A Γ-CONVERGENCE APPROACH TO NON-PERIODIC HOMOGENIZATION OF STRONGLY ANISOTROPIC FUNCTIONALS

Transcription

1 Mathematical Models and Methods in Applied Sciences Vol. 14, No. 12 (2004) c World Scientific Publishing Company A Γ-CONVERGENCE APPROACH TO NON-PERODC HOMOGENZATON OF STRONGLY ANSOTROPC FUNCTONALS ANNALSA BALD and MARA CARLA TES Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, Bologna, taly baldi@dm.unibo.it tesi@dm.unibo.it Received 9 February 2004 Revised 31 March 2004 Communicated by M. Fabrizio n this work we present a homogenization result for a class of degenerate elliptic functionals mimicking strongly anisotropic media. We study the limit as ε 0 of the functionals α ε(x, u)(x) u, u dx, where, for any ε > 0, α ε : R n R n R, α ε(x, ξ) (x)ξ, ξ p/2 1, M n n (R) being measurable non-negative matrices such that A t ε(x) = (x) almost everywhere, p > 1. To take into account the anisotropy of the media we consider two families of weight functions reasonably different, λ ε and Λ ε, possibly degenerate or singular, such that: λ 2/p ε (x) ξ 2 (x)ξ, ξ Λ 2/p ε (x) ξ 2. The convergence to the homogenized problem is obtained by a classical approach of Γ-convergence. Keywords: Homogenization; anisotropic operators; Γ-convergence; weighted Sobolev spaces. AMS Subject Classification: 35B27, 35J70 1. ntroduction Composites are materials containing more than one constituent, finely mixed. 21 They are widely used in several branches of industry, due to their interesting properties. ndeed it is known that in general a composite performs better than a material made of a single component since it combines the attributes of the constituent materials, as in the case for example of ceramics or reinforced concrete. Common examples of composites are bones, which are porous composites, porous rock, in which the pores are often filled with salt water or oil, construction materials, such as wood and concrete, martensite, which is typical of a shape memory material, 1735

2 1736 A. Baldi & M. C. Tesi with a laminar-type structure comprised of alternating layers of the two variants of martensite. Usually in a composite the heterogeneities are very small compared with the global dimension of the sample. The smaller are the heterogeneities, the better is the mixture, which from a macroscopic point of view looks like a homogeneous material. t is then crucial to understand the relationship between the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective (electrical, thermal, elastic) moduli which govern the macroscopic behavior. The aim of the homogenization theory is to describe the macroscopic properties of the composite by taking into account the properties of the microscopic structure. Homogenization covers a wide range of applications, such as the study of composites, 21 optimal design problems, 1 neutron transport problems 23 and many other fields, see for example Refs. 4 and 10 and references therein. n this paper we are interested in the study of a homogenization problem for highly anysotropic non-periodic composites. We want to emphasis that we are considering a non-periodic modellization. ndeed in many situations a simple and appropriate way to model evenly distributed heterogeneities is to consider as a first approximation a periodic distribution, see the vast literature on the topic, e.g. Refs. 10, 13 and 25. However, this approximation is not always appropriate, for example to the case of random media. 27 n many physical situation a certain energy (thermal, electric, elastic) of a system is modelled by a nonlinear integral of the form F ε (u) = f ε (x, u) dx, (1.1) where ε is a scale parameter, small compared to the size of the set R 3, describing in some way the heterogeneities of the medium, and u is a physical field such as for example the displacement or the temperature. The aim of the homogenization in this case is to describe the overall properties of the medium by a simpler homogenized energy integral of the form F hom (u) = f hom ( u) dx, (1.2) obtained by appropriately taking the limit of F ε for ε 0. n the homogenization of multiple integrals, the right notion of variational convergence to be used turns out to be the Γ-convergence one, introduced in Ref. 16, whose natural framework is that of lower semicontinuous functionals. This is precisely the tool we use in this paper to obtain the convergence to the homogenized problem. For comprehensive accounts on this techique see e.g. Refs. 3 and 14. We recall that there are many other different techniques available to treat homogenization problems which arise in other mathematical contexts, for example in the context of PDEs or of boundary values problems, for which we refer to Refs. 1, 10 and 13 and references therein.

3 Non-Periodic Homogenization of Strongly Anisotropic Functionals 1737 n this paper we consider the following functionals, J ε (u) = f ε (x, u) dx + gu dx, (1.3) defined on any bounded open set with Lipschitz boundary in R n, for ε > 0 and g L (), with u belonging to some space where u exists. The family of functions f ε : R n R n R + is defined by f ε (x, ξ) := α ε (x, ξ) (x)ξ, ξ, ε > 0, (1.4) where : R n M n n (R) is a family of measurable matrix-valued functions such that A t ε (x) = (x) 0 a.e. in R n for all ε > 0. We shall assume that there exist two families of weight functions (i.e. non-negative locally summable functions) (λ ε ) ε>0, (Λ ε ) ε>0 such that: λ 2/p ε (x) ξ 2 (x)ξ, ξ Λ 2/p ε (x) ξ 2 (1.5) for a.e. x R n and for all ξ R n, with p > 1. Moreover, we assume (i) α ε (x, ) : R n R continuous for a.e. x R n, α ε (, ξ) measurable on R n for any ξ R n ; (ii) b 1 (x)ξ, ξ p/2 1 α ε (x, ξ) b 2 (x)ξ, ξ p/2 1, where b 1 and b 2 are positive constants ; (iii) α ε (x, ξ) (x)ξ, ξ convex in ξ. (1.6) Up to changing the constants in (1.5), it holds λ ε (x) ξ p f ε (x, ξ) Λ ε (x) ξ p, (1.7) for a.e. x R n, for any ξ R n, p > 1. Concerning the weights we will make the following assumptions: (w 1 ) λ ε A p (A), p > 1, uniformly with respect to ε, i.e. there exists A > 0 such that ( ) ( p 1 λ ε dx λε dx) 1/(p 1) A (1.8) for all cubes with faces parallel to the coordinate planes and for all ε > 0; in addition, for any given cube R n, there exists a constant C 1 = C 1 () > 0 such that λ ε C 1 (1.9) for all ε > 0;

