Andrea Braides, Anneliese Defranceschi and Enrico Vitali. Introduction

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1 HOMOGENIZATION OF FREE DISCONTINUITY PROBLEMS Andrea Braides, Anneliese Defrancesci and Enrico Vitali Introduction Following Griffit s teory, yperelastic brittle media subject to fracture can be modeled by te introduction, besides te elastic volume energy, of a surface term wic accounts for crack initiation. In its simplest formulation, te energy of a deformation u will be of te form (1) E(u, K) = f( u) dx + λh n 1 (K), Ω\K were u is te deformation gradient, Ω te reference configuration, and K is te crack surface. Te bulk energy density f accounts for elastic deformations outside te crack, wile λ is a constant given by Griffit s criterion for fracture initiation (see [49], [50], [54], [53], [14]). Te existence of equilibria, under appropriate boundary conditions, can be deduced from te study of minimum pairs (u, K) for te energy (1), and a description of crack growt can be obtained by a limit of successive minimizations at fixed time steps, as outlined in [36] (see also [27], and [40]). Te presence of two unknowns, te surface K and te deformation u, can be overcome by a weak formulation of te problem in spaces of discontinuous functions. Te space of special functions of bounded variation SBV (Ω; R m ) as been introduced by De Giorgi and Ambrosio [37] as te subset of R m -valued functions of bounded variation on te open set Ω R n, wose measure first derivative can be written in te form (2) Du = u L n Ω + (u + u ) ν u H n 1 S u, were u is now te approximate gradient of u, S u is te complement of te set of Lebesgue points of u, wic admits a unit normal ν u, and u +, u are te approximate values of u on bot sides of S u. Te measures L n and H n 1 are te n-dimensional Lebesgue measure and te (n 1)-dimensional Hausdorff measure, respectively. Te energy in (1) can be rewritten as (3) E(u) = Ω f( u) dx + λh n 1 (S u ), wic makes sense on SBV (Ω; R m ). If f is quasiconvex and satisfies some standard growt conditions, ten we can apply te direct metods of te calculus of variations to obtain minimum points for problems involving E, using Ambrosio s lower semicontinuity and compactness teorems (see [4] [7]). A complete regularity teory for minimum points u for E as not been developed yet, but in some cases it is possible to prove tat te jump set S u is H n 1 -equivalent to its closure (see [38], [31]) or even more regular (see [12], [11]), and tat u is smoot on Ω \ S u, tus obtaining minimizing pairs (u, K) = (u, S u ) for te functional E. 1

2 Te functionals F on SBV (Ω; R m ) wic ave bulk and surface parts, and satisfy te translation invariance condition F(u) = F(u + c) for all constant vectors c, can be written in te form (4) F(u) = f(x, u) dx + g(x, (u + u ) ν u ) dh n 1 Ω S u (we will adopt te equivalent notation g(x, u + u, ν u ) in te course of te paper). Necessary and sufficient conditions for te lower semicontinuity of suc functionals F are described in [6], [9], [7]. In te formulation (4) are included non-isotropic, nonomogeneous Griffit materials, wen (5) g(x, a ν) = g(x, ν), were te condition g(x, ν) = g(x, ν) must be imposed to ave a good definition of te surface integral. We can also include in tis setting fracture problems in te framework of Barenblatt s models, taking (6) g(x, a ν) = g( a ). We sall not treat directly Barenblatt materials, but we remark tat teir study can be carried on by a singular perturbation approac from te study of models of te type (5) (see [24]). Many oter problems in Matematical Pysics and Computer Vision involve minimum pairs wit a free discontinuity set K and an unknown function u as above (see, e.g., [52], [13], [2], [25], [8], [9], [32], [33]). We sall be content to interpret our results in te framework of nonlinear fracture mecanics. In tis paper we study te asymptotic beaviour of functionals of te type (4) modeling cellular elastic materials wit fine microstructure. Te study of tis kind of nonlinear media, but witout considering te possibility of fracture (i.e., in te framework of Sobolev functions), as been carried on by S. Müller [51] and A. Braides [16] (see also [17], [18], [19], [21], [26], [47]; a wide literature exists for te linear case, or wen u is scalar-valued; we refer te interested reader to te ric bibliograpy of [35]). Here we consider functionals (7) F ε (u) = f( x ε S, u) dx + g( x u ε, (u+ u ) ν u ) dh n 1, Ω were f and g are Borel functions, periodic in te first variable, wic model te response of te material to elastic deformation, and to fracture, respectively, at a microscopical scale (wic is given by te small parameter ε). Te beaviour of sequences of minima for problems involving F ε, and of te corresponding minimizers, can be deduced from te Γ-convergence of tis sequence (see [39], [35]). Tis analysis is usually referred to as omogenization. Te main result of tis paper is sowing tat, under te growt conditions (p > 1, α, β > 0) (8) α ξ p f(x, ξ) β(1 + ξ p ) α g(x, ξ) β for all x R n, ξ M m n, we obtain, in te limit wen ε 0, a minimum problem for te functional (9) F om (u) = f om ( u) dx + g om ((u + u ) ν u ) dh n 1. Ω S u 2

3 Te integrands f om and g om can be caracterized by some asymptotic formulas. Te omogenized bulk energy density is te same integrand as obtained in [16] in te case witout fracture: (10) f om (ξ) = lim T + inf { 1 T n } f(x, u + ξ) dx : u W 1,p ]0,T [ n 0 (]0, T [n ; R m ), wile te function g om is given on rank-one matrices by 1 g om (z ν) = lim inf{ g(x, (u (11) T + T n 1 + u ) ν u ) dh n 1 : T Q ν S u u SBV (T Q ν ; R m ), u = 0 a.e., u = u z,ν on (T Q ν ) }, were Q ν ν, and is any unit cube in R n wit centre in te origin and one face ortogonal to (12) u z,ν (x) = { z if x, ν 0 0 if x, ν < 0. Note tat by (9) it is sufficient to define g om on rank-one matrices. From (9) (11) we obtain tat te overall beaviour of te medium described by (7) at te scale ε is te one of a omogeneous material wose bulk elastic response is given by te study of F ε only on elastic deformations witout cracks, and wose response to fracture can be derived by te examination of stiff deformations (i.e., were u = 0). In particular, note tat te omogenized surface energy density is not influenced by f ; tis penomenon is particular of te process of omogenization, since in general we do ave an interaction (see [3] Teorem 4.1). We mention also tat te omogenization under SBV-growt conditions (8) gives rise to a different type of penomena tan wen a growt of order one is allowed; i.e., (13) f(x, ξ) γ ξ or g(x, ξ) γ ξ (e.g., if g(x, ) is positively omogeneous of degree one), in wic case te omogenized functional is defined and finite on te wole BV (Ω; R m ) (see [19]). Te paper is organized as follows. In Section 1 we recall te main definitions and preliminaries on SBV functions, and we introduce te space SBV p (Ω; R m ) of SBV - functions wose approximate gradient is p-summable and wose jump set is H n 1 -finite. Section 2 is devoted to te statement of te omogenization result. In Sections 2 7 we deal wit functionals as in (7), wit g satisfying te tecnical assumption (14) α(1 + ξ ) g(x, ξ) β(1 + ξ ), wic allows us to limit our analysis to SBV p (Ω; R m ). Te treatment of te case wit g satisfying te growt condition (8) is carried on in Section 8 by a singular perturbation approac. Te proof of te omogenization teorem relies on several tecnical results. In Section 3 we give a compactness teorem wit respect to Γ-convergence for functionals defined in SBV p (Ω; R m ). Its proof is based on a fundamental estimate (Proposition 3.1), wic allows te application of te localization tecniques of Γ-convergence (see [35]). We also prove a truncation lemma (Lemma 3.5), wic, in several cases, permits to deal wit equibounded sequences. In Section 4 we apply te tecniques of Buttazzo and Dal Maso [29] and of Ambrosio and Braides [8] to give an integral representation on 3

