The two-scale convergence method applied to generalized Besicovitch spaces

10.1098/rspa.2002.1003 The two-scale convergence method applied to generalized Besicovitch spaces By J u an C a sa d o-D¶i a z a n d Inmacul ad a Ga y t e Departamento de Ecuaciones Diferenciales y An¶alisis Num¶erico, Facultad de Matem¶aticas, C/ Tar¯a s/n, Sevilla, CP 41012, Espa~na ([email protected]; [email protected]) Received 10 August 2001; revised 19 March 2002; accepted 16 April 2002; published online 4 October 2002 The two-scale convergence method has proved to be a very useful tool for dealing with periodic homogenization problems. In the present paper we develop this theory to generalized Besicovitch spaces, which include the almost-periodic functions. The main di¯ culty comes from the fact that these spaces are not separable. We also show how to apply these results to the homogenization of partial di¬erential problems in this framework. Keywords: partial di®erential equations; homogenization; two-scale convergence 1. Introduction It is usual in homogenization theory to deal with composite periodic materials and structures with very small periods. In order to study their physical behaviour (electrical or thermal conductivity, elastic behaviour, etc.), we need to solve a partial di¬erential equation that, in a model case, can be written in the form ¡div Aµx "¶ru"=f; (1.1) besides some boundary conditions. Here the matrix Ais periodic and "is a small parameter. From the numerical point of view, it is very di¯ cult to calculate u"from this problem. We need to use a discretization of size smaller than ", and therefore solving a very large system of equations, which requires a lot of computer memory, is time-consuming and involves several stability problems. Homogenization theory seeks to obtain an approximation of u"through the resolution of simpler partial di¬erential equations. The theory of asymptotic expansions (see Bensoussan et al. 1978; S´anchez Palencia 1980) provides us with u"(x)¹u0(x) + "u1µx; x "¶+"2u2µx; x "¶+¢¢¢ ;(1.2) where the functions uiare obtained as the solutions of partial di¬erential problems much easier to solve than (1.1). A rigorous way of obtaining this expansion and showing its convergence is the two-scale convergence method of Nguetseng and Allaire (see Allaire 1992; Arbogast et al. 1990; Nguetseng 1990). It has proved to be very useful in the homogenization of periodic problems. Proc. R. Soc. Lond. A (2002) 458, 2925{2946 2925 c°2002 The Royal Society on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from 2926 J. Casado-D¶³ az and I. Gayte We note, however, that although periodic materials are common in engineering applications, they do not usually occur in nature. However, we can see a recurrence in the structures which suggests that a better approximation is to consider these materials as almost-periodic. The aim of the present paper is to extend the two-scale convergence method to the case of almost-periodic (or more general) coe¯ cients, in particular, the sum of periodic functions with di¬erent periods, in order to be able to treat more general composite materials than the periodic ones. Similarly to the periodic case, it is necessary to have a characterization of the limit of the expression Z« v"(x)Áµx; x "¶dx(1.3) when v"is a bounded sequence in a Lebesgue space Lp(«), p > 1, and Áis an almost-periodic smooth function or, in a wider sense, in a generalized Besicovitch space (see Casado D´± az & Gayte 2002; Jikov et al. 1994; Zhikov & Krivenko 1983) in its second variable. To prove the corresponding result, the  rst step is to show the existence of a