OPTIMAL DESIGN OF FRACTURED MEDIA WITH PRESCRIBED MACROSCOPIC STRAIN

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1 OPTIMAL DESIGN OF FRACTRED MEDIA WITH PRESCRIBED MACROSCOPIC STRAIN JOSÉ MATIAS, MARCO MORANDOTTI, AND ELVIRA ZAPPALE Abstract. In this work we consider an optimal design problem for two-component fractured media for which a macroscopic strain is prescribed. Within the framework of structured deformations, we derive an integral representation for the relaxed energy functional. We start from an energy functional accounting for bulk and surface contributions coming from both constituents of the material; the relaxed energy densities, obtained via a blow-up method, are determined by a delicate interplay between the optimization of sharp interfaces and the diffusion of microcracks. This model has the far-reaching perspective to incorporate elements of plasticity in optimal design of composite media. Keywords: Structured deformations, optimal design, relaxation, disarrangements, interfacial energy density, bulk energy density. 200 Mathematics Subject Classification: 49J45 (74A60, 49K0, 74A50).. Introduction Starting with the pioneering papers by Kohn and Strang [8, 9, 20], much attention has been drawn to optimal design problems for mixtures of conductive materials. The variational formulation of these problems, particularly useful for finding configurations of minimal energy, entails some technical problems from the mathematical point of view, in particular the non-existence of solutions. In [3, 7] this issue is addressed by introducing a perimeter penalization in the energy functional to be minimized, which has also the effect of discarding configurations where the two materials are finely mixed. For a related problem, leading to a similar energy functional in the context of brutal damage evolution, see [] and [4]. In the spirit of [2, 22] we want to study an optimal design problem which can incorporate elements of plasticity, in a way that it is suited to treat both composite materials (made of components with different mechanical properties) and polycrystals (where the same material develops different types of slips and separations at the microscopic level). In order to do so, we extend the framework introduced in [3], by considering a material with two components each of which undergoes an independent (first-order) structured deformation, according to the theory developed by Del Piero and Owen [2]. The generalization of our model to account for materials with more than two components, or to polycrystals, is straightforward. Structured deformations set the basis to address a large variety of problems in continuum mechanics where geometrical changes can be associated with both classical and non-classical deformations for which an analysis at macroscopic and microscopic level is required. For instance, in a solid with a crystalline defective structure, separation of cracks at the macroscopic level may compete with slips and lattice distortions at the microscopic level preventing the use of classical theories, where deformations are assumed to be smooth. The objective of the theory of structured deformations is to generalize the theoretical apparatus of continuum mechanics as a starting point for a unified description of bodies with microstructure. It also turns out to be relevant to describe phenomena as plasticity, damage, creation of voids, mixing, and fracture in terms of the underlying microstructure (see [2]). We discuss now in more detail the application to polycrystals, which consist of a large number of grains, each having a different crystallograpic orientation, and where the intrinsic elastic and plastic response of each portion may vary from point to point. The anisotropic nature of crystal slip usually entails reorientation and subdivision phenomena during plastic straining of crystalline matter, even under homogeneous and gradient-free external loadings. This leads to spatial heterogeneity in terms of strain, stress, and crystal Date: July 29, 206. Preprint SISSA 42/206/MATE.