4 W 1,p (Ω; R m ) and on spaces of partitions BV (Ω; T ) (T R m is any fixed finite set) of te functionals given by te compactness argument of Section 3. Te caracterization by formula (10) of te volume energy density wic describes te integral representation on W 1,p (Ω; R m ) is obtained in Section 5. In order to use te omogenization results of [16] and [51], we apply a tecnique introduced by Ambrosio (see [7]), wic allows to pass from sequences of SBV -functions wit vanising surface energy to sequences of Lipscitz functions in te description of te process of Γ-limit. Te construction of minimizing sequences wit surface energy tending to 0 is obtained by a scaling argument, wic is based on te periodic structure of te problem. A similar procedure leads in Section 6 to te caracterization by formula (11) of te omogenized surface energy density: after a scaling argument, wic again is possible by te periodicity assumptions, we can pass from sequences in SBV p (Ω; R m ) wit vanising bulk energy to sequences wit u = 0. Tis passage is carried on by a careful use of te coarea formula. In Section 7 we prove te integral representation (9) on SBV p (Ω; R m ) L (Ω; R m ), from wic te general result follows by approximation. Te two key points are te application of te strong convergence results in SBV p (Ω; R m ) of piecewise smoot functions proven by Braides and Ciadò Piat [23], wic gives an inequality in (9) (Proposition 7.1), and, for te opposite inequality, a blow-up argument wic locally reduces te problem to te case of linear or piecewise constant functions. Te caracterization of te Γ-limits troug formulas (10) and (11), togeter wit te compactness argument of Section 2 conclude te proof. Finally in Section 8 we describe te applications of te omogenization teorem to problems in fracture mecanics. 1. Notation and preliminaries Let m 1 and n 1 be fixed integers. If Ω is an open subset of R n we denote by A(Ω) and B(Ω) te families of te open and Borel subsets of Ω, respectively; moreover, we set A = A(R n ) and B = B(R n ), wile A 0 stands for te family of te bounded open subsets of R n. If x, y R n ten x, y denotes teir scalar product; B ρ (x) is te open ball wit centre x and radius ρ, and S n 1 te surface of te unit ball B 1 (0); M m n is te space of te m n real matrices. Te usual product of a matrix ξ M m n and a vector x R n is denoted by ξ x. Te Lebesgue measure and te (n 1)-dimensional Hausdorff measure in R n are denoted by L n and H n 1, respectively, but we also write E in place of L n (E). Moreover, ω n = B 1 (0). If Ω A, we use standard notation for te Lebesgue and Sobolev spaces L p (Ω; R m ) and W 1,p (Ω; R m ). Functions of bounded variation. For te general teory of te functions of bounded variation we refer to [43], [48], [42] and [56]; ere we just recall some definitions and results we sall use in te sequel. Let Ω be an open subset of R n and u: Ω R m be a Borel function. We say tat z R m is te approximate limit of u in x and we write z = ap- lim y x u(y) if for every ε > 0 lim ρ 0 ρ n {y B ρ (x) Ω : u(y) z > ε} = 0. We define S u as te subset of Ω were te approximate limit of u does not exist. It turns out tat S u is a Borel set, S u = 0 and u is approximately continuous a.e. in Ω; more precisely, u(x) = ap- lim y x u(y) for a.e. x Ω \ S u. 4

5 We say tat u = (u 1,..., u m ) L 1 (Ω; R m ) is a function of bounded variation if its distributional first derivatives D i u j are (Radon) measures wit finite total variation in Ω. Tis space will be denoted by BV (Ω; R m ). We sall use Du to indicate te matrix-valued measure wose entries are D i u j. If u BV (Ω; R m ), ten S u is countably (n 1)-rectifiable, i.e. (1.1) S u = N ( ) K i, were H n 1 (N) = 0 and (K i ) is a sequence of compact sets, eac contained in a C 1 ypersurface Γ i. Moreover, tere exist Borel functions ν u : S u S n 1 and u +, u : S u R m suc tat for H n 1 -a.e. x S u i N lim ρ 0 B ρ n u(y) u + (x) dy = 0, ρ + (x) Ω lim ρ n u(y) u ρ 0 B (x) dy = 0, ρ (x) Ω were B + ρ (x) = {y B ρ (x) : y x, ν u (x) > 0} and B ρ (x) = {y B ρ (x) : y x, ν u (x) < 0}. Hence, for H n 1 -a.e. x S u lim ρ 0 ρ n {y B ρ (x) Ω : y x, ±ν u (x) > 0, u(y) u ± (x) > ε} = 0 for every ε > 0. Te triplet (u + (x), u (x), ν u (x)) is uniquely determined up to a cange of sign of ν u (x) and an intercange between u + (x) and u (x). Te vector ν u is normal to S u, in te sense tat, if S u is represented as in (1.1) ten ν u (x) is normal to Γ i for H n 1 -a.e. x K i. In particular, it follows tat ν u (x) = ±ν v (x) for H n 1 -a.e. x S u S v and u, v BV (Ω; R m ). If x / S u we define u + (x) = u (x) = ap- lim y x u(y). We denote by u te density of te absolutely continuous part of Du wit respect to te Lebesgue measure. u(x) turns out to be te approximate differential of u at x for a.e. x Ω, in te sense tat u(y) u(x) u(x) (y x) lim ρ 0 B ρ n dy = 0. ρ (x) Ω y x We point out tat if u, v BV (Ω; R m ) ten u(x) = v(x) for a.e. x Ω suc tat u(x) = v(x). It is easy to verify tat if u, v BV (Ω; R m ) and ϕ is a smoot real function on Ω, ten (u + v) ± = u ± + v ±, (ϕu) ± = ϕu ±, u ± v ± u v L (Ω;R m ) and (ϕu) = u ϕ + ϕ u. We say tat a function u BV (Ω; R m ) is a special function of bounded variation if te singular part of Du is given by (u + u ) ν u H n 1 S u, i.e. Du = u L n + (u + u ) ν u H n 1 S u. We denote te space of te special functions of bounded variation by SBV (Ω; R m ). Te introduction of tis space is due to De Giorgi & Ambrosio [37]. For te properties of te functions u SBV (Ω; R m ) we refer to [5] and [6]. Here we mention te following result (see [10]): if u SBV (Ω; R m ) and ϕ: R m R m is a Lipscitz function wit Lipscitz constant L, ten ϕ(u) SBV (Ω; R m ), S ϕ(u) S u, (ϕ(u)) ± = ϕ(u ± ), and ϕ(u) L u a.e. in Ω. 5

6 Let p > 1; te space SBV p (Ω; R m ) is defined as te space of te functions u SBV (Ω; R m ) suc tat H n 1 (S u Ω) < + and u L p (Ω; M m n ). Sets of finite perimeter. Let Ω A and E B. We say tat E as finite perimeter in Ω if te caracteristic function χ E of E belongs to BV (Ω; R). Define te essential boundary of E as E = {x R n : lim sup ρ 0 If E is a set of finite perimeter in Ω, ten ρ n B ρ (x) E > 0 and lim sup ρ n B ρ (x) \ E > 0}. ρ 0 Ω Dχ E = H n 1 (Ω E) ; tis value is te perimeter of E in Ω. If u BV (Ω; R), ten {x Ω : u(x) > t} as finite perimeter in Ω for a.e. t R, and te following Fleming & Risel coarea formula olds: (1.2) B Du = + H n 1 (B {x Ω : u(x) > t}) dt for every B B(Ω). For an exposition of te teory of sets of finite perimeter see te books quoted above for te functions of bounded variation. Approximation of BV functions by Lipscitz functions. Let µ be a non-negative finite Radon measure on Y =]0, 1[ n. For every x Y let us define { µ(bρ (x)) M(µ)(x) = sup B ρ (x) : ρ > 0 suc tat B ρ (x) Y M(µ) is called te (local) maximal function of µ. If µ is absolutely continuous wit respect to te Lebesgue measure and is its density, we also set M() = M(µ). In [7] M(µ) is defined wit respect to te unit ball B 1 (0) instead of Y. However, it is easy to see tat te analogous of Proposition 2.2 and Teorem 2.3 in [7] still old, as in te following two statements. }. Proposition 1.1. Let µ be as above. Ten tere exists a constant c(n) > 0 suc tat {x Y : M(µ)(x) > λ} c(n) µ(y ) λ for every λ > 0. Moreover, if µ is absolutely continuous wit respect to te Lebesgue measure and its density belongs to L p (Y ) for some p > 1, ten wit C(n, p) = p 2 p c(n)/(p 1). Y (M(µ)(x)) p dx C(n, p) ((x)) p dx, Y 6