4 1738 A. Baldi & M. C. Tesi (w 2 ) Λ ε L 1+µ loc (Rn ), µ > 0, uniformly with respect to ε, i.e. for any given cube R n there exists a constant C 2 = C 2 () > 0 such that ( ) 1 Λ 1+µ 1+µ ε dx C2 (1.10) for all ε > 0; (w 3 ) for every cube R n there exists a constant K = K() > 0 such that Λ ε (x) dx K, (1.11) λ ε (x) uniformly with respect to ε. We give some examples of admissible weights at the end of the section. Given the functionals J ε (u) = α ε(x, u) (x) u, u dx + gu dx we want to prove that there exists a subsequence α εk (x, ξ)k (x)ξ, ξ and a function f (x, ξ), whose properties will be specified later, such that the functionals J εk (u) Γ-converge in the L 1 () topology to a functional that can be written, for suitably regular functions u, as f (x, u) dx + gu dx. n addition we will show that converge in the L 1 () topology to { min f (x, u) dx + min{j εk (u) : u = 0 on } } gu dx : u = 0 on (1.12) with the minima taken in suitable function spaces. Functionals as in (1.3), with a weighted growth condition on the energy density such as the one considered in (1.7), can be used to describe some fine properties of a wide class of degenerate anisotropic structures. Notice that we do not make any periodicity requirement on the energy density, in order to be able to describe nonperiodic structures. For example one could think of models of porous media, 11 or models of mixtures of materials with different nonlinearities, showing a particular different behavior along preferred directions. 20 As a simple model case of a physical situation with an energy density such as (1.4) we can consider an electromagnetic material in R 3 with a mesoscopic structure given by alternate layers of two materials, a metal with high conductivity and a plastic that is electrically insulating. The constitutive equation for this medium is given by j(x) = σ(x)e(x), x = (x 1, x 2, x 3 ), where j(x) is the current field, e(x) is the electric field and the conductivity tensor field (σ i,j (x)) i,j=1,2,3 is such that σ 1,3 (x) = σ 3,1 (x) = σ 2,3 (x) = σ 3,2 (x) = 0 for any x R 3, and σ 3,3 (x) = λ(x 3 ), λ being a periodic A 2 -weight vanishing at a finite number of points. The energy density of such a medium is modelled by f ε (x, u) = σ ε (x) u, u and the energy is described by the functional { 2 } J ε (u) = σ i,j xi u xj u + λ ε (x 3 ) x3 u 2 dx + gu dx, (1.13) i,j=1

5 Non-Periodic Homogenization of Strongly Anisotropic Functionals 1739 where is a bounded connected open subset of R 3, the sub-matrix (σ i,j (x)) i,j=1,2 is bounded and positively definite (again to simplify the picture) and the functions λ ε have the form λ ε (x 3 ) = λ( x3 ε ). The homogenization process leads to a highly anisotropic composite that has the conducting properties of the metal in the directions parallel to the layers and the insulating properties of the plastic normal to the layers. Functionals like (1.13) can model the energy of several different physical situations in addition to the one mentioned above, like for instance thermal conduction in a body containing at the microscopic scale a horizontal bundle of thin layers characterized by a very low thermal conductivity, or again the behavior of heterogeneous materials with some kind of stiff degeneracies along a given direction. Clearly, in (1.13), Λ ε is a constant, but we might easily modify the model to take into account opposite behavior of the inclusions, taking now λ ε 1 and possibly Λ ε = + along some direction. Concerning previous studies of models of possibly degenerate anisotropic situations as the ones we are interested in here, we are basically aware of two papers. n Ref. 15 the author consider functionals with density energy f ε with a growth condition as in (1.7), restricted to the case Λ ε = cλ ε, that is, describing possibly degenerate phenomena basically isotropic in all directions. On the contrary, since we are interested in studying strongly anisotropic situations, we consider two different families of weights λ ε and Λ ε, which can differ each other in the sense that the ratios Λ ε /λ ε can blow-up at some point, provided (1.11) is satisfied. A possibly degenerate anisotropic nonlinear elliptic problem with the same structure of (1.4) and with the same structural hypotheses was treated in Ref. 17, with a completely different approach based on weighted compensated compactness techniques. However, in Ref. 17 the authors were somehow forced to restrict themselves to the periodic context. We want to stress explicitly that the anisotropy of the problem treated here, i.e. the presence of two different families of weights controlling the structure of the functionals, leads to face several problems that we try to explain now. We have to be vague for a while (for precise definitions see the next section). Let be a bounded open set, and p > 1. We denote by W 1,p () a two-weight Sobolev space related to the matrix (satisfying (1.5) for any ε > 0), endowed with a norm induced by, in a sense specified in Definition 2.3 below. Then we consider the space given by the closure of regular functions in with respect to that norm, and we denote it by H 1,p (). Now, a Meyers Serrin type result reading H 1,p () = W 1,p () is not true in the two-weight case (see Ref. 9 and also some examples in Sec. 6). This means that H 1,p () does not coincide with the finiteness domain of the functional J ε, and therefore a Lavrentev-type phenomenon can occur. 22 n addition, when dealing with the Γ-lim ε 0 J ε, it is not even obvious in which way to define the spaces of type H and W connected with the finiteness domain of the Γ-limit. One of the features of this note is to produce consistent definitions

6 1740 A. Baldi & M. C. Tesi of suitable H and W -type space where to represent the Γ-limit. This problem, representing one of the main difficulties of this work, is extensively discussed in Secs. 4 and 5. Moreover, in Sec. 6 we show that these spaces H and W might be different. Examples of admissible weights The simplest situation one could think of is the case of equal periodical A p weights λ = Λ, if we set λ ε (x) = λ( x ε ), which clearly satisfy hypotheses (w1), (w2) and (w3) (notice that a periodic extension of an A p weight is again in the A p class). A less trivial example (however still periodic) is provided by a pair (λ, Λ) of periodic weights with the same period, if we put λ ε (x) = λ( x ε ), Λ ε(x) = Λ( x ε ), with λ A p and Λ L 1+µ loc (Rn ) provided Λ/λ L 1 loc (Rn ). n order to provide simple examples of non-periodic weights, consider first for instance the case λ ε 1. Then (w 1 ) is obvious, whereas (w 2 ) and (w 3 ) read straightforwardly as Λ ε L 1+µ loc (Rn ) uniformly in ε. On the other hand, it is easy to produce non-periodic A p weights with controlled behavior in ε as required in (w 1 ) and (w 3 ) just starting from periodic A p weights with non-rationally comparable periods. We stress also explicitly that, even if we assume λ ε, Λ ε periodic, the function f ε might not be periodic. Our notation is standard. The Lebesgue measure of sets is denoted by and the scalar product in R n by,. For any Lebesgue measurable set E, if ω L 1 loc (Rn ) we denote by ω(e) := E ω(x) dx, and if S denotes a set in, u S is the average of the function u in the set S, i.e. u S = 1 S u(y) dy = S S u(y) dy. We denote by Lip loc (R n ) the space of locally Lipschitz functions on R n and, for any bounded open set we denote by Lip 0 () the space of locally Lipschitz functions with support compactly contained in. The plan of the work is the following. Section 2 briefly recalls the definition of Γ-convergence and some basic properties of A p weights. n Sec. 3 we prove some preliminary results about the Γ-limit of the functionals J ε. Section 4 is devoted to the definition of suitable Sobolev-type spaces, denoted by H () and W (), for studying problem (1.12). n Sec. 5 we obtain a representation theorem for the Γ- limit of J ε in H and we give a result concerning the convergence of the minimum points of J ε. n Sec. 6 we produce two examples concerning the relations between the spaces H and W used in the paper. n the first example we provide weights such that H 1,p () W 1,p () but when we pass to the Γ-limit H () = W (). On the contrary, in the second one we choose weight functions so that H 1,p () = W 1,p () but H () W (). 2. Basic Tools The notion of convergence used in this paper is the so-called Γ-convergence (see Ref. 16).

