R), p 2 0, we have (3.9) where the constant n is defined in Lemma 1.2 and is increasing with respect to M. TOMt 76 - 1997 - IV’ 5 HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 453 Proof. - Applying (1.19), with 0 = GE, 2 = GE, 6 = li’, F = w’(G - 6) and ; = 6’ to the difference between the two equations of (3.3), (and writing the integrals on the whole of s2) we have - I v (w’(ii - 5)) v S(q) cp - .i S(ij’) v (2 - GE) v cp .R n + CA1 .I[ (1 + IV GE12 + IV C”l*)[&(G - C)I CP + (1 + IVC”l + lVCEl)lV (WE@ - ;,))I] l~s^l”cp. Using in this inequality the property (2.1) of fE and $, and taking into account (P6), (P3), (3.4), (3.5), that 7j” converges almost everywhere to 0 and (3.10) II 7je IIIPqql CM, we have: Inequality (2.12) with $’ = 17jEIVL, cp = Iu - VIP, (3.8) and (3.10) implies Writing IV7fI 5 IV (7f - 7L)I + puI, lVv”I < lV(7f - ‘(I)[ + pul and IV (w”(C - G))I 5 Iw’(lV (ii - $)I + Ii2 - 611VwEI, JOURNAL DE MATHfiMATIQUES PURES ET APPLIQU6ES 454 J. CASADO-DfAZ and taking into account (3.10) and then using (3.4), Cauchy-Schwarz’s inequality and tinally (2.6) with $J” = 1 and cp = Iu - VIP, we obtain: I . (/ IVwy21U - +y + 0, R I: CM I IV wE12)?A - tlcp + 0,. .R Inequalities (3.11), (3.12) and (3.13) now give (3.9). STEP 5. -For any function cp E H1(b2)nL”(fl), cp > 0 and for any fixed Ic 2 1, we have where we have written CM (AY) to remark that the constant depend on k. Proof. - Let Sk(s) = Isllc-‘s. Using Sk(rj’)cp as test function in the difference between the two equations of (3.3), we have I v (ii” - 5’) v Sk(7jd) cp + * IF .I’ Sk@&) v (3 - aEj v cp IF ‘%@i? + ii’, v 2) - B(z, ‘li’, v P))Sk(ij&)(p = (f’, #)QC - (g’, -@$&) which using (1.14) and (2. l), implies .I’ S~(ijyV fjEJ2(P + J p v (WC@ - G))V Sk(jjd) cl R 5 CM I[ (1 + Iv&“l2+ IVli”12)1&& -iYI * Q + (1+ p?Y -I- lv?Yl)p ( 2 - q] ISk(7jh)l~ + 0,. Using in this inequality, (P6) and (3.5), .we have: k IVrj’121rjElk-lp J’ n (3.15) 5 C&f I[ (IVu”12 + IV?J”12)I&E - P .R + (lV?fI + IVv”I)IV (ii’ - OE)l]ljjElkq + 0,. TOME76- 1997-No5 HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 455 To estimate the first term of the right-hand side of (3.15) we use the triangle inequality, (P3) and (2.12) with $J” = l?j’l’+‘. Therefore we have + CM(k) (i,lc-l121”-‘:?)~(l,lv621i-I’:““l(I-i.li))+~; Inequalities (3.10) and (3.8) then give (puE12 + p?q2)liY - ;i"lliyl"(p For what concerns the second term of the right-hand side of (3.15) we use the triangle inequality I’ (lV7q + p?J”I)p(fiE - 7qll7j”l”p .R and we estimate the two terms of the right-hand side. For the first term, we use the triangle inequality, Cauchy-Schwarz’s inequality and (2.6) with 4~” = 1. This gives /w4 + IV~‘l)IV7jeIlrjeIkP = ~lclv( 78 - u)I + IV (WC - II)()prjyijclk(p + 0, ~[(~lv(~~-~l.