subsequence of v", still denoted by v", such that there exists the limit of (1.3), for every Áas above. This is easy, using a diagonal argument, if the space of functions Áis separable. However, the generalized Besicovitch spaces are not separable in general. This is the main di¯ culty in obtaining our result. To solve this problem, we propose an abstract theorem generalizing the well-known result about the weak sequential compactness of the unit ball in a re®exive space. In a simpler situation this was carried out in Casado D´± az & Gayte (1996). The case where the almost-periodic functions are in a separable space has been considered by Nguetseng (2000). As an example of how our results can be used in the study of the asymptotic behaviour of composite materials, we study the nonlinear problem ¡div aµx "; u";ru"¶=fin « ; u"= 0 on @« ; where ais a Carath´eodory function which de nes a pseudomonotone operator of order pand belongs to a Besicovitch space in its second variable. In this case we obtain the limit equation and a corrector result related to (1.2). To complete this introduction, we mention that an adaptation of the two-scale convergence to stochastic homogenization problems has been given in Bourgeat et al. (1994) (assuming separability). The notion of a stochastic weak derivative given in this article is strongly related to the mean derivative we use in the present paper. To the study of homogenization problems in a stochastic frame, we also refer to Dal Maso & Modica (1986) and Abddaimi et al. (1997). 2. A compactness theorem It is well known that, for a bounded sequence ffngin the dual space X0of a re®exive space X, there exists a subsequence of ffngwhich converges weakly-¤to some f2X0, i.e. ffngpointwise converges to f. The purpose of the present section is to generalize this result, by showing that it is necessary to assume neither fncontinuous nor X complete. Proc. R. Soc. Lond. A (2002) on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from Two-scale method applied to Besicovitch spaces 2927 Theorem 2.1. Let Xbe a subspace (not necessarily closed) of a re°exive space Y and let fn:X7! Rbe a sequence of linear functionals (not necessarily continuous). Assume there exists a constant C > 0which satis¯es lim sup n fn(x)6Ckxk;8x2X: (2.1) There then exist a subsequence fnkgof fngand a functional f2Y0such that 9lim kfnk(x) = hf; xi;8x2X: (2.2) Remark 2.2. If Xis complete and fnis continuous, theorem 2.1 easily follows from the Banach{Steinhauss theorem and the weak-¤sequential compactness of the unit ball in a re®exive space. It is also clear that theorem 2.1 holds if we replace the hypothesis Xincluded in a re®exive space by Xseparable. The aim of theorem 2.1 is precisely the application to spaces that are not separable. Remark 2.3. Theorem 2.1 has been established in Casado D´± az & Gayte (1996) when Yis a Hilbert space. In order to prove theorem 2.1, we need to recall some results about smooth norms (see Cioranescu 1990). De¯nition 2.4. Let Ybe a Banach space. The norm in Yis called smooth if for every y2Ywith kyk= 1 there exists a unique f2Y0such that kfk= 1 and hf; yi= 1. The following theorem is due to Asplund and Lindenstrauss (see Cioranescu 1990; Lindenstrauss 1966). Theorem 2.5. Every re°exive Banach space has an equivalent smooth norm. Proof of theorem 2.1. By theorem 2.5, it is not restrictive to assume that the norm in Yis smooth. First step. Let us prove that there exist a subsequence fnkgof fng, a constant · C>0 and a sequence fzjg » Xsuch that kzjk= 1;(2.3) lim sup k fnk(x)6· Ckxk;8x2X; (2.4) 9lim kfnk(zj)>· C¡1 j;8j2N:(2.5) To this end, we de ne C1= supnlim