7 Non-Periodic Homogenization of Strongly Anisotropic Functionals 1741 Definition 2.1. Let X be a metric space and for ε > 0 consider the functional F ε : X [, + ]. We say that F ε Γ-converges to F on X as ε goes to zero if the following two conditions hold: (1) for every u X and every sequence (u ε ) which converges to u in X there holds lim inf ε 0 F ε(u ε ) F (u) ; (2.1) (2) for every u X there exists a sequence (u ε ) which converges to u in X and lim F ε(u ε ) = F (u). (2.2) ε 0 Many properties about Γ-convergence can be found for example in Refs. 3 5 and 14. n particular we remark explicitly that, if the functionals F ε are convex, the Γ- limit F is convex. Moreover Γ-convergence turns out to be stable with respect to perturbation with a continuous function and composition with an increasing function. We briefly recall some results on Muckenhoupt weights: Definition We say that a non-negative mesurable function λ on R n is a weight in the class A p (A), p > 1, for a given constant A 1, if it satisfies the following condition: ( ) ( p 1 λ dx λ dx) 1/(p 1) A, for all cubes R n. Moreover, we denote by A p := A 1 A p(a). We need the following result, see Refs. 12, 15 and 18: Theorem 2.1. Let λ A p (A), p > 1. Then there exist two positive constants σ = σ(n, p, A) and C = C(n, p, A) such that ( 1/(1+σ) λ dx) 1+σ C λ dx, (2.3) ( 1/(1+σ) λ dx) (1+σ)/(p 1) C λ 1/(p 1) dx (2.4) for every cube with faces parallel to the coordinate planes. Let M be the Hardy Littlewood maximal function operator defined by M u(x) = sup x u(y) dy, where u is a locally integrable function and is a cube as above. The following weighted norm inequality for M holds: Theorem Let 1 < p <. Given a weight function λ A p (A) there exists a constant c = c(a) such that (M u ) p λ(x) dx c u p λ(x) dx u L p (R n, λ), (2.5) R n R n where L p (R n, λ) = {u L 1 loc (Rn ) : R n u(x) p λ(x) dx < }.

8 1742 A. Baldi & M. C. Tesi Let us now introduce some function spaces suitable to our problem. Let be a bounded open set in R n. f p > 1 and λ A p we denote by W 1,p (, λ) = {u L p (, λ) W 1,1 loc () : u Lp (, λ)}, by W 1,p (, λ) := W 1,1 () W 1,p (, λ), by H 1,p (, λ) the closure of C 1 () W 1,p (, λ) in W 1,p (, λ) and by H 1,p (, λ) the closure of C0 () in W 1,p (, λ). n addition, if has Lipschitz boundary, then the following characterization has been proved (see Ref. 15 and also Ref. 9): H 1,p (, λ) = W 1,p (, λ). Let be the matrix defined in (1.5) and the pair of weights (λ ε, Λ ε ) satisfying hypothesis (w 1 ), (w 2 ) and (w 3 ) listed in the ntroduction. n complete analogy with Ref. 17, we define the following spaces: Definition 2.3. f 1 < p <, we define for all ε > 0 { } W 1,p () = u L p (, λ ε ) W 1,1 loc () : u, u p/2 dx < +, (2.6) endowed with the norm 1/p ( 1/p u W 1,p Aε ( () = u p λ ε dx) + u, u dx) p/2. (2.7) With suitable minor changes in the proof of Theorem 1 in Ref. 17, it is possible to show that W 1,p () is a reflexive Banach space continuously embedded in W 1,p (, λ ε ) and hence continuously embedded in W 1,1 (). We stress explicitly that the embedding in W 1,1 () is uniform in ε since the constant controlling the norm W 1,1 () depends only on the constant A of Definition 2.2. Definition 2.4. f 1 < p <, we define for all ε > 0 H 1,p () = Lip 0 () 1,p W Aε (). (2.8) Due to the assumptions on f ε, the functionals J ε defined in (1.3) have minima in W 1,p () := W 1,1 () W 1,p () for any ε > 0 (see e.g. Ref. 19). n general W 1,p () and H 1,p () do not coincide, as shown in Sec. 6. Therefore the result concerning the convergence of minimum points of J ε when passing to the Γ-limit is restricted to only the spaces H 1,p () (see Theorem 5.2 for a detailed statement). 3. Preliminary Results The following theorem states that for functions in Lip loc the Γ-limit of (1.3) has an integral representation. More precisely we have: Theorem 3.1. (Theorem 3.4 of Ref. 7) Let (f ε ) be the family of functions defined in (1.4) satisfying hypothesis (1.6), and (Λ ε ) be a family of weights satisfying (1.10). Then there exists a weight Λ L 1+µ loc (Rn ) such that, up to subsequences, Λ ε w L 1+µ () Λ, (3.1)

9 Non-Periodic Homogenization of Strongly Anisotropic Functionals 1743 for every given cube in R n. Moreover, there exist a subfamily (f εk ) (f ε ) and a function f (x, ξ) convex in ξ such that and 0 f (x, ξ) Λ (x) ξ p, a.e. x R n, ξ R n (3.2) Γ- lim f εk (x, u) dx = f (x, u) dx < +, (3.3) k in the L 1 () topology, for any bounded open set, u Lip loc (R n ). Now if we assume a uniform A p condition and a local uniform lower bound on the weights λ ε, it is possible to obtain, locally, a sharper estimate for f. This result is largely inspired by Proposition 4.1 in Ref. 15. Theorem 3.2. Let (f ε ) be the family of functions defined in (1.4) satisfying hypothesis (1.6). Let be a given cube, and (λ ε ) and (Λ ε ) be as in (1.8) (1.10) respectively. Then there exist a subfamily (f εk ) (f ε ) and two weights Λ L 1+µ loc (Rn ) and λ in A p such that, and there exists σ > 0 such that λ 1/(p 1) ε Λ ε w L 1+µ () Λ, (3.4) w L 1+σ () λ 1/(p 1). (3.5) Moreover, there exists a function f (x, ξ), convex in ξ, satisfying λ (x) ξ p f (x, ξ) Λ (x) ξ p (3.6) for almost every x, for any ξ R n and such that Γ- lim f εk (x, u) dx = f (x, u) dx (3.7) k in the L 1 ()-topology, for every open set and u Lip loc (R n ). Proof. Let be a given cube such that (1.9) holds. By hypothesis (1.10), we have, up to a subsequence, Λ ε w L 1+µ () Λ. (3.8) By Theorem 3.1 the estimate (3.2) holds almost everywhere in, and we have the related Γ-convergence result (3.3) for all bounded open set for every u Lip loc (R n ). Due to hypotheses (1.8) and (1.9), and by Theorem 2.1 there exist σ > 0 and C 3 > 0 such that, for all ε > 0, we have ( λ (1+σ)/(p 1) ε ) 1/(1+σ) ( ) 1/(p 1) A C 3 ; (3.9) C 1