~)~+(~lv(7~&-7~),2~)’](i;~v~~2,~~~2~-)-+~~~ For the second term we make again the computation that we did in (3.13) with now 7~ replaced by k. We obtain I (puEI + pw’l)p (w&(72 - q)llrjcl”p 5 C,,(k) IV,WE121U, - wlp + 0,. .R J n The estimates we obtained for each term of the right-hand side of (3.15) now give (3.14). STEP 6. - For any function cp E H1(R) n L”(G), cp 2 0, we have: (3.16) J Iv7j”12v 5 CM Cl JOURNAL DE MATHeMATIQLJES PURES ET APPLIQUBES 462 .I. CASADO-DfAZ 5. Construction of the function T We have seen in the previous Section that the function E is of the form E(z) = T(z, U(X)) for some function T. However the function T is only defined for the pairs of the form (2, U(X)) where u is such that there exists f’, f and u&, where fE, f, Q and u satisfy (2.3). We begin this Section by showing that for every s E R there exists a sequence of such functions u (which we denote by s,) which converges to s in Hfo,(R) n L”(s2, dp) where H&,(R) is endowed with its strong topology and L” (0, db) with its weak-* topology. LEMMA 5.1. - Assume that H satis$es (1.2) and (4.3). Consider s E IR. For any n E N, define s”, as the solution of the problem (5.1) - AS: + nsi i- H(z, si, V 3:) = ns in 2S(V), s’, E Hi(W) n L”(W). Then, there exists a subsequence of E (which in order to simplify the notation we still denote by E), two sequences of functions s, and S, and a function S such that (5.2) S, E H;(R) n L”(o)! s, E r(o,dp), s E ryot,dp), (5.3) - As, + S,P + ns, + H(z, s,, V s,) = ns in D’(R), (5.4) for any n E N fixed, si - s, in Hi(R) weak as E -+ O> (5.5) II s’, IIP(Q)l Cl81 and thus II sn lb(~)< Cl+ (5.6) s, -+ s in H&(a) and P(R, c&u) (1 5 p < $00) strong, (5.7) sn - S in L”(R,dp) weak- * and in L*(R,dp) (1 5 p < +oo) strong, (5.8) 11 Sn lb=(n,cip,< Cl.31 and thus II S ll~m(n,ci~)l Cl+ Proof. STEP 1. - Let n E N be fixed. By Theorems 1.1 and 1.2, there exists a unique solution s”, of problem (5.1) such that II si I(H;CnJ,-,L- (o) is bounded by a constant which is independent on E but could depend on n. In fact the L”(Q) norm of s”, is bounded independently of E and n since in view of (1.4), we have: (5.9) II 4 lIP( w0(0) + +I < w0(0) - + Is( = cl+ X+n - X By the diagonal process, we can thus assume that there exists a subsequence of E and a sequence 5, such that s, belongs to Hi(R) n Lm(G) and that (5.4) and (5.5) hold true. By Theorem 2.2, there exists for each n E N a function S, E L”(R, dp), with (5.10) II sn II P(n,dp)i $1; such that s, satisfies (5.3). TOME 76 - 1997 -No 5 HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 463 STEP 2. - For cp E D(n), inequality (1.7) with 0 = R, u = s,, r = s, cp = (p2 and f = n(s - sn) - S,p ( o b serve that S,+ E H-l(R) + L1(R) by (5.3)) yields J IVSn12P2 5 72 J n(s - sJh(s, - s)(p2 - S,h(s, - s)(p2 dp R J R -2 h(s,- J s)'pV 37% v cp + CIsl J Ih(Sn - 4b2, cl R where we used (2.19) with E = S,, x = h(s - s,)(p2. Since h’ > 1 and h(0) = 0 (see (1.6)) we have (sn - s)h(s, - s) 2 Is, - s12 almost everywhere in R, which implies: (5.11) i J IV s,j2p2 + n J jsn - s12p2 < - J S,h(s, - s)p2 dp n R n -2 h(s,- J s)cpV sn v P + CIsl 1% - 41'~~. R J R By (5.5) and (5.10), the two terms Is - Snh(sn - 4'p2 dp , R I II Ih(Sn - s)l'p2 R are bounded independently on n, while for the remaining term, we have II h(sn - s)cpV sn V cp 111 h(sn - s> llL-(rz,II cpvsn lIP(Q) v cp IILyny . R Thus for each cp E D(R) there exists two positive constants a(v), b(cp) such that: II cpvsn Il&q~ +n II (sn - sh’ II;‘(n)< a(p) + @‘) 11 ~vsn llP(n)N This implies that s, is bounded in H&,( 0) and that fi(s, - s) is