sup n fn(x) : x2X; kxk= 1o: This supremum is  nite because of (2.1). By de nition of C1there exist z12Xwith kz1k= 1 and a subsequence fn1(k)gk of fngsuch that 9lim kfn1(k)(z1)>C1¡1: Proc. R. Soc. Lond. A (2002) on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from 2928 J. Casado-D¶³ az and I. Gayte Then we de ne C2by C2= supnlim sup k fn1(k)(x) : x2X; kxk= 1o: Obviously, C26C1and there exist z22Xwith kz2k= 1 and a subsequence fn2(k)gkof fn1(k)gksuch that 9lim kfn2(k)(z2)>C2¡1 2: Repeating this reasoning, we deduce that for every j2Nthere exist Cj2R,zj2X and fnj(k)gksuch that, denoting n0(k) = kfor every k2N, we have kzjk= 1;(2.6) Cj= supnlim sup k fnj¡1(k)(x) : x2X; kxk= 1o;(2.7) fnj(k)gkis a subsequence of fnj¡1(k)gk;(2.8) 9lim kfnj(k)(zj)>Cj¡1 j;(2.9) 06Cj+ 1 6Cj:(2.10) Taking the diagonal subsequence fnk(k)g, which we denote by fnkg, we have lim sup k fnk(x)6Cjkxk;8x2X; 8j2N; 9lim kfnk(zj)>Cj¡1 j;8j2N: Therefore, for · C= limjCjstatements (2.4) and (2.5) hold. Second step. Let us now prove that there exists f2Y0such that fnkgand fsatisfy (2.2). Note that we can suppose · C > 0 because if not, by (2.4) we immediately get (2.2) with f= 0. Since Yis re®exive and fzjgis bounded, there exist a subsequence, still denoted by fzjg, and z02Ysuch that zj* z0in Y: (2.11) By (2.6), z0satis es kz0k61:(2.12) Let x2Xbe arbitrary. Since ffnk(x)gkis bounded, there exists a subsequence fnk(j)gjof fnkgk, depending on x, such that 9lim jfnk(j)(x):(2.13) Denoting S= span(fxgSfzn:n2Ng), statements (2.13), (2.5) and fnk(j)linear imply 9lim jfnk(j)(s);8s2S: We de ne f:S7! Rby f(s) = lim jfnk(j)(s);8s2S: Proc. R. Soc. Lond. A (2002) on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from Two-scale method applied to Besicovitch spaces 2929 By (2.4) and because nk(j)is a subsequence of nkfor every s2S»X, we have f(s) = lim jfnk(j)(s)6lim sup k fnk(s)6· CkskX; and so fbelongs to S0and satis es kfkS06· C. On the other hand, by (2.5) and (2.6), for every j2Nwe have f(zj)>· C¡1 j=µ· C¡1 j¶kzjk:(2.14) Thus kfk>· C¡1=j for every j2Nand so kfkS0=· C: (2.15) By the Hahn{Banach theorem, we can extend fto a functional of Y0, still denoted by f, which satis es kfkY0=· C. By (2.14), (2.15), (2.11) and (2.6), we deduce · C= lim jf(zj) = hf; z0i:(2.16) Using (2.12) and (2.15), we then have hf; z0i=· C; kfk=· C; kz0k= 1:(2.17) Since Yis smooth there exists a unique element f2Y0satisfying (2.17), so in (2.13) it is not necessary to take a subsequence, and the whole of the sequence ffnkg pointwise converges to fin X.¥ Following the idea of theorem 2.1, we can also prove the following theorem, which generalizes theorem 2.1 and contains the case where Xis separable. The proof can be found in Gayte Delgado (1998). Theorem 2.6. Let Xbe a normed space (not necessarily complete) such that the unit sphere of X0endowed with the weak-¤topology is ¯rst countable. Let fn:X7! Rbe a sequence of linear functionals (not necessarily continuous) satisfying (2.1). There then exist a subsequence fnkgof fngand a functional f2X0such that (2.2) holds. Remark 2.7. It can be proved that the unit sphere of X0endowed with the weak-¤ topology is  rst countable if and only if for every f2X0,kfk= 1, there exists a sequence fzng » Xsuch that if g2X0satis es kgk= 1 and hg; zni=hf; zni, for every n2N. Then g=f. 3. Preliminaries on generalized Besicovitch spaces In this section we recall some results on generalized Besicovitch spaces we will need later. They have been proved in Casado D´± az & Gayte (2002) (see also Gayte Delgado (1998); Jikov et al. (1994)). We recall the de nition of the mean value. De¯nition 3.1. We say that a function f:RN!Rhas a mean value if there exists a real number Mffgsuch that for every bounded measurable set K»RN with jKj>0 we have Mffg= lim T!