10 1744 A. Baldi & M. C. Tesi therefore, up to a subsequence, there exists a function λ 1/(p 1) that L 1+σ () such λ 1/(p 1) ε w L 1+σ () λ 1/(p 1). (3.10) As for the proof that λ A p, we argue as in the proof of Proposition 4.1 in Ref. 15, since by (1.10), we have λ ε dx C 2 µ 1 1+µ. Now, to get the estimate from below in (3.6) we proceed as follows. By Hölder s inequality, if u Lip loc (R n ) is given, it is trivial to see that ( p u dx) ( u p λ ε dx ) ( p 1 λε dx) 1/p 1. Applying the lower estimate in (1.7) to the right-hand side, it follows that ( p u dx) ( f ε (x, u) dx ) ( p 1 λε dx) 1/p 1. (3.11) By Theorem 3.1 there exists u ε u in L 1 (), such that lim ε 0 f ε(x, u ε ) dx = f (x, u) dx. On the other hand, the functional u dx is L1 ()-lower semicontinuous on W 1,1 loc (Rn ) (and hence in Lip loc (R n )), hence u dx lim inf ε 0 u ε dx. Therefore, using (3.11) for u ε, we have ( u dx)p lim inf ε ( u ε dx) p lim inf ε ( f ε(x, u ε ) dx)( λ 1/p 1 ε dx) p 1, and we get ( p u dx) ( f (x, u) dx ) ( p 1 λ dx) 1/p 1. Now take u(x) = ξ, x. Choosing = B( x, r) and letting r 0 we get the estimate λ ( x) ξ p f ( x, ξ) at any Lebesgue point x of λ 1/p 1 and f (, ξ). Remark 3.1. f we choose α ε (x, ξ) = (x)ξ, ξ p/2 1, we can apply Theorem 3.2 to the functions f ε (x, ξ) = (x)ξ, ξ p/2. More precisely, there exists a function Q(x, ξ), convex in ξ, satisfying λ (x) ξ p Q(x, ξ) Λ (x) ξ p (3.12) for almost every x, for any ξ R n and such that Γ- lim k (x) u, u p/2 dx = Q(x, u) dx (3.13) k in the L 1 () topology, for every open set and u Lip loc (R n ). Even if the functions f ε have a polynomial structure, as in previous remark, the the Γ-limit might have a different structure (for a counterexample see Ref. 6). Nevertheless, if f ε (x, ξ) = (x)ξ, ξ p/2, we show in the next section that Q(x, u) dx produces a norm in Lip 0 ().

11 Non-Periodic Homogenization of Strongly Anisotropic Functionals Definition of the Space H n the sequel is a given cube, where hypotheses of Theorem 3.2 are satisfied and we consider a Lipschitz domain, compactly contained in. For ε > 0, we consider the functionals (x) u, u) p/2 dx u Lip loc (R n ), Ĝ ε (u) = (4.1) + u (W 1,1 loc (Rn ) Lip loc (R n )). By a Γ-convergence property, up to subsequences, it is known that the Γ-limit in the L 1 () topology is well defined for all u W 1,1 loc (Rn ); we denote it by ĜΓ(u). Using Remark 3.1, and because of Proposition 6.15 in Ref. 14 the previous Γ-limit is represented by the functionals Ĝ Γ (u) = Q(x, u) dx (4.2) for every u Lip loc (R n ), where Q satisfies (3.12). We introduce the following notation: Definition 4.1. For u W 1,1 loc (Rn ), we denote by [u] the following Γ-limit ( ) [u] := Γ- lim u dx + (Ĝε(u)) 1/p ε 0 in the L 1 () topology. (4.3) Recalling that Γ-convergence is stable under continuous perturbations and under composition by an increasing function, the following proposition follows easily: Proposition 4.1. For u Lip loc (R n ), ( 1/p [u] = u dx + Q(x, u) dx), (4.4) where Q(x, ξ) is given by (3.13) and satisfies (3.12). We now restrict ourselves to the space Lip 0 () and show that [ ] is a norm on Lip 0 (). To achieve this result we first recall the following statement proved in Ref. 8, Theorem 1.2: Proposition 4.2. Let be a bounded Lipschitz domain, u W 1,1 loc () L1 (), and λ ε A p (A), then u u p λ ε dx C u p λ ε dx, (4.5) where u = u dx and the constant C = C (A) is independent of u and ε. Let us introduce a new norm on W 1,p () equivalent to (2.7):

12 1746 A. Baldi & M. C. Tesi Lemma 4.1. Let (λ ε ), (Λ ε ) satisfying (1.8) and (1.10); in addition let be a given cube, assume that (1.9) holds in for all ε > 0. Let be a bounded Lipschitz domain,. The norm (2.7) and the following ( u W 1,p Aε () := u dx + 1/p u, u dx) p/2 (4.6) are equivalent on W 1,p (), and the constants appearing in the equivalence are independent of ε. Proof. By Hölder s inequality, since λ ε are uniformly A p and (1.9) holds, we get u dx c( u p λ ε dx) 1/p with c independent of ε. On the other hand, using Poincaré inequality (4.5) and (1.10), we easily obtain ( ( 1/p u p λ ε dx) C 1/p C 1/p u u p λ ε dx ( ( where C is a constant independent of ε. u p λ ε dx ) 1/p + u λ ε () 1/p ) 1/p + Λ ε() 1/p ) 1/p u, u p/2 dx + C u dx u dx, By previous lemma [u] can be seen as a Γ-limit of norms, hence we prove: Proposition 4.3. [ ] is a norm on Lip 0 (). Proof. First [u] < + on Lip 0 (); indeed Q(x, u) dx Λ (x) u p dx u p Lip 0 () Λ (x) dx C 1+µ 2 u p Lip 0 () by (3.12) and since Λ L 1+µ loc (Rn ). To simplify notations in the remaining of the proof we will indicate the norm (4.6) by ε. Let u be in Lip 0 () and t R, t 0. By definition of Γ-convergence, there exists a sequence of functions {v ε } in Lip loc (R n ) such that v ε tu in L 1 (), v ε ε [tu]. Without loss of generality we can assume v ε = tu ε, with u ε u in L 1 (). Then [u] lim inf u ε ε = 1 lim inf ε 0 t t u ε ε ε 0 = 1 lim inf t tu ε ε = 1 ε 0 t [tu]. Hence, for t 0, t [u] [tu] and analogously 1 t [tu] [u], which imply [tu] = t [u].