bounded in L:‘,(R). Thus s, - s converges to zero strongly in L&(R), and weakly in H,1,,(R). By Theorem A6 in [Cl], the weak convergence in Hfo,(R) implies the p-almost everywhere convergence. It is now easy to see that the right-hand side of (5.11) converges to zero. This implies that (5.6) holds true. STEP 3. - By Lemma 4.1, the sequence S, satisfies IS, - S,l 5 Cjslls, - s,( CL-a.e. in R and then, since sn is a Cauchy sequence in L1 (R, dp) we deduce that S, converges strongly to a function S in L1 (0, dp) . Since I I S, I I ~~ (02,dP)i Clsl, this proves (5.7) which completes the proof of Lemma 5.1. n REMARK 5.1. - In order to define the function T, the idea is now to set: T(z,s) = S(x) CL-a.e. z E s2, V’s E W, where for s E R, S = S(z) is the function defined in Lemma 5.1. The problem in this definition is the fact that the subsequence of E given by Theorem 5.1 depends on s. In JOlJRNAL DE MATHeMATIQUES PURES ET APPLIQIJl?ES 464 J. CASADo-DiAZ order to avoid this problem we thus define T(z, s) only when s is a rational number, and then extend the definition to any real number s by a limit argument. This will be carried out in Theorem 5.1. Moreover, we will prove in Section 6 that the subsequence E given in Lemma 5.1 may be choosen independently on s and that the functions S,, and S which appear in Lemma 5.1 satisfy S,, = T(.r:. s,?), S = T(x:. s) jL-a.e. :x’ E c2. But in order to prove this result, we need a uniqueness result for the limit problem which cannot be proved at this stage. THEOREM 5.1. - Assume that H satisjies (1.2) and (4.3). For any q E Q and for any u, E N, define qf by (5.12) { - AyE + ,nf& + H(:r, q,E,. C &) = nq in O(V) fg E Hi(W) n L”(W). Then there exist u subsequence of c’ which does not depend neither on q nor on U, (and which to simplify the notation we still denote by E), two sequences of functions q7, and C),,, nnd a function Q such that (5.2) ,..., (5.8) hold true with s = (I, s,, = q,,, S,, = QT1, S = Q. De&e the function T : It x Q H R b? (5.13) T(:r. q) = Q(Y) p-a.e. .I: in (2. Vq E (2. If s E 88 and if qk is a sequence in Q which converges to s, then the sequence T(x, qk) converges in Lp(R, dp) strong (1 < p < +30) and in L”(i2. ~,LL) weak-*. Dejine now 7’ : 52 + R by (5.14) T(:r;, s) =hJg;T(:~, qk) in U’(S2, dir) strong (l<p<+x) and in L”(i2, dp) weak-* The ,function T : (2 x I&’ H R’ dejined in this way, satisjies: (5.15) T(., s) E L”(b2,dp). Vs E R. with 11 T(., ,s) IILX,(f2.C~l,)< C(l.sl) (5.16) IT(:c, sl) - T(:r:> +)I < C(s)ls, - s#. s = IIIHX{ /sl I. 1cs21}, where C : [O, +oo) H [O. +CO) and X : [l. +3c) H [O, +CO) are increasing ,functions. Proof: STEP 1. - Since Q is a countable set, by Lemma 5.1 and the diagonal process we can extract a subsequence E such that for any Q E Q the results of Lemma 5.1 hold true. This proves the first part of Theorem 5.1. STEP 2. - Define the function T : (2 x Q H R by (5.13). By (5.8), the function T satisfies (5.17) ~T(:I:. q)l < Cini Ic-a.e. .I’ in 62, Vq E Q. Consider q and y’ in Q. By the definitions of Q1?, Q:, and Lemma 4.1, we have where kl = rnax{ 141, /q’l} and then, passing to the limit in n, we obtain: (5.18) IT(:r:, q) - T(:c, q’)I < C,ll(q - ,‘I* kr-a.e. :I: in 62. where M = max{lql, 14’1). Th’ IS uniform continuity of the mapping q E Q H T( . . y) E Ll(R,dp) and (5.17) allows one to define T(.. s) for any s E R by (5.14). By (5.17) and (5.18), the function T satisfies (5.15) and (5.16). H T"M~ 76 - 1997 -No 5 HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 465 6. The homogenization result and a property of the function T 6.1. A first homogenization result In this subsection we prove that the function T : $2 x W H R defined in Theorem 5.1 is such that the function E defined by (2.15) may be expressed in the form E = T(z, u). Indeed, we will prove: THEOREM 6.1. - Assume that H satisjes (1.2) and (4.3). Then for the subsequence & and the function T defined in Theorem 5.1 we have the following homogenization result: If f”, f, ~~~ and IL satisfy (2.3), the function 11. satisjes (6.1) C - A IL + T(z, u)p + H( IC, u, V 7~) = f in D’(O), IL E H;(O) n L”(R), or equivalently Proof. - Consider fE, f, 1~~ and u which satisfy (2.3) and let E be defined by (2.15). By applying (4.4) to the problems (2.2) and (5.12) with f’ = f’ and gE = ny - nq:, we have for any q E Q and any 71, E N JE - QnI < G,I~I~ - qT1/* p-a.e. in R, M = max{sup{ll uE IILm(n)}, Iql} and then by passing to the limit in 71, we obtain that for any q E Q IE - T(:I:: q)l < CA~/U - qI* b-a.e. z E 0, M = max{sup{l( ?L’ JIL=(o)}, lql}. If now s belongs to R, taking a sequence of rational numbers qi which converges to s and using the continuity (5.16) of T, we get (6.3) (E-T(:c:s)) 5 C~~~~YL--S~~, /A-a.e. n; E 0; M = max{sup{ll 7f (ILOO(n IsI}. Considering the points IC where u(x) and E(z) are defined by their representatives and then taking s = U,(X), inequality (6.3) implies that REMARK 6.1. - Using Theorem 1.1, we can prove that there exists a subsequence (still denoted by E) of the subsequence extracted in Theorem 5.1, such that the corresponding subsequence of ?L”, solution of (0. l), converges weakly in Hi (R) to a function u which is a solution of (6.1). Once uniqueness will be proved for problem (6.1), we will deduce that for the subsequence E extracted in Theorem 5.1, the whole subsequence 2, solution of (O.l), converges to ?I,, without extracting another subsequence. JOURNAL DE MATHBMATIQUES PURES ET APPLlQU6ES 466 J. CASADO-DiAZ 6.2. The uniqueness of the limit problem We will prove in this Subsection the uniqueness of the solution of the limit problem (6.1). LEMMA 6.1. - Assume that H satisjes (1.2) and (4.3) and consider the function T dejned in Theorem 5.1. For any M > 0, there exist two constants A and K which only depend on M and are increasing in M such that ,for the functions 8 and $ dejined by (1.1 I) and (1.12) we have (6.4) T(x, s) for any s and t in R such that rnax{lsl, ItI} <_ M. Since 4 = tip1 and 19 are increasing functions, the result of Lemma 6.1 states that the function s -+ $$$ (or equivalently w) is increasing. Proof. STEP 1. - We will first prove the following result: Consider f”, f, *(I,‘, u and gE, !J, ?f, II which satisfy (2.3). Then, there exist two constants A and K (which are increasing with respect to M) such that for the functions 41 