7 Teorem 1.2. Let λ > 0, u BV (Y ; R m ) L (Y ; R m ), and let E = {x Y : M( Du )(x) > λ}. Ten, for every 0 < ε < 1 we can find a Lipscitz function v : Y ε R m, were Y ε =]ε, 1 ε[ n, suc tat u = v a.e. on Y ε \ E, and te Lipscitz constant Lip(v, Y ε ) of v on Y ε satisfies te inequality for a suitable positive constant c (n). ( Lip(v, Y ε ) m c (n)λ + 2 ) ε u L (Y ;R m ), Γ-convergence. We recall briefly te notion of Γ-convergence ([39]). Let (X, d) be a metric space and let F : X R be a sequence of functionals on X, and let F : X R. We say tat (F ) Γ-converges to F at te point x X wit respect to te topology induced by d, if te following conditions are satisfied: (i) for every sequence (x ) in X suc tat d(x, x) 0, we ave F (x) lim inf (x ); + (ii) tere exists a sequence (x ) in X suc tat d(x, x) 0 and F (x) = lim (x ). + We say tat (F ) Γ-converges to F on te space X wit respect to te topology induced by d if (i) and (ii) old for every x X. In tis case F is called te Γ-limit of (F ), and we write F = Γ- lim. + For a complete treatment of te subject we refer to [35]. Here we only recall te following facts. If (F ) Γ-converges to F ten F is lower semicontinuous. If (F ) is a constant sequence, i.e. if F is equal to te same functional G for every N, ten te Γ-limit exists and coincides wit te lower semicontinuous envelope (or relaxed functional) G of G on te space X wit respect to te topology induced by d (see [28]). Under suitable coercivity conditions, Γ-convergence guarantees te convergence of te minimum values of te functionals F to te minimum value of teir Γ-limit. Quasiconvexity. We finally recall tat a continuous function f : M m n R is quasiconvex if for every open set Ω and ξ M m n we ave Ω f(ξ) f(ξ + Du) dx for Ω all u C0 1(Ω; Rm ). Quasiconvexity is a well-known necessary and sufficient condition for te weak lower semicontinuity of integral functionals defined on Sobolev spaces (see [15], [1], [34], [28]). 2. Statement of te main result Let f : R n M m n [0, + [ and g : R n R m S n 1 [0, + [ be two Borel functions. We suppose tat f satisfies i) for every ξ M m n te function f(, ξ) is 1-periodic, i.e., f(x + e i, ξ) = f(x, ξ) for every i = 1,..., n and x R n ; ii) tere exist two constants c 1 > 0 and c 2 > 0 suc tat c 1 ξ p f(x, ξ) c 2 (1 + ξ p ) for a.e. x R n and for every ξ M m n, and g satisfies 7

8 i) g(x, s, ν) = g(x, s, ν) for every (x, s, ν) R n R m S n 1 ; ii) g(, s, ν) is 1-periodic for every (s, ν) R m S n 1 ; iii) tere exist a function ω : [0, + [ [0, + [, continuous and non decreasing wit ω(0) = 0, and a constant L > 0, suc tat ω(t) Lt for t 1 and g(x, s, ν) g(x, t, ν) ω( s t ) for every x R n, s, t R m, ν S n 1 ; iv) tere exist two constants c 3 > 0 and c 4 > 0 suc tat c 3 (1 + s ) g(x, s, ν) c 4 (1 + s ) for every (x, s, ν) R n R m S n 1. For every ε > 0, A A, u SBV loc (A; R m ) and B B(A) we define (2.1) F ε (u, B) = f( x ε, u) dx + g( x ε, u+ u, ν u ) dh n 1. B S u B We remark tat tere exists a one-to-one correspondence between (R m \ {0}) S n 1 modulo te equivalence relation (s, ν) ( s, ν) and te space of matrices of rank equal to 1. Hence we could as well write te functional in (2.1) in te form F ε (u, B) = f( x ε, u) dx + g( x ε, (u+ u ) ν u ) dh n 1, B S u B wit te identification g(x, s ν) = g(x, s, ν), to ave a symmetric notation in te two integrals. However, in te sequel we sall always use te notation (2.1) to igligt te different beaviour of te surface integral wit respect to u + u and ν u. Te following propositions introduce te functions f om and g om wic will appear in te integral representation of te limit functional of te family (F ε ) ε>0. Proposition 2.1. For every ξ M m n tere exists f om (ξ) = lim inf { f( x 1,p, u + ξ) dx : u W 0 ε 0 ]0,1[ n ε (]0, 1[n ; R m ) }. Te function f om is quasiconvex, and for every ξ R m c 1 ξ p f om (ξ) c 2 (1 + ξ p ). Moreover, for every infinitesimal sequence (ε ) of positive numbers and for every Ω A 0 te sequence u Ω f( x ε, u) dx Γ-converges to te functional u Ω f om( u) dx on W 1,p (Ω; R m ) wit respect to te L p -topology. Proof. For te proof see [16] and [17] (Teorem 2.3 and te subsequent remark, Proposition 1.8 and Remark 1.7). Proposition 2.2. For every (z, ν) R m S n 1 tere exists g om (z, ν) = lim ε 0 inf { S u Q ν g( x ε, u+ u, ν u ) dh n 1 : u SBV (Q ν ; R m ), u = 0 a.e., u = u z,ν on Q ν }, 8

9 were Q ν is any unit cube in R n wit centre in te origin and one face ortogonal to ν (te limit being independent of suc a coice), and { z if x, ν 0 u z,ν (x) = 0 if x, ν < 0. Te function g om is continuous on (R m \ {0}) S n 1, and c 3 (1 + z ) g om (z, ν) c 4 (1 + z ). for every (z, ν) (R m \ {0}) S n 1 Te proof of Proposition 2.2 is postponed to Section 6. For every A A, u SBV loc (A; R m ) and B B(A) we define F om (u, B) = f om ( u) dx + g om (u + u, ν u ) dh n 1. B S u B Te main result of te paper is te following omogenization teorem. Teorem 2.3. Let (F ε ) ε>0 and F om be as above. Let (ε ) be an infinitesimal sequence of positive real numbers. Ten for every A A 0 F om (, A) = Γ- lim F ε (, A) + on te space SBV p (A; R m ) wit respect to te L 1 (A; R m )-topology, and on te space SBV p (A; R m ) L p (A; R m ) wit respect to te L p (A; R m )-topology. In te case wen f and g are constant wit respect to te first variable, we immediately obtain te following relaxation result (see also [44]). Corollary 2.4. Let Ω A 0 and F (u) = f( u) dx + Ω S u Ω g(u + u, ν u )H n 1 for u SBV p (Ω; R m ). Ten te lower semicontinuous envelope of F on SBV p (Ω; R m ) wit respect to te L 1 (Ω; R m )-topology (or on SBV p (Ω; R m ) L p (Ω; R m ) wit respect to te L p (Ω; R m )-topology) is given by F (u) = f( u) dx + g(u + u, ν u )H n 1, were Ω S u Ω f(ξ) = inf { ]0,1[ n f( u + ξ) dx : u W 1,p 0 (]0, 1[n ; R m ) }, i.e., f is te quasiconvex envelope of f (see [34]), and g(z, ν) = inf { S u Q ν g(u + u, ν u ) dh n 1 : u SBV (Q ν ; R m ), u = 0 a.e., u = u z,ν on Q ν }, for every ξ M m n and (z, ν) R m S n 1, wit Q ν and u z,ν as in Proposition