+1 1 jT KjZT K f(y) dy: (3.1) In this case, we say that Mffgis the mean value of f. Proc. R. Soc. Lond. A (2002) on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from 2930 J. Casado-D¶³ az and I. Gayte Following Jikov et al. (1994, 7.5, p. 242), we now give the de nition of an algebra with mean value. De¯nition 3.2. A linear space, X, of real-valued functions de ned in RNis a Banach algebra with mean value if the following conditions are satis ed. (i) The elements of Xare bounded, uniformly continuous and possess a mean value (see (3.1)). (ii) The constant functions belong to X. (iii) Xis an algebra. (iv) Xendowed with the uniform convergence topology is complete. (v) For every f2Xand s2R, the function f(¢+s) belongs to X. Remark 3.3. As examples of X, we have the space of continuous (0;1)N-periodic functions and the space of uniformly almost-periodic functions. The above de nition allows us to de ne the generalized Besicovitch spaces in the following way. De¯nition 3.4. We de ne the generalized Besicovitch space of order p(relative to X), with 1 6p < +1, and we denote it by Bp, as the closure of Xfor the seminorm [f]p=µlim sup T!+1 1 jBTjZBT jf(x)jpdx¶1=p ;(3.2) i.e. Bp=ff:RN!Rmeasurable: 8" > 0;9’2Xwith [f¡’]p< "g: The generalized Besicovitch space of order 1(relative to X), B1, is de ned by B1=nf2B1such that [f]1= sup p>1 [f]p<+1o: The spaces Bpare seminormed spaces. The quotient of Bpwith the kernel of [¢]pis denoted by Bpand it is a normed space. Remark 3.5. When Xis the space of continuous (0;1)N-periodic functions, Bpis the space of functions in Lp loc(RN) which are (0;1)N-periodic. The space of almostperiodic functions in the sense of Besicovitch (see, for example, Besicovitch 1954; Bohr 1951) is obtained by taking Xas the space of uniformly almost-periodic functions. The following theorem shows that the spaces Bpare analogous to the spaces Lpfor a probability measure (see Casado D´± az & Gayte (2002) and Gayte Delgado (1998) for the proof). Theorem 3.6. The spaces Bpsatisfy the following properties. (i) For 16p6+1,Bpand then Bpare complete. (ii) For every f2Bp,16p < +1, there exist Mffgand Mfjfjpg. Besides, [f]p=Mfjfjpg1=p. Proc. R. Soc. Lond. A (2002) on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from Two-scale method applied to Besicovitch spaces 2931 (iii) For f2B1and ¬= [f]1, the function T¬(f)2L1(RN)satis¯es kT¬kL1( R N)= [f]1;[T¬(f)¡f]1= 0; where T¬is de¯ned by T¬(s) = 8 > < > : ¬if s > ¬ ; sif jsj6¬ ; ¡¬if s < ¡¬ : (iv) If p < q, then Bq»Bpand [¢]p6[¢]q. Moreover, if f; g 2Bqsatisfy [f¡g]p= 0, then [f¡g]q= 0, and we can also then see that Bqis a subspace of Bp. (v) The dual space of Bp, for 16p < +1, can be identi¯ed with Bp0through the following isometric isomorphism: F:Bp0!(Bp)0 hF(f); gi=Mf~ f~gg 8 ~ f2f2 Bp0;8~g2g2 Bp;if 1<p<+1; and hF(f); gi=MfT[f]1(~ f)~gg 8 ~ f2f2 B1;8~g2g2 B1;if p= 1: To  nish this section, we recall some results related to the derivation theory for generalized Besicovitch spaces (see Casado D´± az & Gayte 2002). We start by introducing the space D1, which plays in the spaces Bpthe same role as the spaces C10(RN) in the distributional theory. De¯nition 3.7. We de ne D1as D1=f’2C1(RN) : D¬’2B1\L1(RN)8¬2(N[ f0g)Ng: Reasoning by convolution (see Casado D´± az & Gayte 2002), we can show the following proposition. Proposition 3.8. The space D1is dense in Bpfor 16p < +1. For every f2B1, there exists a sequence ffngin D1which converges to fin B1and is bounded in B1. Analogously to distributional