2 2 JOSÉ MATIAS, MARCO MORANDOTTI, AND ELVIRA ZAPPALE orientation. Beyond the aim of gaining fundamental insight into polycrystal plasticity, an improved understanding of grain-scale heterogeneity is important, and this is the main motivation for our work. As noted in [24], structural and functional devices are increasingly miniaturized. This involves size reduction down to the single crystal or crystal-cluster scale. In such parts, crystallinity becomes the dominant origin of desired or undesired anisotropy. In miniaturized devices plastic heterogeneity and strain localization can be sources of quality loss and failure. Thus, optimized design of small crystalline parts requires improved insight into crystal response and kinematics at the grain and subgrain scale under elastic, plastic, or thermal loadings. Moreover, the better understanding of the interaction between neighbouring grains, namely the quantification of its elastoplastic interaction, is in itself relevant for the verification and improvement of existing polycrystals homogenization models. These models are often considered to capture the heterogeneities on material response for a polycrystals, see, e.g., [23]. In this spirit, this work can also be viewed as a first step towards the derivation of a homogenization result for a polycrystalline material in the context of plasticity. To minimize our functional from a variational point of view, we rely on the energetics for structured deformations first studied by Choksi and Fonseca [9], where the problem is set in the space of special functions of bounded variation. Given an open bounded subset Ω R N, a structured deformation (in the context of [9]) is a pair (g, G) SBV (Ω; R d ) L (Ω; R d N ), where g is the microscopic deformation and G is the macroscopic deformation gradient. The energy associated with a structured deformation is then defined as the most effective way to build up the deformation using sequences u n SBV (Ω; R d ) that approach (g, G) in the following sense: u n g in L (Ω; R d ) and u n G in L p (Ω; R d N ), for p > a given summability exponent. The convergences above imply that the singular parts D s u n converge, in the sense of distributions, to Dg G. To have a better understanding of this phenomenon, consider the simpler case of a deformation g W, (Ω; R d ), that is, without macroscopic cracks. Then, Du n = u n L N D s u n with D s u n absolutely continuous with respect to the Hausdorff measure H N and supported in S(u n ), the jump set of u n. Since Du n g in the sense of distributions and u n G, we conclude that D s u n g G in the sense of distributions. This tells us that the difference between microscopic and macroscopic deformations is achieved through a it of singular measures supported in sets S(u n ) such that H N (S(u n )). The tensor M := g G is called the disarrangements tensor and embodies the fact that the difference between the microscopic and the macroscopic deformations in the bulk are achieved as a it of singular measures. The results obtained in [9] show that the bulk density of the energy of a structured deformation can be influenced by both the bulk and interfacial densities of the energy of these approximating sequences, and the interplay is characterized by means of precise relations between them. The energy functional that we consider (see (.)) will feature (i) different bulk densities associated with each of the two components, (ii) surface energy densities to account for the jumps in the deformations inside each component, (iii) a perimeter penalization (which measures the boundary between the two components independently on the discontinuities on the deformation), and finally (iv) a surface energy term that accounts for the interaction between neighbouring components (where both discontinuities in the deformation and in the components are counted). More precisely, in order to take the presence of two components into account, we consider a set of finite perimeter E Ω, describing one of them, and let χ BV (Ω; 0, ) be its characteristic function. Denoting by χ = the set of points in Ω with density (see [4]), by χ = 0 the set of points in Ω with density 0, and letting u SBV (Ω; R d ), we consider the following energy F od-sd : BV (Ω; 0, ) SBV (Ω; R d ) [0, [, defined as F od-sd (χ, u) := (( χ)w 0 ( u) χw ( u)) dx Ω g([u], 0 ν(u)) dh N g([u], ν(u)) dh N (.) Ω χ=0 S(u) Ω χ= S(u) g 2 (χ, χ, u, u, ν(u)) dh N Dχ (Ω), Ω S(χ) S(u) where ν(u) S N denotes the normal of the function u to the jump set S(u) of u, S N being the unit sphere in R N. For i = 0,, W i : R d N R are the bulk energy densities associated with the two components, g i : R d S N [0, [ are the surface energy densities associated with jumps in the deformation in the