13 Non-Periodic Homogenization of Strongly Anisotropic Functionals 1747 As for the triangular inequality, let u, v be in Lip 0 (). Then there exists two sequences of functions {u ε } and {u ε } in Lip loc (R n ) such that Since u ε + v ε u + v in L 1 (), we have and we are done. u ε u in L 1 (), u ε ε [u], v ε v in L 1 (), v ε ε [v]. [u + v] lim inf ε 0 u ε + v ε ε lim inf ε 0 ( u ε ε + v ε ε ) lim sup ε 0 = [u] + [v] ( u ε ε ) + lim sup( v ε ε ) ε 0 Thanks to previous results it makes sense to define: Definition 4.2. f 1 < p <, we define the space H () = Lip 0 () [ ]. (4.7) The space H () will turn out to be the suitable one where to treat the minimum problem (1.12) stated in the ntroduction and proved in Theorem 5.2 of Sec. 5. n the sequel we show how to represent the Γ-limit (4.3) in H (). First we prove that Lemma 4.2. The space H () is continuously embedded in W 1,1 (). Proof. Let u H (), then there exists a sequence of functions u n Lip 0 () such that [u n u] 0. This implies u n u in L 1 (). On the other hand, for any v Lip 0 (), by (3.12) and (4.5) we have ( [v] 1/p λ (x) v dx) p + c v W 1,1 () v dx C v dx + v dx (4.8) since λ A p. The sequence u n is a Cauchy sequence in H (), hence by (4.8) it is a Cauchy sequence in W 1,1 (), thus u n ũ in W 1,1 (). By uniqueness of the limit ũ = u, that is H () W 1,1 (). We show now how to represent the norm [ ] in H (); indeed we show that the representation result given in Proposition 4.1 holds not only on Lip 0 () but also in H (). Theorem 4.1. f u H (), then we have [u] = u dx + ( Q(x, u) dx)1/p.

14 1748 A. Baldi & M. C. Tesi Proof. Since u H () there exists a sequence (u n ) n N in Lip 0 () such that u n u in H () and hence, by Lemma 4.2, in W 1,1 (). Thus, without loss of generality, we may assume that u n (x) u(x), u n (x) u(x) for a.e. x and in particular we notice that Q(x, u) = lim n Q(x, u n ) since Q is continuous in the second variable. Hence, Q(x, u) dx lim inf n Q(x, u n) dx lim inf n [u n ] <, by Fatou s lemma and since (u n ) n N is bounded in H (). Hence Q(x, u) dx < +. We show now that, if u n Lip 0 (), u n u in H () then ( 1/p [u n ] u dx + Q(x, u) dx) as n. (4.9) To prove (4.9) it is enough to show that the assertion holds for a subsequence. Let u H () be given, and let (u n ) n N be a sequence in Lip 0 () converging to u in H (). Again, without loss of generality, we may assume that u n (x) u(x), u n (x) u(x) for a.e. x. On the other hand, mimicking the proof of Riesz Fischer theorem, we choose a subsequence (u ni ) i N such that [u ni+1 u ni ] < 1/2 i. Now we can write (putting u n0 = 0) m u nm (x) = ( u ni u ni 1 ). Thus for a.e. x Q(x, u nm (x)) 1/p i=1 m Q(x, u ni (x) u ni 1 (x)) 1/p i=1 Q(x, u ni (x) u ni 1 (x)) 1/p, (4.10) i=1 where the first inequality is proved in detail in Remark 4.1 below. Since [u nm ] = ( Q(x, u n m (x)) dx) 1/p + u n m (x) dx, (4.9) follows by dominated convergence theorem once we prove that Q(, u ni ( ) u ni 1 ( )) 1/p L p (). This holds since ( i=1 ( ) p Q(x, u ni (x) u ni 1 (x)) 1/p dx i=1 ( Q(x, u ni (x) u ni 1 (x)) dx i=1 1 [u ni u ni 1 ] 2 i = 1. i=1 i=1 ) 1/p ) 1/p

15 Non-Periodic Homogenization of Strongly Anisotropic Functionals 1749 Now the representation result follows easily. ndeed, let u H (), by definition there is a sequence (u n ) n N in Lip 0 () converging to u, hence, in particular, [u n ] [u] as n. By uniqueness of the limit (in R), (4.9) the representation of [u] in H (). n the following remark we show in detail how to get formula (4.10) used in the previous proof. Remark 4.1. For a.e. x, for any ξ, η R n it holds Q( x, ξ + η) 1/p Q( x, ξ) 1/p + Q( x, η) 1/p. (4.11) Proof. Let B = B( x, r) be a ball centered in x with radius r, B. Let u ε, v ε Lip loc (R n ) such that u ε, ξ v ε, η in L 1 (B) and ( 1/p ( (x) u ε, u ε dx) p/2 B ( 1/p ( (x) v ε, v ε dx) p/2 B B B Q(x, ξ) dx) 1/p, Q(x, η) dx) 1/p ; this is possible since the Γ-limit has the integral representation (3.13), the functions, ξ, η being in Lip loc (R n ). For the same reason, ( 1/p Q(x, ξ + η) dx) lim inf B ε 0 ( B ( (x) (u ε + v ε ), (u ε + v ε ) p/2 dx B ( 1/p Q(x, η) dx). Q(x, ξ) dx) 1/p + B ) 1/p Letting r 0, (4.11) is proved if x if a Lebesgue point of Q(, ξ + η), Q(, ξ) Q(, η). On the other hand, the set 0 ={ x ; x is a Lebesgue point of Q(, ξ + η), Q(, ξ), Q(, η) for any ξ, η Q n } is such that \ 0 = 0. Eventually (4.11) holds for ξ, η R n and x \ 0, thanks to a limit argument. Let be as above. The representation Theorem 4.1 allows us to give a representation result for the Γ-limit of the following functionals defined for functions u 0 Lip loc (R n ), f ε (x, u) dx u u 0 + Lip 0 (), F ε (u) = (4.12) + u u 0 + ( W 1,1 () Lip 0 ()).