theory, we use the spaces D1to give a de nition of the derivative in Bp. De¯nition 3.9. For f2B1, we de ne the mean partial iderivative of f, 1 6i6 N, and we denote it by @i;mf, as the linear application of D1in Rgiven by @i;mf(’) = ¡M½f@’ @xi¾;8’2D1:(3.3) We also de ne the mean gradient of f2B1,rmf, as rmf= (@1;mf; : : : ; @N;mf) and the mean divergence of F2(B1)N, divmF, as divmF=PN i= 1 @i;mFi:Clearly, these de nitions can also be extended to B1. The following result, which relates the distributional derivative with the mean derivative, is shown in Casado D´± az & Gayte (2002). Proc. R. Soc. Lond. A (2002) on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from 2932 J. Casado-D¶³ az and I. Gayte Proposition 3.10. If f2B1is such that there exists i2 f1; : : : ; Ngwith @f=@xi2B1, then @i;mf=@f=@xiin the following sense: h@i;mf; ’i=M½@f @xi ’¾;8’2D1: The following space plays a very important role in applications. De¯nition 3.11. For 1 6p < +1we de ne Wp=ff2W1;p loc (RN) : rf2(Bp)N; Mfrfg= 0g and rWp=frf:f2Wpg: Identifying an element of rWpwith its class in (Bp)N,rWpwill be considered as a subspace of (Bp)N. Moreover, we identify w1; w22Wpif [r(w1¡w2)]p= 0, and then we can consider Wpas a normed space for the norm kwk= [rw]p. The following theorem gives some interesting properties of Wp(see Casado D´± az & Gayte (2002) and Gayte Delgado (1998) for the proof). Theorem 3.12. The subspace rWpis closed in (Bp)N(and then Wpis Banach). If f2Bp,16p < +1, is such that rmfbelongs to (Bp)Nthen there exists g2Wp such that rmf=rgin (Bp)N. To obtain further properties of Wpthe algebra must satisfy another property. De¯nition 3.13. An algebra Xis called ergodic if for every f2B1such that [f¡f(¢+s)]1= 0 for every s2RN(equivalently rmf= 0), we have [f¡Mffg]1= 0. Proposition 3.14. An algebra is ergodic if and only if lim R!+1·1 jBRjZBR f(x+y) dy¡Mffg¸p = 0;8f2Bp;16p < +1:(3.4) In the ergodic algebras we have the following density result. Theorem 3.15. If the algebra is ergodic, then rD1is dense in rWp. 4. The two-scale convergence method In this section, we present the extension of the two-scale convergence theory (see Allaire 1992; Nguetseng 1990) to the generalized Besicovitch spaces Bprelative to an algebra with mean value X. We start by giving the de nition of two-scale convergence. De¯nition 4.1. Let «»RNbe open. We say that a sequence fu"g » L1 loc(«) two-scale converges to u2L1 loc(«;B1) if for every g2B1\L1(RN) and every Proc. R. Soc. Lond. A (2002) on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from Two-scale method applied to Besicovitch spaces 2933 E»» «bounded, measurable, we have 9lim "!0Z« u"(x)Áµx; x "¶dx=Z« Myfu(x; y)Á(x; y)gdx; (4.1) where Á(x; y) = g(y)XE(x) for x2«,y2RN. We will denote u" 2e * u. Remark 4.2. The left-hand side of (4.1) makes no sense if gis only in B1since two representatives of gmay di¬er in every point of RN. The right-hand side does not depend on the representative of uchosen. Remark 4.3. Since B1can be identi ed with the dual of B1, it is easy to deduce that the two-scale limit, if that exists, is unique. Remark 4.4. Our de nition of two-scale convergence can seem di¬erent to the usual one for the periodic case, which establishes that (4.1) holds for every Áperiodic in the second variable and smooth enough (in general, in the space of admissible functions (see Allaire 1992)). As established in proposition 4.6, this is equivalent to our de nition, because if (4.1) holds for Áas in de nition 4.1, then it holds for Áin all of the spaces which appear with the usual de nition. We have chosen the de nition given above because it makes it easier to check