and 8 = ii/-l defined by (1.11) and (1.12) the functions u and 11 satisfy (6.5) T(z, u) T(x, v) @(6u)) - Yww) > (6(u) - d(v)) > 0 p-a.e. z E 0. Proof. - By Lemma 1.2 applied to equation (2.2), with 0 = RE, ‘u = U’ and f = f’ (respectively u = uE and f = ,9’) there exist two constants A and K which are increasing with respect to M, such that the functions GLE = 19(u’) and 6’ = In satisfy f” - AC + B(z,,$, v 6”) = - I)‘( ii&) in D’(V), (6.6) -A? + B(z,G”JW) = in D’(V), where the function B satisfies properties (1.14) and (1.15). Estimate (1.18) applied to these equations with 0 = a’, 6 = ?, G = P, .f^ = p and jj = jje, implies that for GE = YY - 6” and for any function cp E D(R), ‘p 2 0, we have (6.7) x,ii’,Vii”) - B(z,iY,V?))S(3) + ;S’(lj’)/GLjc/2 1 cp S(Z) ’ (SE> dl,(,,)hF - J R s(;7 VLf v P, where the integrand of the second term is nonnegative. By Theorem 2.1, this integrand converges almost everywhere to (B(z,Wii) - B(z.,C,VG))S(lj) + ;S(;)lVSi cp, 1 TOME 76 - 1997 - No 5 HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 467 with ?I = 6(u), fi = G(w) and 2 = & - 6. Therefore Fatou’s lemma permit us to pass to the limit in (6.7) and to obtain On the other hand, by Theorem 6.1 the functions u and w satisfy the equations: 1 -Au+T(x,u)P++( x, u, VU) = f in D’(R), -Av+T(z,+++( x, 21, VU) = g in D’(n), where f - T(z, w) and g - T(z, w)~ belong to H-l(R) + L1(R). Applying Lemma 1.2 to these two equations with the same functions 6 and II, as above (since 11 u [IL-(o)< M and II 71 IIL-(o~~ M) and 0 = 0 implies that & and 6 satisfy: I* + B(x,C, Vii) = - (6.9) -AG+# - A$ + T;‘;) ’ in D’(R”) Ijl’(G) ’ Ap + B(x, 6, v a) = --E- ‘6 Tw) in D’(W). Taking S(G)(p with cp E D(n), cp > 0 as test function in the difference of the two equations of (6.9) and applying (2.19) with E = $#h - $$#b and z = S(Lj)(p, we get J s'(LqvLJ~2qi + J S(lj)VLY7cp+ .I( T(x,u) T(x, w) ~ - ~ R R n 1/lvL) $J'(C) > S(+P dP + JB( J x,6, v 2) - B(z, 6, v q]s(qp = (f, ~~)~ - (9, ~~)ci. Comparison with (6.8) implies that Since the sign of S(G) coincides with the sign of &, we have proved that (6.6) holds true. STEP 2. - Let 4 and q’ be rational numbers with max{ (ql, lq’l} 5 AI. From (6.5) applied to the sequences uE = q; and uE = (q’)E, defined by (5.12), we deduce that there exists two constants A and K which are increasing with respect to A4 such that for the functions $ and 19 defined by (1.11) and (1.12) we have T(x, qn) T(x> qb) !b'w7n)) - 'WJMJ) > (t!J(q,) - S(qL)) > 0 p-a.e. in 0, Vn E N, where the functions qn and qk are defined by Theorem 5.1. Taking in this expression the limit in n we deduce that T(x, d T(x’q’) 4’Md) - !wJW) > (79(q) - d(q’)) > 0 p-a.e. in (1. JOURNAL DE MATHBMATJQUES PURES ET APPLIQUBES 468 .I. CASADO-DiAZ The continuity (5.16) of T then implies (6.4). n We are now in position to prove a maximum principle. THEOREM 6.2.- Let I? : $1 x R x R” H R be a Carathkodory function (note that H can be different of H) which sati@es hypotheses similar to (1.2) and (1.91, i.e.: i) For almost every z E R the .function I?