10 3. A compactness result on SBV p (Ω; R m ) In tis section we prove some general properties for functionals of te form F ε (u, A) = f ε (x, u) dx + g ε (x, u + u, ν u ) dh n 1, A S u A were f ε : R n M m n [0, + [ and g ε : R n R m S n 1 [0, + [ are Borel functions satisfying c 1 ξ p f ε (x, ξ) c 2 (1 + ξ p ), c 3 (1 + s ) g ε (x, s, ν) c 4 (1 + s ), for a.e. x R n and for every ξ M m n for every (x, s, ν) R n R m S n 1 for suitable positive constants c i. Moreover, we suppose tat g ε (x, s, ν) = g ε (x, s, ν) for every (x, s, ν) R n R m S n 1. In particular we can ave f ε (x, ξ) = f( x ε, ξ) and g ε (x, s, ν) = g( x ε, s, ν), were f and g are te functions introduced in Section 2. For every A A, u SBV loc (A; R m ), and B B(A) let (3.1) H(u, B) = u p dx + (1 + u + u ) dh n 1. B S u B Te functional H(, A) is lower semicontinuous on SBV loc (A; R m ) wit respect to te L 1 loc (A; Rm )-topology: see [6] Teorems 2.2 and 3.7 or [7] Teorem 4.5 and Remark 4.6. In view of te growt conditions satisfied by f ε and g ε, tere exist γ 1, γ 2 > 0 suc tat for every ε > 0, (3.2) γ 1 H(u, A) F ε (u, A) γ 2 (H(u, A) + A ). Hence, for every A A 0 te Γ-limit of any sequence (F ε (, A)) (ε 0) on a subspace of SBV loc (A; R m ) wit respect to a topology stronger tan L 1 loc (A; Rm ) is finite exactly on SBV p (A; R m ). Tus in te sequel we sall restrict ourselves to te space SBV p (A; R m ). Te crucial properties of te Γ-limit are based on te so-called fundamental estimate ([35]), of wic we give now an SBV -version. Proposition 3.1. (fundamental estimate) Let (F ε ) be te family of functionals defined in (2.1). For every η > 0 and for every A, A, B A, wit A A, tere exists a constant M > 0 wit te following property: for every ε > 0 and for every u SBV p (A ; R m ), v SBV p (B; R m ) tere exists a function ϕ C0 (A ) wit ϕ = 1 in a neigbourood of A and 0 ϕ 1 suc tat F ε (ϕu + (1 ϕ)v, A B) (1 + η) [ F ε (u, A ) + F ε (v, B) ] + M u v p L p (S;R m ) + η, were S = (A \ A ) B. Remark 3.2. From te proof it follows tat te cut-off function ϕ can be cosen in a finite family depending only on η, A, and A. Proof. Let η > 0, A, A and B be fixed as in te statement. Let k N satisfy (3.3) 1 k max( 2 p 1 c 2 c 1, c 4 c 3, c 2 (A \ A ) B ) η. 10

11 Let A 1,..., A k+1 be open subsets of R n suc tat A A 1 A 2 A k+1 A. For every i = 1,..., k let ϕ i be a function in C 0 (A i+1) wit ϕ i = 1 on a neigbourood V i of A i. Define M = 2 p 1 c 2 k max ϕ i p L. 1 i k Fixed ε > 0, u SBV p (A ), and v SBV p (B), define on A B te function w i = ϕ i u + (1 ϕ i )v (were u and v are extended arbitrarily outside A and B, respectively). Ten for i = 1,..., k (3.4) F ε (w i, A B) F ε (u, (A B) V i ) + F ε (v, B \ sptϕ i ) + F ε (w i, B (A i+1 \ A i )) F ε (u, A ) + F ε (v, B) + F ε (w i, B (A i+1 \ A i )). Set T i = B (A i+1 \ A i ), we estimate te last term: F ε (w i,t i ) c 2 (1 + w i T p ) dx + c 4 (1 + w + i w i ) dhn 1 i S wi T i ) c 2 ( T i + ϕ i u + (1 ϕ i ) v + ( ϕ i )(u v) p dx T i ( + c 4 (1 + u + u ) dh n 1 + (1 + v + v ) dh n 1 (S u \S v ) T i (S v \S u ) T i ) + (1 + ϕ i (u + u ) + (1 ϕ i )(v + v ) ) dh n 1 S u S v T i ( c 2 T i + 2 T p 1 ( u p + v p + ϕ i p u v p ) dx i ( + c 4 (1 + u + u ) dh n 1 + S u T i ) (1 + v + v ) dh n 1 S v T i 2 c ( p 1 2 f( x, u) dx + f( x ), v) dx c 1 T i ε T i ε + c ( 4 g( x c 3 S u T i ε, u+ u, ν u ) dh n 1 + g( x ) S v T i ε, v+ v, ν v ) dh n 1 + c 2 ( T i + 2 p 1( ) ) p ϕ i L u v Lp (T i ) γ(f ε (u, T i ) + F ε (v, T i )) + c 2 ( T i + 2 p 1( max 1 i k ϕ i p L ) u v p L p (T i ) were γ = max(2 p 1 c 2 c 1, c 4 c 3 ). Hence tere exists i 0 {1,..., k} suc tat F ε (w i0, T i0 ) 1 k γ k k F ε (w i, T i ) i=0 were S = (A \ A ) B. From (3.4) it follows tat ( Fε (u, A ) + F ε (v, B) ) + c 2 k S + M u v p L p (S;R m ) F ε (w i0, A B) (1 + γ k )( F ε (u, A ) + F ε (v, B) ) + c 2 k S + M u v p L p (S;R m ). 11 ) ),

12 By (3.3) te proof of Proposition 3.1 is accomplised. Proposition 3.3. Let (ε ) be an infinitesimal sequence of positive numbers. Ten tere exist a subsequence (ε σ() ) of (ε ) and a functional F 0 defined on te set {(u, A) : A A, u SBV p (A; R m ) L p (A; R m )} and wit values in [0, + ] suc tat for every A A 0 F 0 (, A) = Γ- lim + F ε σ() (, A) on te space SBV p (A; R m ) L p (A; R m ) endowed wit te L p (A; R m )-topology. Moreover, for every Ω A 0 and u SBV p (Ω; R m ) L p (Ω; R m ) te set function F 0 (u, ) is te restriction to A(Ω) of a Borel measure on Ω. Proof. For every ε > 0 let G ε : L p (R n ; R m ) A [0, + ] be defined by { Fε (u, A), if u A SBV p (A; R G ε (u, A) = m ) +, oterwise. By Teorem 16.9 in [35] tere exist a subsequence (ε σ() ) of (ε ) and a functional G 0 : L p (R n ; R m ) A [0, + ] suc tat G 0 = Γ(L p (R n ; R m ))- lim G ε σ(). Adopting + te te notation of [35], wom we refer to for details and precise definitions, tis means tat G 0 is te inner regular envelope of bot te Γ-lower and te Γ-upper limit of te sequence (G εσ() ). By (3.2) γ 1 H(u, A) G ε (u, A) γ 2 ( H(u, A) + A ) for every ε > 0, A A and u L p (R n ; R m ) wit u A SBV p (A; R m ). Taking into account Proposition 3.1 we can apply te same metod of proof as in Teorem 18.7 in [35]. Tus we obtain tat for every A A 0 te sequence (G εσ() (, A)) of functionals on L p (R n ; R m ) Γ-converges to G 0 (, A) wit respect to te L p (R n ; R m )-topology at all points u L p (R n ; R m ) wit u A SBV p (A; R m ). For every A A and u SBV p (A; R m ) L p (A; R m ) define F 0 (u, A) = G 0 (ũ, A), were ũ is any L p -extension of u to R n. Tis definition is well posed since from te Γ- convergence of (G εσ() ) to G 0 it follows tat for every u, v L p (R n ; R m ), if u A = v A ten G 0 (u, A) = G 0 (v, A). Now te stated convergence of F εσ() (, A) is easily proved. Observe now tat G ε (u, ) is te restriction to A of a Borel measure on R n for every u. Ten, by Proposition 3.1 and by Teorem 18.5 in [35] (wic olds wit te same proof also for vector valued L p -functions), for every u L p (R n ; R m ) te set function G 0 (u, ) is te restriction to A of a Borel measure on R n. From tat we obtain te stated measure property of F 0. We prove now some furter properties of te Γ-limit F 0. Proposition 3.4. Let (ε ) be an infinitesimal sequence of positive numbers, and A A 0 suc tat te limit F 0 (, A) = Γ- lim F ε (, A) exists on te space SBV p (A; R m ) + L p (A; R m ) endowed wit te L p (A; R m )-topology. Ten for every sequence (u ) in SBV p (A; R m ) converging in L 1 (A; R m ) to a function u SBV p (A; R m ) L (A; R m ) we ave F 0 (u, A) lim inf F ε (u, A). + For te proof we need a tecnical lemma (see also [30]). 12