if a sequence two-scale converges, and when u"is bounded in Lp(«) for some p2(1;+1) (usual situation), it does not depend on p. Although we have de ned the two-scale convergence merely for a sequence fu"gin L1 loc(«), in the applications we will usually have a bounded sequence in Lp(«) for some p2[1;+1]. In this case, we have the following result. Proposition 4.5. Let fu"gbe a bounded sequence in Lp(«)for some p2 [1;+1], which two-scale converges to a function u2L1 loc(«;B1). Then ubelongs to Lp(«;Bp), the sequence fu"gconverges weakly in Lp(«) (weakly-¤if p= +1) to u0=Myfu(¢; y)gand we have lim inf "!0ku"kLp(«)>kukLp(«;Bp)>ku0kLp(«):(4.2) Proof . We denote by Stc(«;Bp0) the set of simple functions which have the support strictly included in «. For Á2Stc(«;Bp0), jÁj6kÁkL1(«;B1)a.e. in «£RN if p= 1, we have ¯¯¯¯Z« u"(x)Áµx; x "¶dx¯¯¯¯ 6ku"kLp(«)° ° ° ° Áµx; x "¶° ° ° °Lp0(«) ; where passing to the limit when "tends to zero, we deduce ¯¯¯¯Z« Myfu(x; y)Á(x; y)gdx¯¯¯¯ 6lim inf "!0ku"kLp(«)kÁkLp0(«;Bp0)8Á2Stc(«;B1): (4.3) From this inequality and theorem 3.6 we easily deduce that ubelongs to Lp(«;Bp) and that (4.2) holds. ¥ The following result, which is easy to prove, extends de nition 4.1 to a wide class of admissible function Á. Proc. R. Soc. Lond. A (2002) on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from 2940 J. Casado-D¶³ az and I. Gayte By (4.18), we have u"¡1 jB"rjZB"r u"(¢+») d»!0 in Lp loc(«) strongly. So, taking the limit in (4.20), and using (4.19) and lemma 4.12(b), we get Z« Myfvr(x; y)rÁ(y)g’(x) dx =¡Z« My½µ¹(x; y)¡1 jBrjZBr ¹(x; y +») d»¶Á(y)¾’(x) dx; for every ’2C10(«), Á2D1. This implies that for a.e. x2«, ry;mvr(x; ¢) = ¹(x; ¢)¡1 jBrjZBr ¹(x; ¢+») d» ; (4.21) and then the right-hand side of (4.21) belongs to rWpfor a.e. x2«. Since it converges a.e. in «to ¹(x; ¢)¡Myf¹(x; ¢)gand since rWpis closed, we deduce that there exists u12Lp(«;Wp) such that ¹¡Myf¹g=ryu1: By proposition 4.5, it is also clear that Myf¹g=ru, which together with the above equality gives (4.17). ¥ Analogously to proposition 4.11, we have the following result, which implies that the regularity of u1in theorem 4.13 is optimal at least for 1 6p < +1. Proposition 4.14. Let u2W1;p(«)and u12Lp(«;Wp), with 16p < +1. Then there exists a bounded sequence fu"ngin W1;p(«), such that u"n* u in W1;p(«)-weak; ru"n 2e *ru+ryu1:)(4.22) Proof . By theorem 3.15 it is easy to check that there exists a sequence fÁng » C1 0(«;C1(RN)) such that ryÁn2C0(«; (Bp)N), ryÁn! ryu1in Lp(«; (Bp)N) (4.23) and lim "!0µZ«¯¯¯¯ ryÁnµx; x "¶¡ ryÁmµx; x "¶¯¯¯¯ p dx¶1=p =µZ« [ry(Án(x; y)¡Ám(x; y))]p pdx¶1=p <1 2m8n>m: (4.24) Thus, for every n2Nthere exists a decreasing sequence f"ng » (0;+1) such that µZ«¯¯¯¯ ryÁnµx; x "n¶¡ ryÁmµx; x "n¶¯¯¯¯ p dx¶1=p <1 2m8m; n 2N;16m6n; (4.25) Proc. R. Soc. Lond. A (2002) on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from Two-scale method applied to Besicovitch spaces 2941 and µZ«¯¯¯¯ "nÁnµx; x "n¶¯¯¯¯ p dx¶1=p <1 2n;µZ«¯¯¯¯ "nrxÁnµx; x "n¶¯¯¯¯ p dx¶1=p <1 2n: (4.26) De ning u"n(x) = u+"nÁnµx; x "n¶ we deduce (4.14). ¥ 5. Applications of homogenization problems Similarly to the `classical’ two-scale convergence for periodic functions (see Allaire 1992; Nguetseng 1990), the main application of the results obtained in the present paper is the homogenization of partial di¬erential problems with coe¯ cients in the spaces Bp, generated by an ergodic algebra. As an example, for 1 <p<+1, let us consider the nonlinear problem ¡div(a(x="; u";ru")) = fin W¡1;p0(«); u"2W1;p 0(«);)(5.1) where «»RNis a bounded open set, fbelongs to W¡1;p0(«) and a:RN£R£RN7! RNsatis es (1) For every (s; ¹ )2R£RN,a(¢; s; ¹ ) belongs to (Bp0)N. For a.e. x2RNand every s2R,a(x; s; ¢) is continuous. (2) There exists ¬ > 0 such that a(x; s; ¹ )¢¹>¬j¹jp;8(s; ¹ )2R£RN;a.e. x2RN:(5.2) (3) There exist h2Bp0and  > 0 such that ja(x; s; ¹ )j6h(x) + (jsj+j¹j)p¡18(s; ¹ )2R£RN;a.e. x2RN:(5.3) (4) We have (a(x; s; ¹ 1)¡a(x; s; ¹ 2)) ¢(¹1¡¹2)>08s2R;8¹1; ¹ 22RN;a.e. x2RN: (5.4) (5) There exist ® > 0, 0 < ¼ 6minfp¡1;1gand k2Bp0such that ja(x; s1; ¹ )¡a(x; s2; ¹ )j6k(x)+®(j¹j+js1j+js2j)p¡1minfjs1¡s2j;1g¼;(5.5) for every s1; s22R, every ¹2RNand a.e. x2RN. The existence of a solution u"of (5.1) can be found in Lions (1969). The next theorem gives the asymptotic behaviour of u". Proc. R. Soc. Lond. A (2002) on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from 2942 J. Casado-D¶³ az and I. Gayte Theorem 5.1. Assume the above hypothesis and let u"be a solution of (5.1) for every " > 0. There then exist a subsequence, still denoted by f"g, a function u2W1;p 0(«)and a function u12Lp(«;Wp)such that u"* u in W1;p 0(«)-weak; ru" 2e *ru+ryu1; where (u; u1)2W1;p 0(«)£Lp(«;Wp)is a solution of the two-scale homogenized system ¡divxMyfa(y; u; ru+ryu1)g=fin W¡1;p0(«); ¡divm;yfa(y; u; ru+ryu1)g= 0 a.e. x2« : )(5.6) Remark 5.2. In general, the problem (5.6) does not have a unique solution, so the convergence of fu"gis only for a subsequence. Assuming a further hypothesis, for example, that a(x; s; ¹ ) does not depend on sand that in (5.4) the inequality is strict for ¹16=¹2, we have the uniqueness of solution of the limit problem which assures that the whole of the sequence fu"gconverges. Remark 5.3. De ning b:R£RN!Rby b(s; ¹ ) = Myfa(y; s; ¹ +ryvs;¹ )g 8s; ¹ 2R£RN; with vs;¹ the solution of ¡divm(a(y; s; ¹ +ryvs;¹ )) = 0 in (Wp)0; vs;¹ 2Wp;¾ the function uin the statement of theorem 5.1 satis es ¡div b(u; ru) = fin W¡1;p0(«); u2W1;p 0(«):) Remark 5.4. In the particular case of a linear equation, a(x="; u";ru") = A(x=")ru"and A, a matrix whose coe¯ cients are almost-periodic functions, the problem has been studied in Oleinik & Zhikov (1982). In the case when the equation is monotone, with a(x="; ru") almost-periodic in the  rst variable, the homogenization of (5.1) has been done in Braides et al. (1992), using approximation results in smoother almost-periodic spaces. Corrector results for these operators are proved in Braides (1991), exploiting the geometric properties of a. Proof . By (5.2) the sequence fu"gis bounded in W1;p 0(«), and then by (5.3), the sequence fg"gde ned by g"=aµx "; u";ru"¶ is bounded in Lp0(«)N. Theorems 4.8 and 4.13 then imply that there exists a subsequence, still denoted by f"g, a function u2W1;p 0(«), a function u12Lp(«;Wp) and a function g02Lp0(«;Bp0)Nsuch that u"* u in W1;p 0(«);(5.7) ru" 2e *ru+ryu1;(5.8) g" 2e * g0:(5.9) Proc. R. Soc. Lond. A (2002) on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from Two-scale method applied to Besicovitch spaces 2943 Let ’2C10(«), Á2C10(«), v2D1, and consider ’(x) + "Á(x)v(x=") as a test function in (5.1). This gives Z« g"(x)µr’(x) + "vµx "¶rÁ(x) + Á(x)ryvµx "¶¶dx=¿f; ’(x) + "Á(x)vµx "¶À; and then, taking the limit when "tends to zero, we deduce Z« Myfg0(x; y)(r’(x) + Á(x)ryv(y))gdx=hf; ’i; for every ’,Á,vas above. Reasoning by linearity and density, g0satis es Z« Myfg0(x; y)(rv(x) + ryv1(x; y)gdx=hf; vi; 8v2W1;p 0(«);8v12Lp(«;Wp);9 = ; (5.10) i.e. ¡divxMyfg0(x; y)g=fin W¡1;p0(«); ¡divm;yfg0(x; y)g= 0;a.e. x2« : )(5.11) Let us use the Minty rule to characterize g0. For ª ; © 2Stc(«; (Bp)N), such that ª(x; ¢), ©(x; ¢) belong to D1for a.e. x2«, and t2(0;1), we de ne ·"=ru(x) + ªµx; x "¶+t© µx; x "¶; which two-scale converges to ·0de ned by ·0(x; y) = ru(x) + ª(x; y) + t© (x; y):(5.12) By (5.4) we have Z«µg"¡aµx "; u"; · "¶¶¢(ru"¡·") dx>0:(5.13) Let us pass to the limit in the di¬erent terms of this inequality. Taking u"as a test function in (5.1) and uas a test function in the  rst equation of (5.11), we get lim "!0Z« g"ru"dx= lim "!0hf; u"i=hf; ui=Z« Myfg0rugdx: (5.14) By (5.9), we have lim "!0Z« g"·"dx=Z« Myfg0·0gdx: (5.15) The hypothesis (5.5) of aimplies ¯¯¯¯ aµx "; u"; · "¶¡aµx "; u; · "¶¯¯¯¯ 6kµx "¶+®(j·"j+ju"j+juj)p¡1minfju"¡uj;1g¼: Proc. R. Soc. Lond. A (2002) on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from 2944 J. Casado-D¶³ az and I. Gayte Taking the power p0and integrating in «, we deduce Z«¯¯¯¯ aµx "; u"; · "¶¡aµx "; u; · "¶¯¯¯¯ p0 dx 621=(p¡1) Z« kµx "¶p0 +®p03p¡1(j·"jp+ju"jp+jujp) minfju"¡uj;1g¼ p0dx: (5.16) Since the power p0of kand the power pof j·"jand ju"jhave a mean value, the sequence fk(x=")p0+®p03p¡1(j·"jp+ju"jp+jujp)gconverges weakly in L1(«). Moreover, the sequence fminfju"¡uj;1g¼ p0gis bounded in L1(«) and converges a.e. in «to zero. Thus (use Egorov’s theorem), we conclude that the right-hand side of (5.16) converges to zero, and then aµx "; u"; · "¶¡aµx "; u; · "¶!0 in Lp0(«):(5.17) So, by (5.8) we deduce lim "!0Z« aµx "; u"; · "¶(ru"¡·") dx= lim "!0Z« aµx "; u; · "¶(ru"¡·") dx =Z« Myfa(y; u; · 0)((ru+ryu1)¡·0)gdx: (5.18) Thus, passing to the limit when "tends to zero in (5.13) we conclude Z« Myfg0grudx¡Z« Myfg0·0gdx¡Z« Myfa(y; u; · 0)((ru+ryu1)¡·0)gdx>0: Replacing ·0by (5.12) and using the second equation of (5.11) we have Z« Myf[g0¡a(y; u; · 0)][ª+t© ¡ ryu1]gdx60:(5.19) Choosing ªconverging to ryu1in Lp(«;Bp)Nand ©converging to a function W2 Lp(«; (Bp)N), we deduce tZ« Myf[g0¡a(y; u; ru+ryu1+tW )W]gdx608W2Lp(«; (Bp)N): Dividing by tand then taking the limit when ttends to zero, we get Z« Myf(a(y; u; ru+ryu1)¡g0)Wgdx>08W2Lp(«; (Bp)N); i.e. g0=a(y; u; ru+ryu1), which by (5.11)  nishes the proof of theorem 5.1. ¥ In order to obtain a corrector result, let us now assume that ais uniformly monotone, i.e. there exists ¯ > 0 such that for every s2R,¹1; ¹ 22RNand a.e. x2«we have (a(x; s; ¹ 1)¡a(x; s; ¹ 2)) ¢(¹1¡¹2)>8 > < > : ¯j¹1¡¹2jpif p>2; ¯j¹1¡¹2j2 (j¹1j+j¹2j)2¡pif 1 <p<2:(5.20) Proc. R. Soc. Lond. A (2002) on November 24, 2016http://rspa.royalsocietypublishing.org/Downloaded from Two-scale method applied to Besicovitch spaces 2945 Theorem 5.5. Under the hypothesis of theorem 5.1 and (5.20), if u1is smooth enough (for example, u12C1(· «£RN)with rxu12L1(«£RN),ryu12 C(· «;B1)\L1(«£RN), then u"(¢)¡u(¢)¡"u1µ¢;¢ "¶!0in W1;p(«):(5.21) Proof . Let Z"=ru(x)+ryu1(x; x="). Reasoning similarly to the proof of (5.17), we can prove aµx "; u"; Z"¶¡aµx "; u; Z"¶!0 in Lp0(«): So, lim "!0Z«µaµx "; u";ru"¶¡aµx "; u"; Z"¶¶¢(ru"¡Z") dx = lim "!0Z«µaµx "; u";ru"¶¡aµx "; u; Z"¶¶¢(ru"¡Z") dx: (5.22) In order to pass to the limit on the right-hand side of this equality, it is enough to use the fact that a(x="; u; ru") two-scale converges to a(y; u; ru+ryu1) (see the proof of theorem 5.1), the fact that ru"two-scale converges to ru+ryu1and (5.14), which imply lim "!0Z«µaµx "; u";ru"¶¡aµx "; u; Z"¶¶¢(ru"¡Z") = 0:(5.23) From (5.22), (5.23) and (5.20) we conclude that ru"¡Z"converges strongly to zero in Lp(«) and then (5.21) holds. ¥ This work has been partly supported by Project PB98-1162 of the DGESIC of Spain. References Abddaimi, Y., Michaille, G. & Licht, C. 1997 Stochastic homogenization for an integral function of a quasiconvex function with linear growth. Asymp. Analysis 15, 183{212. Allaire, G. 1992 Homogenization and two-scale convergence. SIAM J. Math. 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