16 1750 A. Baldi & M. C. Tesi By a property of Γ-convergence, up to a subsequence, the Γ-limit in L 1 () of (4.12) is well defined for all u u 0 + W 1,1 (); we denote it by F Γ (u). The problem we are concerned with in next section is to find a representation for F Γ (u). Let us first define the space W () := {u W 1,1 (); [u] < + }; clearly H () W () but in general the two spaces do not coincide (see counterexamples in Sec. 6). The lack of equality between H () and W () involves a representation problem for the Γ-limit. ndeed, it is not possible to prove that f (x, u) dx u u 0 + H (), F Γ (u) = (4.13) + u u 0 + ( W 1,1 () H ()), for every u u 0 + W 1,1 () with u 0 Lip loc (R n ), since as we already noticed the finiteness domain of F Γ is in general bigger than H (). For this reason we can get a representation theorem only in H (), i.e. we prove that F Γ (u) = f (x, u) dx for u u 0 + H (). 5. Representation Results on Γ-Limits At the end of previous section we explained the problem of representation of the limit of the functionals (4.12). As already mentioned, their Γ-limit F Γ (u) is well defined for all u u 0 + W 1,1 (). Recalling Theorem 3.2, it is easy to prove (see Ref. 14, Proposition 6.15) that F Γ is represented by the functional F Γ (u) = f (u, u) dx, (5.1) for every u u 0 + Lip 0 (), where f satisfies (3.6). n fact, we prove in Theorem 5.1 that F Γ (u) = f (u, u) dx for every u u 0 + H (). Let us introduce the functionals f ε (u, u) dx u Lip loc (R n ), ˆF ε (u) = (5.2) + u (W 1,1 loc (Rn ) Lip loc (R n )). Again up to subsequences, it is known that the Γ-limit in the L 1 () topology is well defined for all u W 1,1 loc (Rn ); we denote it by ˆF Γ (u). Using Remark 3.1, and because of Proposition 6.15 in Ref. 14 the previous Γ-limit is represented by the functional ˆF Γ (u) = f (x, u) dx, (5.3) for every u Lip loc (R n ), where f satisfies (3.6).

17 Non-Periodic Homogenization of Strongly Anisotropic Functionals 1751 The proof of Theorem 5.1 is based on Theorem 3.1 and an approximation result stated in Proposition 5.1 below will be overcome via an approximation. Proposition 5.1. Let p > 1 and let be a bounded open set compactly contained in a cube. Let (f ε (x, ξ)) be a family of functions measurable in x and convex in ξ satisfying λ ε (x) ξ p f ε (x, ξ) Λ ε (x) ξ p for a.e. x R n for any ξ R n. Let (λ ε ) be as in (1.8) and (1.9); let (Λ ε ) satisfy (1.10); in addition suppose that the pair (λ ε, Λ ε ) satisfies (1.11). Consider u W 1,1 () and a sequence (u ε ) Lip 0 () such that u ε u in L 1 () and (x) u ε, u ε p/2 dx < C 4 (5.4) for all ε > 0. Then for every u 0 Lip 0 () and τ > 0 there exist τ so that τ < 3τ, β τ > 0, α τ > 0, such that β τ 0, α τ, as τ 0, and there exist a sequence (v ε,τ ) Lip loc (R n ), and a function v τ Lip loc (R n ) verifying v ε,τ L (R n ) c(n)α τ, v ε,τ v τ in L () as ε 0 and lim inf ε 0 n addition, setting we have V τ < 2τ. f ε (x, u 0 + u ε ) dx lim inf ε 0 τ f ε (x, u 0 + v ε,τ ) dx β τ. (5.5) V τ := {x R n : v τ (x) u(x)}, (5.6) Proof. The scheme of the proof is essentially the one of Theorem 3.1 in Ref. 15. However since we are dealing with two different weights, appropriate changes must be taken into account. For the reader convenience we will report the proof, obviously stressing the differences. To avoid cumbersome notation we will use the same label for sequences and subsequences. Without loss of generality, consider (u ε ) C0 () and identify u and (u ε ) with their zero extension outside. For N > 0, let us define the sets CN ε = {x Rn : Λε λ ε > N}. By hypothesis (1.11), it follows that N CN ε = N C ε N dx Λ ε λ ε dx < K (5.7) uniformly in ε. Then, for a fixed τ > 0 there exists a N τ such that, for N > N τ, CN ε < τ, uniformly in ε. On the other hand, since the weights λ ε are in the A p (A) class we can apply Theorem 2.2 to the functions u ε which gives (M u ε ) p λ ε (x) dx c u ε p λ ε (x) dx. R n

18 1752 A. Baldi & M. C. Tesi By (1.5) and (5.4) the integrals (M u R n ε ) p λ ε (x) dx are equibounded, therefore the sequence ((M u ε ) p λ ε (x)) is bounded in L 1 (R n ). Then, as proved in Ref. 2, for every η > 0 we can find a set S η with S η < η and a constant δ η such that, up to a subsequence (M u ε ) p λ ε (x) dx < η (5.8) B for any measurable set B with B S η = and B < δ η. By Hölder s inequality, the uniform condition A p on λ ε and (1.5) yield ( u ε dx) p = ( u ε λ 1/p ε dx) p ( λ 1/(p 1) ε where in the last inequality we also use (1.9) and (5.4). By the one-to-one weak type property of the maximal function (see e.g. Theorem 1 of Ref. 26), there exists a constant c(n) depending only on n such that {x R n : (M u ε ) > γ} c(n) γ u ε L1 (R n ). λ 1/p ε Therefore we can choose α η c(n) η We define the sets dx) p 1 (x) u ε, u ε p/2 dx < C 4 A C 1 ( AC4 C 1 ) 1/p such that {x R n : (M u ε ) α η } min{η, δ η }. (5.9) Z η ε = {x Rn : (M u ε ) α η } ; we have u ε Lip(Z η ε ) with Lipschitz constant c(n)α η (see Lemma -11 of Ref. 2). We denote by v ε,η the Lipschitz extension of u ε out of Z η ε to the whole R n, with the same Lipschitz constant c(n)α η. n summary we have and v ε,η (x) = u ε (x) and v ε,η (x) = u ε (x) a.e. in Z η ε (5.10) v ε,η L (R n ) c(n)α η ; (5.11) moreover, without loss of generality we can assume that v ε,η (x) = 0 if dist(x, ) > 1. Up to a subsequence, it holds that v ε,η v η in L () and v η L (R n ) c(n)α η, {x : v η (x) u(x)} 2η, and by (5.9) ( S η ) Zε η < min{η, δ η } (see Ref. 15 for a detailed proof). Finally, we can proceed as follows: f ε (x, u 0 + u ε ) dx = C ε N C ε N f ε (x, u 0 + u ε ) dx + f ε (x, u 0 + u ε ) dx CN ε f ε (x, u 0 + u ε ) dx. (5.12)