13 Lemma 3.5. Let A A 0 and let (u ) be a sequence in SBV p (A; R m ) wic is bounded in L 1 (A; R m ) and suc tat (H(u, A)) is bounded. Ten for every η > 0, M 0 > 0 and N tere exists a Lipscitz function ϕ : R m R m wit Lipscitz constant less tan or equal to 1 satisfying te property { y if y a ϕ (y) = 0 if y > b. for suitable constants a,b R wit M 0 a < b, suc tat F ε (ϕ (u ), A) F ε (u, A) + η for every N. Te function ϕ can be cosen in a finite family independent of. Proof. Fix η > 0, M 0 > 0. Let (a j ) be a strictly increasing sequence of positive real numbers suc tat for every j N tere exists a Lipscitz function ϕ j : R m R m wit Lipscitz constant less tan or equal to 1 satisfying { y if y aj ϕ j (y) = 0 if y > a j+1. Te sequence (a j ) will be determined subsequently in a suitable way (see (3.6)) and will depend only on η and M 0. For every N and j N let w j = ϕ j(u ). Consider te volume part of F ε (w j, A); we ave f ε (x, w j ) dx = f ε (x, u ) dx + f ε (x, 0) dx A + A { u a j } A {a j < u a j+1 } f ε (x, w j ) dx A { u >a j+1 } f ε (x, u ) dx + c 2 A { u > a j+1 } A + c 2 (1 + u p ) dx. A {a j < u a j+1 } As for te surface part, it is not restrictive to assume tat u u+ Hn 1 -a.e. on S u. Since (w j )± = ϕ j (u ± ) we ave S j A w g ε (x,(w j )+ (w j ), ν w j ) dh n 1 (S u \{a j+1 u }) A g ε (x, ϕ j (u + ) ϕ j(u ), ν u ) dh n 1. 5 Te set S u \ {a j+1 u } can be decomposed as S j i, were i=1 S j 1 = { u+ < a j}, S j 2 = { u < a j, a j+1 u + }, S j 3 = { u < a j u + < a j+1}, S j 4 = {a j u, u+ < a j+1}, S j 5 = {a j u < a j+1 u + }. 13

14 Hence, taking into account te Lipscitz continuity of ϕ j, we ave g ε (x,(w j )+ (w j ), ν w j ) dh n 1 S j A w g ε (x, u + S j 1 A u, ν u ) dh n 1 + c 4 (1 + u S j 2 A 5 + c 4 (1 + u S + ji A u ) dhn 1. i=3 ) dhn 1 We now use tese inequalities to estimate 1 N N j=1 F ε (w j, A), for every fixed N and N N. Note tat eac of te families ({a j < u a j+1 }) j N, (S j i ) j N(i = 3, 4, 5) consists of pairwise disjoint sets. Ten (3.5) 1 N N F ε (w j, A) F ε (u, A) j=1 + 1 N + 1 N N ( c2 A { u > a j+1 } + c 4 (1 + u S j2 A j=1 ( c2 A (1 + u p ) dx + 3c 4 S u A ) dhn 1) (1 + u + u ) dhn 1). By assumption tere exists a constant c > 0 suc tat c 2 (1 + u p ) dx + 3c 4 (1 + u + u ) dhn 1 c A S u A for every N. Coose N N suc tat c/n η/3. Moreover, we ave (we can suppose c 4 1) c u + S j 2 A u dhn 1 ( u + S j 2 A u ) dhn 1 wence S j 2 A (1 + u ) dhn 1 c (a j+1 a j ) H n 1 (S j 2 A), 1 + a j a j+1 a j. Te sequence (a j ) is now defined recursively by te following conditions (3.6) c 2 A { u > a 1 } η/3 for every N, a 1 M a j c 4 c η/3 for every j N, a j+1 a j wic is possible by te assumed boundedness of (u ) in L 1 (A; R m ). From (3.5) we now obtain 1 N F ε (w j N, A) F ε (u, A) + η. j=1 14

15 Terefore for every N tere exists j() {1,..., N} suc tat F ε (w j(), A) F ε (u, A) + η. Te function ϕ = ϕ j() is te Lipscitz function we were looking for. Note tat N is independent of. Remark 3.6. From its proof it follows immediately tat te previous lemma still olds for te functionals of te type F ε (u, A) = A f ε (x, u) dx or F ε (u, A) = S u A g ε (x, u + u, ν u ) dh n 1, wit f ε and g ε as above. Proof of Proposition 3.4. We can assume tat (F ε (u, A)) converges to a finite value. Fix η > 0. By applying Lemma 3.5 to te sequence (u ) wit M 0 = u, we obtain a sequence (v ) in SBV p (A; R m ) L (A; R m ), bounded in L (A; R m ), suc tat v u in L p (A; R m ) and lim inf F ε (v, A) lim inf F ε (u, A) + η. By te + + Γ-convergence of (F ε ) we ave F 0 (u, A) lim inf F ε τ() (v, A). Te arbitrariness of η + concludes te proof. In te following lemma we assume f ε (x, ξ) = f( x ε, ξ) and g ε (x, s, ν) = g( x ε, s, ν) were f and g are te functions introduced in Section 2. Lemma 3.7. Let (ε ) be an infinitesimal sequence of positive numbers suc tat for every A A 0 te limit F 0 (, A) = Γ- lim F ε (, A) exists on SBV p (A; R m ) L p (A; R m ) + wit respect to te L p (A; R m )-topology. Ten for every u SBV p (A; R m ) L p (A; R m ), a R m and y R n we ave (i) F 0 (u + a, A) = F 0 (u, A), (ii) F 0 (τ y u, τ y A) = F 0 (u, A), were (τ y u)(x) = u(x y) and τ y A = A + y. Proof. First we prove (i). Let (u ) in SBV p (A; R m ) L p (A; R m ) be a sequence converging to u in L p (A; R m ) and suc tat (F ε (u, A)) converges to F 0 (u, A). Ten (u + a) converges to u + a in L p (A; R m ) and F 0 (u + a, A) lim inf + F ε (u + a, A) = lim inf + F ε (u, A) = F 0 (u, A). On te oter and, F 0 (u, A) = F 0 ((u + a) + ( a), A) F 0 (u + a, A). We prove now (ii). Tere exists a sequence (z ) in Z n suc tat y = ε z converges to y. Let (u ) be a sequence as in te proof of (i). Set v = τ y u : A + y R m ; by taking te periodicity assumptions on f and g into account we get F ε (u, A) = A f( x + y ε, u ) dx + = f( x, v ) dx + A+y ε S u A S v (A+y ) g( x + y, u + ε u, ν u ) dh n 1 g( x ε, v + v, ν v ) dh n 1. Let B A; for sufficiently large we may assume A + y B + y ; ence F ε (u, A) f( x, v ) dx + g( x, v + ε ε v, ν v ) dh n 1, B+y S v (B+y) wic yields F 0 (u, A) F 0 (τ y u, B+y), since (v ) converges to τ y u, By te arbitrariness of B A we ave also F 0 (u, A) F 0 (τ y u, τ y A). We conclude te proof of (ii) by noticing tat F 0 (τ y u, τ y A) F 0 (τ y (τ y u), τ y (τ y A)) = F 0 (u, A). 15