(jc, . . .) is continuously derivable and there exists a constant ?I > 0, such that .for almost eve? :c E 61 we have (6.10) ii) There exists an increasing,function iv : [O, +x) I-+ [O, +w) such that (W(.,O,O) E Lrn(0), Assume that H satisjies (1. 5 4l.W + K12). a.e. :I: E it, Y(s,[) E 63 x W“ 2 W4)(l + Ill). a.e. :r E 0, V (s: <) E R X R”. 2 ) and (4.3) and let T be the function de$ned in Theorem 5. I. Consider u and II in H’(n) n L”(R) such that there exist f and y in H-l (0) + L1(0) which satisfy (6.12) - au + T(:e; u)p, + H(, S, U, Vu) = f 5 0 in D’(Q), - Au + T(:c, ?I)/” + H( :I:. 11. V 71) = g 2 0 in D’(Q). Then inequality u 5 II in X2 (i.e. (U - $11)~ E Hi(Q)) implies that u < II almost everywhere in R. In particular for f E H-l (62) + L1 ((I), the problem (6.13) ~ - Au + T(:c, U)/L + jjT( :c, u, Vu) = f in D’(O) TL E Hi(Q) n L”(O). has at most one solution. Pro05 - By Lemma 1.2 with 0 = 12 and H = l?, there exist two constants A and K such that for the functions $ and ?Y = $-l defined by (1.11) and (1.12), the functions G = 19((u) and ,i, = S(U) respectively satisfy (recall that by (6.12) T(x, ,u)/J and T(.J:.v)~ belong to H-l (0) + L1(R) and observe that $J’ > 0) TOME76- 1997-N” 5 HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 469 where the function B satisfies properties analogous to (1.14) and (1.15). Moreover, by (6.4) we have (6.15) where 6 = d(u) and li = 8(u). Define ij = 6, - fi and apply estimate (1.20) to the two equations of (6.14). By (6.15) we have 1 . T(x, u) T(x, w) i i s’(;+)pij+(2 5 - ~ - ~ .R s( R uil’@) lilw > S(G+)dp 5 0 h-a.e. in 12 and thus ij+ = 0 almost everywhere in (2, i.e. 6 < 6 and thus u < II almost everywhere in R. H 6.3. The homogenization result As a consequence of Theorem 6.2, we can now prove the following homogenization Theorem. THEOREM. - Assume that H satisfies (1.2) and (4.3) and consider the subsequence E and the function T dejined in Theorem 5.1. Let f : R x R x RN H R’ be a Carathkodory function such that: i) There exist two increasing functions To, Yl : [O, +oo) ++ [0, +co) and there exists Q E [0,2) such that (6.16) ii) For every x E R the function f(x, ., .) is continuously derivable and there exists an increasing function < : [0, +m) +-+ [0, +oc) such that (6.17) iii) There exists o > 0 such that for almost every x E R (6.18) A-g(x,s,<)>u, a.e.inR,V(s,<)ERxRN, where X is dejined by (1.2). Then the unique solution uE of the problem: (6.19) - Ad + H(z, uE, V tf) = f(x, 18, V d), in D’(V), uE E H;(V) n L”(W), lOlJRNAL DE MATHBMATIQUES PURES ET APPLlQlJtES 470 J. CASADO-DLAZ converges weakly in H,1 (O), strongly in W,‘,p(0) (1 5 p < 2) and weakly-* in L”( $1) to the unique solution u of the problem (6.20) - Au + T(z,u)p + H(n:, U, Vu) = f(x, II,, Vu) in ‘D’(Q) u E H;(U) n L”(0). REMARK 6.1. - Taking f( 2, s, <) = f(z) E L”(R) and taking into account Remark 6.3 below, we inmediately deduce Theorem 0.1 from Theorem 6.3. REMARK 6.2. - As announced in Remark 5.1, Theorem 6.3 implies that the subsequence E which appears in the statement of Lemma 5.1 may be chosen as the subsequence E given in Theorem 5.1, and thus independently of U. Moreover Theorem 6.3 implies that the functions S, and S defined in Lemma 5.1 satisfy S, = T(z, So,), S = T(z. s), IL-a.e. :c E 0. Proof of Theorem 