19 Non-Periodic Homogenization of Strongly Anisotropic Functionals 1753 Let us set = CN ε, clearly ( S η ) Zε η < min{η, δ η } uniformly in ε; then, f ε (x, u 0 + u ε ) dx f ε (x, u 0 + v ε,η ) dx ( S η) Zε η = f ε (x, u 0 + v ε,η ) dx f ε (x, u 0 + v ε,η ) dx. (5.13) ( S η) ( S η) Z η ε We want to show that the second integral in the last line is small with respect to τ. We stress that the following chain of inequalities involves two weight estimates, requiring the use of our assumptions on the functionals, listed in the order they have to be used ((1.7), (5.10), (5.11), (5.9), (1.10), (5.7) (5.8)). Let L 0 = u 0 L (), we have: f ε (x, u 0 + v ε,η ) dx ( S η) Z η ε ( S η) Z η ε 2 p 1 L p 0 2 p 1 L p 0 Z η ε ( u 0 + v ε,η p Λ ε (x) dx Z η ε + 2 p 1 c(n) p Λ ε (x) dx + 2 p 1 (c(n)α η ) p ) 1 Λ 1+µ 1+µ ε (x) dx Z η ε µ 1+µ ( S η) Z η ε (M u ε ) p Λ ε (x) dx 2 p 1 L p 0 C 2 Z η ε µ 1+µ + 2 p 1 c(n) p ( S η) Z η ε ( S η) Z η ε Λ ε (x) dx (M u ε ) p Nλ ε (x) dx 2 p 1 L p 0 C 2η µ 1+µ + 2 p 1 c(n) p ηn. (5.14) Set γ η := 2 p 1 L p 0 C 2η µ 1+µ + 2 p 1 c(n) p ηn. For a fixed τ > 0 there exists η = η(τ) such that γ η < τ for η < η(τ). Since from now on η = η(τ) we will denote the quantities v ε,η, v η, α η, γ η, S η by v ε,τ, v τ, α τ, γ τ, S τ to stress their τ dependence. Clearly γ τ 0 as τ 0. For any measurable subset E of we have ( ) 1 Λ ε (x) dx Λ ε (x) 1+µ 1+µ µ µ dx E 1+µ C2 E 1+µ. E This inequality and (5.11) allows us to choose an open set τ such that S τ τ for which f ε (x, u 0 + v ε,τ ) dx f ε (x, u 0 + v ε,τ ) dx < τ ; (5.15) τ S τ it is easy to see that τ < 3τ.

20 1754 A. Baldi & M. C. Tesi Using (5.12) (5.15) we have f ε (x, u 0 + u ε ) dx f ε (x, u 0 + v ε,τ ) dx γ τ τ. τ We define β τ := γ τ + τ. Obviously β τ 0 as τ 0 and the thesis follows. Theorem 5.1. Let be an open set with Lipschitz boundary, compactly contained in a cube. Let f ε be the family of functions defined in (1.4) satisfying hypothesis (1.6). Let (λ ε ) and (Λ ε ) be as in (1.8) (1.10) respectively. Then for all u u 0 + H () with u 0 in Lip loc (R n ), it holds F Γ (u) = f (x, u) dx, (5.16) where F Γ is the Γ-limit of F ε and F ε are defined in (4.12). Proof. n the sequel we always argue up to subsequences. Let u 0 be in Lip loc (R n ). Let us prove that F Γ (u) f (x, u) dx (5.17) for all u u 0 + H (). As remarked above, F Γ is well defined for any u u 0 + consider u, u ε H ()( W 1,1 ()) such that u ε u in L 1 () and W 1,1 (). Hence if we lim sup F ε (u 0 + u ε ) F Γ (u 0 + u) <, (5.18) by definition (4.12) we deduce that u ε Lip 0 () for all ε and so F ε (u ε ) = f ε(x, u ε ) dx. Therefore F Γ (u 0 + u) lim inf f ε (x, (u 0 + u ε )) dx. (5.19) ε 0 By Proposition 5.1, for a fixed τ > 0, there exist τ, β τ > 0, and (v ε,τ ), v τ Lip loc (R n ) such as in Proposition 5.1, so that lim inf f ε (x, (u 0 + u ε )) dx lim inf f ε (x, (u 0 + v ε,τ )) dx β τ ε 0 ε 0 τ ˆF Γ (u 0 + v τ ) β τ = f (x, u 0 + v τ ) dx β τ, (5.20) τ where the functional ˆF Γ is now restricted to τ and the last equality follows by the representation given in (5.3). Let V τ be defined as in (5.6), by Proposition 5.1 we

21 Non-Periodic Homogenization of Strongly Anisotropic Functionals 1755 know that V τ < 2τ. Moreover f (x, u 0 + v τ ) dx β τ f (x, u 0 + v τ ) dx β τ τ τ V τ = f (x, u 0 + u) dx β τ. (5.21) τ V τ n conclusion, for all u H () such that (5.18) holds we find that F Γ (u 0 + u) f (x, u 0 + u) dx β τ. (5.22) τ V τ Since ( τ V τ ) 5τ we get (5.17) from (5.22) letting τ 0. Let us prove now that F Γ (u) F (u) in u 0 + H (). Let u u 0 + H () and u k u 0 +Lip 0 () such that [(u k u 0 ) (u u 0 )] 0 in H ()( W 1,1 ()). n particular u k u in L 1 () and u k u in L 1 (), thus, without loss of generality we may assume that u k (x) u(x) for a.e. x. By the convexity of f this implies f (x, u k (x)) f (x, u(x)) for k for a.e. x. By the very definition of f ε we have the estimate b 1 (x)ξ, ξ) p/2 f ε (x, ξ) b 2 (x)ξ, ξ) p/2. (5.23) Using (3.3) and (3.13), passing to Γ-limits we obtain b 1 Q(x, u) dx f (x, u) dx b 2 Q(x, u) dx (5.24) for every u Lip loc (R n ). Since for u Lip loc (R n ) both f (x, u) and Q(x, u) belong to L 1 loc (Rn ) (by (3.2) and (3.12)), passing to the Lebesgue points we obtain b 1 Q(x, u) f (x, u) b 2 Q(x, u) a.e. x R n (5.25) for every u Lip loc (R n ). On the other hand, recalling that [ ] is a norm on H (), we have Q(x, u k) dx Q(x, u). Hence, there exits a subsequence, still denoted by Q(x, u k ) and a function h L 1 () such that Q(x, u k ) h(x) a.e. in. By (5.25) we have f (x, u k (x)) b 2 Q(x, u k (x)) b 2 h(x). The dominated convergence theorem implies f (x, u) dx = lim f (x, u k ) dx. k By (5.1) F Γ (u k ) = f (x, u k ) dx. Eventually, using the lower semicontinuity property of the Γ-limit, we get