16 4. Integral representation on W 1,p (Ω; R m ) and on partitions On account of Proposition 3.3 we sall try to identify te Γ-limits of convergent sequences of functionals F ε. Terefore, up to Section 7 we sall assume tat an infinitesimal sequence (ε ) of positive numbers is given, suc tat for every A A 0 te limit (4.1) F 0 (, A) = Γ- lim + F ε (, A) exists on te space SBV p (A; R m ) L p (A; R m ) endowed wit te L p (A; R m )-topology. In particular F 0 (, A) is lower semicontinuous wit respect to te L p (A; R m )-topology. As we ave seen, for every Ω A 0 and u SBV p (Ω; R m ) L p (Ω; R m ) te set function F 0 (u, ) can be extended to a Borel measure on Ω. Suc a measure is given by (see [35] Teorem 14.23) (4.2) F 0 (u, B) = inf{f 0 (u, A) : A A(Ω), B A} for every B B(Ω). Moreover, from te considerations at te beginning of Section 3 it follows tat tere exist two constants γ 1, γ 2 > 0 suc tat for every u SBV p (A; R m ) L p (A; R m ) we ave γ 1 H(u, A) F 0 (u, A) γ 2 ( H(u, A) + A ), were H is defined in (3.1). By (4.2) we obtain immediately (4.3) γ 1 H(u, B) F 0 (u, B) γ 2 (H(u, B) + B ) for every B B(Ω). Proposition 4.1. Tere exists a unique quasiconvex function f : M m n [0, + [ wit te following properties: (i) tere exist γ 1, γ 2 > 0 suc tat γ 1 ξ p f(ξ) γ 2 (1 + ξ p ) for every ξ M m n ; (ii) for every A A 0 and u W 1,p (A; R m ) we ave F 0 (u, A) = f( u) dx. Proof. Let Ω A 0 and consider F 0 : W 1,p (Ω; R m ) A(Ω) [0, + [. Tis functional satisfies te assumptions of Teorem 1.1 in [29], i.e., for every u, v W 1,p (Ω; R m ) and A A(Ω): a) F 0 (u, A) = F 0 (v, A) wenever u A = v A ; b) te set function F 0 (u, ) is te restriction to A(Ω) of a Borel measure on Ω; c) F 0 (u, A) c A (1 + Du p ) dx, wit c a positive constant; d) F 0 (u + a, A) = F 0 (u, A) for every a R m ; e) F 0 (, A) is sequentially weakly lower semicontinuous on W 1,p (Ω; R m ). In fact, properties b), c), and d) follow from Proposition 3.3, estimate (4.3) and Lemma 3.7, respectively, wile a) and e) can be obtained immediately from te fact tat F 0 (, A) is te Γ-limit (4.1). By [29] Teorem 1.1, te Caratèodory function f : R n M m n [0, + [ defined by F 0 (u ξ, B ρ (x)) (4.4) f(x, ξ) = lim sup ρ 0 B ρ (x) 16 A

17 (u ξ is te linear function defined by u ξ (x) = ξ x) gives te integral representation F 0 (u, Ω) = f(x, Du) dx Ω for every u W 1,p (Ω; R m ). Te function f(x, ) is quasiconvex for a.e. x R n, and, from (4.4) and Lemma 3.7, we obtain tat f(x, ξ) is constant wit respect to x R n. Consequently we can drop te dependence on x. Finally, te uniqueness of f follows from (4.4), wile (i) follows from b) and (4.3). Te next step is to obtain an integral representation formula for F 0 on finite partitions, i.e., on tose BV functions wic take only a finite set of values. It will be acieved by applying a teorem due to Ambrosio and Braides. Given Ω A 0 and a finite subset T of R m we denote by BV (Ω; T ) te set of functions u : Ω T wic belong to BV (Ω; R m ). It turns out tat BV (Ω; T ) SBV p (Ω; R m ) L (Ω; R m ). Proposition 4.2. Tere exists a unique function g : R m S n 1 [0, + [ continuous in te second variable and suc tat (i) g( s, ν) = g(s, ν) for every (s, ν) R m S n 1 ; (ii) for every finite subset T of R m F 0 (u, S) = g(u + u, ν u ) dh n 1 for every A A 0, u BV (A; T ) and S Borel subset of S u A. S Proof. Let T be a finite subset of R m and let Ω A 0. For every A A(Ω) and u BV (Ω; T ) we define G T (u, A) = F 0 (u, S u A), were F 0 (u, S u A) is defined in (4.2). Let us sow tat te assumptions of Teorem 3.1 in [8] are satisfied by G T : BV (Ω; T ) A(Ω) [0, + [ in te following form: (i) 0 G T (u, A) ΛH n 1 (A S u ) wit Λ R fixed; (ii) G T (u, ) is te restriction to A(Ω) of a Borel measure on Ω; (iii) G T (u, A) = G T (v, A) wenever u = v a.e. in A; (iv) if u u a.e. in A, ten G T (u, A) lim inf G T (u, A); + (v) for every y R n G T (τ y u, τ y A) = G T (u, A) (were (τ y u)(x) = u(x y) and τ y A = A + y ) wenever τ y A Ω. Property (i) follows immediately from te definition of G T and from estimate (4.3). As to (ii) te Borel measure F 0 (u, ) on Ω (see (4.2)), restricted to S u, is an extension of G T (u, ). Te proof of (iii) follows immediately from (4.1) and te definition (4.2) of F 0 (u, ) on Borel sets. Let us come to (iv). If u u a.e. on A, by te equiboundedness of (u ) it turns out tat u u in L p (A; R m ). For every E open subset of A wit S u A E we ave F 0 (u, E) lim inf F 0(u, E). + Furtermore, by (4.3) F 0 (u, E) = F 0 (u, S u E) + F 0 (u, E \ S u ) F 0 (u, S u A) + γ 2 E. Tus, F 0 (u, E) lim inf + F 0(u, S u A) + γ 2 E. 17

18 By passing to te infimum over all open sets E S u A, we get F 0 (u, S u A) lim inf + F 0(u, S u A). Finally, for te proof of (v) it is enoug to refer to te property of translation invariance sown for F 0 in Lemma 3.7. At tis point we can apply Teorem 3.1 in [8], wic yields te existence of a unique continuous function gt Ω : Ω T T Sn 1 [0, + [ suc tat gt Ω (x, a, b, ν) = (x, b, a, ν) and g Ω T (4.5) F 0 (u, S u A) = G T (u, A) = S u A for every u BV (Ω; T ) and A A(Ω). Define g Ω T (x, u+, u, ν u ) dh n 1 { u a,b a if x x0, ν 0 x 0,ν(x) = b if x x 0, ν < 0, Π x0,ν = {y R n : y x 0, ν = 0} wenever x, x 0 R n, a, b R m tat and ν S n 1. Te continuity of gt Ω (, a, b, ν) yields (4.6) gt Ω(x F 0 (u a,b x 0, a, b, ν) = lim 0,ν, Π x0,ν B ρ (x 0 )) ρ 0 H n 1 (Π x0,ν B ρ (x 0 )) for every (x 0, a, b, ν) Ω T T S n 1. Tis allows us to replace te integrand function in (4.5) by a function g(x, a, b, ν), valid for all Ω and T, defined on R n R m R m S n 1. From (4.6) and Lemma 3.7 we also obtain tat g is independent of x and depends on (a, b) troug te difference a b. Terefore we can write F 0 (u, S u A) = S u A g(u + u, ν u ) dh n 1 for every finite subset T of R m, A A 0 and u BV (A; T ). Since F 0 (u, ) S u is a regular Borel measure, tis immediately yields te integral representation on te Borel subsets of S u stated in te proposition. 5. Caracterization of te omogenized bulk energy density Te goal of tis section is to prove tat te function f given in Proposition 4.1 is precisely te function f om introduced in Proposition 2.1. Tis will be acieved in te next two propositions. We sall use te notation Y =]0, 1[ n. Proposition 5.1. For every ξ M m n we ave f(ξ) f om (ξ). Proof. Fix ξ M m n. From te definition of f om, for every σ > 0 tere exists ε = ε(σ) > 0 and a function v W 1,p 0 (Y ; Rm ) suc tat f( x ε, ξ + v(x)) dx f om(ξ) + σ. Y 18