6.3. - Theorems 1.1 and 1.2 applied to 0 = R’ and H = H - f, imply that there exists a unique solution uE of (6.19) and that ‘1~~ is bounded in Hi (0) n L”(R). Therefore there exists a subsequence E’ such that u”’ converges weakly in H,1 (Q) and weakly-* in L” (0) to a function U. By Theorem 2.1, applied to uE = u”, H = H - f and f” = f”’ = 0, we also have that u”’ converges strongly in W1,p(0) (1 5 p < 2) to ‘~1. Inequality (6.16) and Lebesgue’s dominated convergence theorem imply thus that f( IC, &, V Q’) converges strongly in L1(R) (and thus in the sense of (2.1)), to f(z, U, Vu). By Theorem 6.1 we then have that u is a solution of (6.20). Theorem 6.2 applied to g = H - f implies the uniqueness of u and therefore the convergence for the whole sequence. n REMARK 6.3. - As a consequence of the results of the present Section, we also could prove the following monotonicity property of the function T(z, .). (6.21) T(z,O) = 0. /L-a.e. :I: E 0. (T(z, sl) - T(:r, s2))(s1 - s2) > 0, /L-a.e. :1: E R, ‘dsi, s2 E Iw. Actually this monotonicity property is not very important (except for esthetic reasons). What is important for uniqueness is property (6.4), i.e. that w is increasing with respect to s, and thus we do not give the proof of (6.21). 7. Corrector In this Section, we use Lemma 3.1 and Lemma 5.1 to give an approximation of V uE in the strong topology of L2 (i2)N. DEFINITION 7.1. - Define P,” : 0 x Iw --+ RN by Pi(z, s) = V s:(z), wherefor any s E R and n E N, SE is de$ned as the unique solution of problem (5.1). THEOREM 7.1. - Assume that H satisfies (1.2) and (4.3) and let E be the subsequence dejned in Theorem 5.1. Consider f E, f, ‘u,” and u which satisfy (2.3). Then, for any step TOME 16 - 1997 - No 5 HOMOGENIZATION OF GENERAL QUASI-LINEAR DIRICHLET PROBLEMS 471 function y(x) = x2”=, six&,(x) with si E R and Qi closed subsets of RN with Q; c i2, which satisfy p(Qi n Qj) = 0 for i # j, we have (7.1) Q = ij Qi and A4 = ma&v{ll uE IIP(12)~. II !I IIL”(R)I i=l and where the constant Cfif does not depend on Q, REMARK 7.1. - The meaning of Theorem 7.1 is the following: If we could take y = II, in (7.1) we would obtain limsuplirnsup [Vu” - Vu - P~(z,u)~~ = 0 TI-CXI i -Q which says that V u + Pi(z, U) is a good approximation of V U” in L2(Q)N strongly. However this choice is not possible, since P,” is not a Caratheodory function in general and since therefore PE(x, U(X)) has not reason to be measurable. This is why we approximate 11, by the step function y. REMARK 7.2. - In the statement of Theorem 7.1, the value of the function y on the set Qi n Qj, i # j, does play any role since j~(Qi n Qj) = 0 by hypothesis. Indeed estimate (7.2) applied to Q = Qi n Qj and s = s1 and s2, together with the triangle inequality, shows that REMARK 7.3. - Consider a closed set Q such that /L(Q) = 0. Applying estimate (7.1) to y = 0, we obtain lim li;:;p (V~L’ - Vu - PJj(z, O)(” = 0. 1?‘03 J Q On the other hand by taking TL~ = ‘u, = 0 and y = 0 in (7.1), which is possible since If = 0 satisfies - A tf + H(z, d, VU’) = H(z, 0: 0) d E H,(V) n L”(F), we have lim li’f;;p IP~(x,O)~~ = 0. 72-CC J Q JOURNAL DE MATHCMATIQUES PURES ET APPLlQUkES