22 1756 A. Baldi & M. C. Tesi f (x, u) dx = lim f (x, u k ) dx k = lim k F Γ(u k ) F Γ (u), that implies together with (5.17) the desired result (5.16). Thanks to result above we can state the following theorem. Theorem 5.2. Let p > 1, a cube in R n and f ε be the family of functions defined in (1.4) satisfying hypothesis (1.6). Let (λ ε ) and (Λ ε ) be as in (1.8) (1.10) respectively, and satisfying (1.11). Then there exist a subfamily (f εk ) (f ε ) and two weights Λ L 1+µ loc (Rn ) and λ A p and a function f satisfying (3.6), such that for any open set with Lipschitz boundary, g L () and u 0 Lip loc (R n ) the sequence of solutions of the problem { min f ε (x, u) dx + } gu dx : u u 0 + H 1,p () converge in L 1 () to the solution of the problem { } min f (x, u) dx + gu dx : u u 0 + H () ; and the convergence of the minimum values holds. Proof. Using Theorem 5.1 the proof is the same as the one in Theorem 4.6 of Ref Examples and Counterexamples n this section we produce two examples concerning the relations between the spaces H and W used in the paper. Let be a bounded open set in R 2, with Lipschitz boundary, and p > 2. n the first example we provide weights such that H 1,p () W 1,p () but when we pass to the limit H () = W (). n the second one we consider weights such that H 1,p () = W 1,p () but H () W (). Example 6.1. Let 0 be the unit cube 0 = ] 1/2, 1/2[ ] 1/2, 1/2[ in R 2. n Ref. 9, Example 2.2, the authors construct an example of a weight λ such that H 1,p ( 0, λ) W 1,p ( 0, λ), where λ is defined as follows. Let p, α, β R, with p > 2 and 0 < α < β < 2(p 1). The authors define a suitable π-periodic smooth function k : R [α, β] (see Ref. 9 for details), and they define the weight λ : 0 [0, + [ as t is clear that λ(x) = { x k(arccos x 1 x ) if x 0, 0 if x = 0. (6.1) x β λ(x) x α for every x 0. (6.2)

23 Non-Periodic Homogenization of Strongly Anisotropic Functionals 1757 Then they define u : 0 R as 1 if x 1, x 2 > 0, 0 if x 1, x 2 < 0, x u(x) = 2 if x 1 < 0 < x 2, x (6.3) x 1 x if x 2 < 0 < x 1 and show that u W 1,p ( 0, λ) but u H 1,p ( 0, λ) for β > p 2 and 0 < α < β < 2(p 1). We now consider a smooth function ψ, with ψ 1 in ] 1/4, 1/4[ ] 1/4, 1/4[ and supp ψ 0, and we define U : 0 R as U = ψu. Arguing as in Ref. 9 with the same choice of α and β, we can show that U W 1,p ( 0, λ) but U H 1,p ( 0, λ). Let us take the matrix (x) := λ 2/p # ( x ε ), where the symbol # denotes the periodic extension of λ to the whole plane and is the unit matrix in R 2. With this definition and by (6.2), the assumption (1.5) becomes x ε 2β/p ξ 2 (x)ξ, ξ x ε 2α/p ξ 2 a.e. in 0. We now extend to the whole R 2 the functions x ε β and x ε α, by periodicity and we denote the periodic extension adding the symbol # to both functions. We choose the weights λ ε (x) = x ε β # and Λ ε(x) = x ε α #. By periodicity, it is easy to show that λ ε and Λ ε satisfy conditions (w 1 ) and (w 2 ) in the ntroduction. As for condition (w 3 ) to hold, we need further relation α > β 2. By a careful analysis of the proof given in Ref. 9 we can also show that for this choice of α and β we still have U W 1,p ( 0, λ) but U H 1,p ( 0, λ). Let = 0 for simplicity. We claim that the periodic function u ε (x) := U # ( x ε ) is in W 1,p () but not in H 1,p (). The first assertion is easy to be verified. ndeed, since supp U 0 then U # = ( U) # and since the number of ε-cells recovering is of the order 1/ε 2 and u W 1,p ( 0, λ), for every ε > 0 we have ( x ) (x) u ε, u ε p/2 = λ # u ε p dx ε = 1 ( x ) ( (ψu)# x ) p λ # dx ε ε ε = 1 ε 3 λ # (x) (ψu) # (x) p dx 1 ε 1 λ(x) (ψu)(x) p dx ε 1 ( ) λ(x) u p dx + λ(x) u p dx <. ε To show that u ε H 1,p (), we follows the same argument used in Ref. 9. By contradiction, suppose there exists a sequence (u ε ) k C 1 () which converges to u ε in

24 1758 A. Baldi & M. C. Tesi W 1,p (). n particular the convergence holds in the ε-cell ε = 1 ε (which can be supposed centered in the origin). n particular we have lim k ε λ # ( x ε ) (u ε) k u ε p dx = 0. Up to rescaling, we have in fact shown that U can be approximated by a C 1 -function in and we have a contradiction since u ε (x) = ( U) # ( x ε ) and U H 1,p (, λ). n addition, if we choose a cutoff function Φ with support contained in, and we consider ū ε := Φu ε, we see that ū ε W 1,p () but ū ε H 1,p (). We want to show that the two families of admissible weights describing the problem in this example induce, via a Γ-limit argument, H () = W (). ndeed, by Remark 3.1, there exist a function Q(x, ξ) such that λ (x) ξ p Q(x, ξ) Λ (x) ξ p for almost every x, for any ξ R n, where the two weights λ and Λ are weak limits of λ ε and Λ ε respectively. On the other hand, since the weights λ ε and Λ ε are periodic they converges to their averages (and hence to two positive constants a, b). By unicity of the limit, λ = a and Λ = b. Therefore the ellipticity condition a ξ p Q(x, ξ) b ξ p implies H () = W () = W 1,1 (). Example 6.2. Let = 0 and λ as in (6.1). Define λ ε = max{λ, ε}, clearly λ ε ε; moreover by (6.2) λ ε 1 in. Define (x) = λ ε (x) 2/p, since for any ε > 0 the ellipticity growth condition ε ξ p (x)ξ, ξ p/2 ξ p holds for a.e. x and every ξ R n, we have H 1,p () = W 1,p (). On the other hand, in Theorem 4.1 (where we take u 0 = 0) it is shown that, up to subsequences, [u] = Γ lim( u dx + ( (x) u, u p/2 dx) 1/p ) = ( u dx + ( Q(x, u) dx)1/p ) for u H (). By definition of λ ε, the sequence ( (x) u, u p/2 ) ε is non-increasing when ε 0, hence using the monotone convergence theorem, lim ε 0 ( (x) u, u p/2 dx) 1/p = ( u p λ dx) 1/p on Lip 0 (). Since ( u p λ dx) 1/p is L 1 ()-lower semicontinuous on Lip 0 (), by Theorem 5.7 and Proposition 6.1 in Ref. 14, the norm [ ] coincides with the norm W 1,p (,λ) in Lip 0 () and hence H () = H 1,p (, λ). Now take U as in Example 6.1, and recall that U H 1,p (, λ) but U W 1,p (, λ) W (). Hence H () W (). Acknowledgments t is a pleasure to thank M. Fabrizio for fruitful discussions and many valuable suggestions. nvestigation supported by University of Bologna. Funds for selected research topics. Partially supported by MUR, taly, and by GNAMPA og NDAM, taly. References 1. G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, Vol. 146 (Springer, 2002). 2. E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rational Mech. Anal. 86 (1984)

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