19 We still denote by v te 1-periodic extension of v to R n. For every N define u (x) = ξ x + ε ε v( ε x) for x R ε n. Since (v( ε ε x)) is bounded in L p (Y ; R m ), we ave tat (u ) converges to ξ x in L p (Y ; R m ). We may assume tat ε = 1/k for a suitable k N, so tat te function x f( x ε, η) is Y -periodic for every η. Hence, by te definition of F 0 : F 0 (ξ x, Y ) lim inf + Y lim inf (ε + ε )n f ( x, ξ + ( v)( ε x) ) dx ε ε ([ε/ε ]+1)Y = lim inf (ε + ε )n([ ε ] ) + 1 n ε f( x, ξ + v(x)) dx ε Y f( x, ξ + v(x)) dx. ε Terefore F 0 (ξ x, Y ) f om (ξ) + σ. Te conclusion follows from Proposition 4.1 and te arbitrariness of σ > 0. Proposition 5.2. For every ξ M m n we ave f(ξ) f om (ξ). Proof. For te sake of clarity, te proof will be divided in tree steps. Fix ξ M m n. Step 1. Tere exist a sequence of positive real numbers (α ) converging to 0 and a sequence of functions (w ) in SBV p (Y ; R m ) L p (Y ; R m ) suc tat (5.1) (i) w ξ x in L p (Y ; R m ) (ii) (iii) lim + Hn 1 (S w Y ) = 0 lim sup f( x, w (x)) dx + α f(ξ). Y Proof. By assumption tere exists a sequence (v ) in SBV p (Y ; R m ) L p (Y ; R m ) suc tat v 0 in L p (Y ; R m ) and lim + F ε (ξ x + v, Y ) = F 0 (ξ x, Y ) = f(ξ). Set u (x) = ξ x + v (x), v L p (Y ;R m ) = σ (n/p)+1. It is easy to see tat tere exists a divergent sequence (k ) of natural numbers suc tat, setting β = k ε, one as β 0, σ β 0. Now, in order to be able to introduce te sequence (w ), let us consider for every N and λ N n te set Q,λ = β (λ + Y ). Let λ() N n be te index of a minimal cube, i.e., F ε (u, Q,λ() ) F ε (u, Q,λ ) for every λ N n wit Q,λ Y, and set Q = Q,λ(), x = β λ(), w (x) = ξ x + 1 β v (x + β x), x Y. 19

20 Let us prove (5.1)(i). Easy computations sow tat w ξ x L p (Y ;R m ) = 1 ( v (β ) (n/p)+1 (y) p dy ) 1/p Q 1 (β ) v (n/p)+1 L p (Y ;R m ) = ( σ ) β (n/p)+1. Te conclusion is now immediate by our assumptions on σ /β. Let us prove (5.1)(ii). For every N F ε (u, Y ) [ 1 β ] nfε (u, Q ) [ 1 β ] nc3 H n 1 (S u Q ). Terefore, since te sequence (F ε (u, Y )) is bounded, tere exists a constant C > 0 suc tat for every N H n 1 (S u Q ) C [1/b ] n. On te oter and S w Y = 1 β ((S v Q ) x ) = 1 β ((S u Q ) x ), wic implies H n 1 (S w Y ) = ( ) n 1 1 1/β (β ) n 1 Hn 1 (S u Q ) C [1/β ] 1 [1/β ]. Since (β ) converges to 0 as tends to +, te proof of (5.1)(ii) is accomplised. Let us now define α = ε /β. Ten α 0 as +. We now prove (5.1)(iii). By taking into account te periodicity assumption on f and te fact tat β /ε N, we ave Y f( x, w (x)) dx = f( x, ξ + ( v )(x + β x)) dx α Y α = 1 f( y x, ξ + v (β ) n (y)) dy = 1 f( y, ξ + v Q ε (β ) n (y)) dy Q ε ( ) n ( ) 1/β [ 1 ] n nfε 1/β (u, Q ) F ε(u, Y ). [1/β ] β [1/β ] Hence, lim sup + Y f( x α, w (x)) dx lim + F ε (u, Y ) = f(ξ), wic proves (5.1)(iii). Step 2. For every fixed η > 0 te sequence (w ) can be cosen in suc a way tat (5.2) (i) Λ > 0 N w L (Y ;R m ) Λ (ii) w ξ x in L p (Y ; R m ) (iii) (iv) lim + Hn 1 (S w Y ) = 0 lim sup f( x, w (x)) dx + α f(ξ) + η. Y Proof. Apply Remark 3.6 to te sequence of functionals u Y f( x α, u(x)) dx. Tus, Lemma 3.5 applied to (w ), wit M 0 = ξ x L (Y ;R m ), furnises a sequence ϕ (w ) 20

21 wic satisfies properties (5.2), as one can easily ceck on account of properties (5.1) and te fact tat S ϕ (w ) S w. Now te functions ϕ (w ) are renamed w. Step 3. We replace te sequence (w ) wit a suitable sequence in W 1,p (actually Lipscitz functions), still satisfying properties similar to (5.2)(ii) and (5.2)(iv), and apply te omogenization results in W 1,p. Proof. We first need a preliminary remark. By Proposition 1.1 for every N we ave Y M p ( w ) dx C(p, n) Y w p dx C(p, n) c 1 Y f( x α, w ) dx. Ten te sequence (M p ( w )) is bounded on L 1 (Y ). Tis ensures, as proved in [1] Lemma I.7, a weak equi-integrability property for (M p ( w )). More precisely, for any ε > 0 tere exist a Borel set C ε Y, δ > 0 and an infinite set S N suc tat C ε < ε and for all C B(Y ) if C C ε =, and C < δ, ten C M p ( w ) dx < ε for all S. Fix ε > 0 and let C ε and δ enjoy tis property. It will not be restrictive to assume S = N. Since (M p ( w )) is bounded in L 1 (Y ) we can coose λ ε 1 suc tat for every λ λ ε and N {x Y : M( w ) > λ} < δ and Λ ε < λ, were Λ is given in (5.2)(i). By Teorem 1.2 for every N and λ λ ε fixed, tere exists a Lipscitz function w,λ : Y ε R m wose Lipscitz constant does not exceed 2m(c (n) + 1)λ, suc tat w,λ = w a.e. on Y ε \ E,λ, were E,λ = {x Y : M( Dw ) > 2λ}. Since M( Dw ) M( w ) + M( D s w ), by Proposition 1.1 we ave E,λ \ C ε {x Y \ C ε : M( w )(x) > λ} + {x Y : M( D s w )(x) > λ} {x Y \ C ε : M p ( w )(x) > λ p } + λ 1 c(n) D s w (Y ) λ p M p ( w ) dx + 2λ 1 c(n) w L (Y ;R m )H n 1 (S w Y ) {M( w )>λ}\c ε ελ p + 2λ 1 c(n)λh n 1 (S w Y ). Tis implies by (5.2)(iii) tat (5.3) lim sup + λp E,λ \ C ε ε. Moreover, for a suitable constant K = K(n, m) > 0 we ave (5.4) 0 f( x α, w,λ ) Kλ p on Y ε. 21

22 Tus, since w,λ = w on Y ε \ E,λ we get f( x, w,λ ) dx f( x, w ) dx + Kλ Y ε \C ε α (Y ε \C ε )\E,λ α p E,λ \ C ε f( x, w ) dx + Kλ α p E,λ \ C ε. Y By taking into account (5.2)(iv) and (5.3) te previous inequality yields lim sup f( x, w,λ ) dx lim sup f( x, w ) dx + Kε + Y ε \C ε α + Y α (5.5) f(ξ) + Kε + η. From (5.4) we also deduce tat for every λ λ ε tere exist an increasing sequence (σ()) of natural numbers and a function ϕ L (Y ε ) suc tat f(x/α σ(), w σ(),λ ) ϕ in w -L (Y ε ). In particular, for all B B(Y ε ) x (5.6) ϕ(x) dx = lim f(, w σ(),λ ) dx. + α σ() B B On te oter and, by passing, if necessary, to a furter subsequence (still depending on λ) we can assume wit no loss of generality tat tere exists w λ W 1, (Y ε ) suc tat w σ(),λ w λ in w -W 1, (Y ε ). By Proposition 2.1 x f om ( w λ ) dx lim inf f(, w σ(),λ ) dx = ϕ(x) dx. + α σ() A A for all A A(Y ε ). It follows tat f om ( w λ (x)) ϕ(x) for a.e. x Y ε. Ten (5.5) and (5.6) wit B = Y ε \ C ε imply f om ( w λ ) dx lim sup f( x, w,λ ) dx Y ε \C + ε Y ε \C ε α f(ξ) + Kε + η. Terefore, (5.7) f om (ξ) Y ε \ C ε,λ = f om (ξ) dx f(ξ) + Kε + η, Y ε \C ε,λ were C ε,λ = C ε {x Y ε : w λ ξ x}, and λ λ ε. Since ( Dw (Y )) is a bounded sequence, by Proposition 1.1 tere exists a constant C > 0 suc tat {x Y ε : w σ(),λ w σ() } E σ(),λ C λ. By te lower semicontinuity of te functional w {x Y ε : w(x) 0} wit respect to te convergence in measure we infer {x Y ε : w λ (x) ξ x} C λ. A Hence, from (5.7), f om (ξ) ( (1 ε) n ε C λ ) f(ξ) + Kε + η. Te conclusion now follows if we consider successively te limits as λ +, ε 0 +, and η

23 6. Caracterization of te omogenized surface energy density Here we first prove Proposition 2.2, were te function g om is introduced. Ten, as we did in te previous section for te volume part, we prove tat te function g, wic represents F 0 on finite partitions according to Proposition 4.2, is actually te function g om. Proof of Proposition 2.2. For every ortonormal basis ν = (ν 0, ν 1,..., ν n 1 ) of R n, set Q ν = { α 0 ν α n 1 ν n 1 : α 0,..., α n 1 ] 1 2, 1 2 [}, and for every z R m and ε > 0 (6.1) I ε (ν) = inf { S u Q ν g( x ε, u+ u, ν u ) dh n 1 : u SBV (Q ν ; R m ), u = 0 a.e., u = u z,ν0 on Q ν }. For te following four steps we consider z R m fixed. Step 1. Let ν = (ν 0, ν 1,..., ν n 1 ) and ν = (ν 0, ν 1,..., ν n 1 ) be two ortonormal bases of R n wit equal first vector. Suppose ν is an ortonormal rational basis, i.e., for all i {0,..., n 1} tere exists r i R \ {0} suc tat r i ν i Z n. Ten lim sup ε 0 I ε (ν ) lim inf ε 0 I ε (ν). Proof. For every i = 1,..., n 1 let γ i > 0 be suc tat v i = γ i ν i Z n. If we set P = { α 1 v α n 1 v n 1 : α 1,..., α n 1 [ 1 2, 1 2 [}, ten te function g is P -periodic in te first variable, in te sense tat g(x+l 1 v l n 1 v n 1, w, µ) = g(x, w, µ) wenever (x, w, µ) R n R m S n 1 and l 1, l 2,..., l n 1 Z. Let ε > 0, η > 0 and σ > 0 be fixed. Let u ε SBV (Q ν ; R m ) wit u ε = 0 a.e. and u ε = u z,ν0 on Q ν be suc tat (6.2) S u ε Q ν For every λ = (λ 1,..., λ n 1 ) Z n 1 set g( x ε, u+ ε u ε, ν uε ) dh n 1 I ε (ν) + σ. x λ = η(λ 1 v λ n 1 v n 1 ), Q λ = x λ + η ε Q ν. We ave to coose properly te centres x λ. Let Λ = Λ(η, ε) = {λ Z n 1 : Q λ Q ν, and l = (l 1,..., l n 1 ) Z n 1, It is easy to see tat i) te cubes of te family (Q λ ) λ are pairwise disjoint; 23 n 1 x λ l i ( η ε + ηγ i)ν i + ηp }. i=1

24 ii) denoting by S = S uz,ν0 te yperplane x, ν 0 = 0, we ave lim η 0 Hn 1 (S (Q ν \ Q λ )) = 0 λ Λ (i.e. lim( η η 0 ε )n 1 #Λ(η, ε) =1). Define u η : Q ν R m by It turns out tat S uη { uε ( ε u η (x) = η (x x λ)) if x Q λ, λ Λ u z,ν0 (x) oterwise. S λ Λ Q λ, and clearly I η (ν ) We can estimate tis integral. We ave S u η Q ν g( x η, u+ η u η, ν uη ) dh n 1. I 1 S g( x u η Q η, u+ η u η, ν uη ) dh n 1 λ λ Λ = (η ) n 1 g( y ε λ S u ε Q ε + λ 1v λ n 1 v n 1, u + ε u ε, ν uε ) dh n 1, ν were te cange of variable x ε η (x x λ) as been applied on Q λ. Ten, by te P -periodicity of g and by (6.2) Let us come to S uη (S \ λ Λ Q λ): I 2 I 1 ( η ) n 1#Λ(η, ε)(iε (ν) + σ). ε S u η (S\ λ Λ Q λ) g( x η, u+ η u η, ν uη ) dh n 1 c 4 (1 + z ) H n 1 (S (Q ν \ λ Λ Q λ )). Te estimates now obtained for I 1 and I 2, togeter wit property ii) satisfied by Λ, yield lim sup I η (ν ) I ε (ν) + σ. η 0 We conclude by taking te lower limit for ε 0 and by considering te arbitrariness of σ > 0. Step 2. Let ν = (ν 0, ν 1,..., ν n 1 ) and ν = (ν 0, ν 1,..., ν n 1 ) be two ortonormal rational bases of R n wit equal first vector. Ten te limits lim ε 0 I ε (ν) and lim ε 0 I ε (ν ) exist and are equal. Proof. By applying Step 1 wit ν = ν we obtain te existence of te limits. By excanging te roles of ν and ν we obtain tat tey are equal. 24

25 Step 3. For every σ > 0 tere exists δ > 0 (independent of z R m ) suc tat if ν = (ν 0, ν 1,..., ν n 1 ) and ν = (ν 0, ν 1,..., ν n 1 ) are two ortonormal bases of Rn wit ν i ν i < δ for every i = 0,..., n 1, ten lim inf ε 0 I ε(ν) Kσ lim inf ε 0 were K = 1 + 2c 4 (1 + z ). Proof. We sall use te notation I ε(ν ) lim sup ε 0 I ε (ν ) lim sup I ε (ν) + Kσ. ε 0 Q ν,η = (1 η)q ν (ν ortonormal basis of R n, 0 < η < 1). Let σ > 0 be fixed and let 0 < η < 1 be suc tat (6.3) 2(1 (1 2η) n 1 ) < σ. It is easy to see tat tere exists δ > 0 wit te property tat for every pair ν = (ν 0, ν 1,..., ν n 1 ) and ν = (ν 0, ν 1,..., ν n 1 ) of ortonormal bases of Rn, if ν i ν i < δ (i = 0,..., n 1), ten (i) Q ν,η Q ν \ Q ν,2η (ii) H n 1 (( Q ν,η) (H H )) < σ, were H and H denote te alfspaces x, ν 0 > 0 and x, ν 0 > 0, respectively. Fix ν and ν wit tis property. Given ε > 0 tere exists u SBV (Q ν ; R m ) suc tat u = 0 a.e., u = u z,ν0 on Q ν and g( x S u Q ν ε, u+ u, ν u ) dh n 1 I ε (ν) + σ. We consider u extended wit value u z,ν0 on te wole R n. Ten we can define v : R n R m, v(x) = u(x/(1 2η)). We ave S v Q ν = ((1 2η)S u ) Q ν, and x ε(1 2η), v+ v, ν v ) dh n 1 ence (6.4) S v Q ν g( S v Q ν g( = (1 2η) n 1 S u ( 1 1 2η Q ν) g( y ε, u+ u, ν u ) dh n 1 S u Q ν g( y ε, u+ u, ν u ) dh n 1 + c 4 (1 + z )[1 (1 2η) n 1 ] ; x ε(1 2η), v+ v, ν v ) dh n 1 I ε (ν)+σ +c 4 (1+ z )[1 (1 2η) n 1 ]. Define w : Q ν R m by w(x) = { v on Qν,η u z,ν 0 on Q ν \ Q ν,η. 25