3 OPTIMAL DESIGN OF FRACTRED MEDIA WITH PRESCRIBED MACROSCOPIC STRAIN 3 two components, and g 2 : 0, 2 (R d ) 2 S N R is the surface energy density associated with the jumps in the deformation at the interface between the two components. The energy contribution of the interface, independently of the discontinuities of the deformation, is carried by Dχ (Ω), the total variation of Dχ in Ω. In (.) we have split the jump set S(χ, u) of the pair (χ, u) into the disjoint union S(χ, u) = (S(χ) S(u)) (S(u)\S(χ)) (S(χ)\S(u)). In this way, we penalize the underlying structured deformation occurring in χ = 0 (S(u) \ S(χ)) and χ = (S(u) \ S(χ)) through g 0 and g, respectively, and we penalize the interface S(χ) through in S(χ)\S(u) (via the perimeter term) and through g 2 in S(χ) S(u). Therefore, when χ jumps, we are accounting for the perimeter of E plus a contribution along S(χ) depending on the discontinuities of u. Our main goal is to find an integral representation for the functional F od-sd : BV (Ω; 0, ) SBV (Ω; R d ) L (Ω; R d N ) [0, [ defined by F od-sd (χ, u, G) := inf inf F od-sd(χ n, u n ) : (χ n, u n ) BV (Ω; 0, ) SBV (Ω; R d ), (.2) χ n χ in BV (Ω; 0, ), un u in L (Ω; R d ), u n G in L p (Ω; R d N ). Our main result (see Theorem 3.3) states that for χ BV (Ω; 0, ), u SBV (Ω; R d ), G L (Ω; R d N ), and F od-sd defined by (.2) for functions W i, g, i i 0, and g 2 satisfying hypotheses (H ) (H 7 ) in Section 3, for some p > (see Section 3), we have that F od-sd (χ, u, G) admits an integral representation of the form F od-sd (χ, u, G) = H(χ, u, G) dx γ(χ, χ, u, u, ν) dh N, Ω Ω S(χ,u) where H and γ are given in (3.5) and (3.6), respectively. We observe that the bulk energy density H : BV (Ω; 0, ) (L (Ω; R d N )) 2 [0, [ depends on the structured deformation on χ = 0 or χ = (see (3.5)) and that the interfacial energy density γ : 0, 2 (R d ) 2 S N [0, [ (see (3.6)) can be further specialized on the various pieces of the decomposition of S(χ, u), as noted in detail in Remark 3.4. We remark also that if we consider the classical deformation setting, that is no jumps in u and G = u, then we recover an optimal design problem studied in [8]; if we consider just one material, then we recover the results in [9]. The overall plan of this work is the following: in Section 2 we fix the notation and recall some basic results used throughout this article. In Section 3 we formulate the problem, with detailed settings and assumptions and state the main result. Section 4 is devoted to proving some auxiliary results and finally we prove the main theorem in Section 5. The proof follows the blow-up method of [6]: we will compute the Radon-Nikodým derivatives of F od-sd with respect to the L N and H N measures and see that they can be bounded above and below by the densities H and γ, respectively. 2. Preinaries In this section we fix the notation used throughout this work and give a brief survey of functions of bounded variation and sets of finite perimeter. 2.. Notation. Throughout the text Ω R N will denote an open bounded set. We will use the following notations: - O(Ω) is the family of all open subsets of Ω; - M(Ω) is the set of finite Radon measures on Ω; - L N and H N stand for the N-dimensional Lebesgue measure and the (N )-dimensional Hausdorff measure in R N, respectively; - µ stands for the total variation of a measure µ M(Ω); - the symbol dx will also be used to denote integration with respect to L N ; - S N stands for the unit sphere in R N ; - denotes the unit cube of R 3 centered at the origin; - := x N > 0 and is defined similarly; - η denotes the unit cube of R N centered at the origin with two sides perpendicular to the vector η S N ; - (x, ) := x, η (x, ) := x η ;

4 4 JOSÉ MATIAS, MARCO MORANDOTTI, AND ELVIRA ZAPPALE - C represents a generic positive constant that may change from line to line; -,n := 0, k,n := k Measure Theory. In the proof of the upper and lower bounds for the blow-up method of [6], it is necessary to work with localizations of the functional F od-sd and see that it is a Radon measure. The following lemma, proved in [5], provides sufficient conditions for a set function Π : O(Ω) [0, ) to be the restriction of a Radon measure on O(Ω). It is a refinement of the De Giorgi-Letta s criterion (see []) and it is of importance to apply the Direct Method as well as for the use of relaxation methods that strongly rely on the structure of Radon measures. Lemma 2. (Fonseca-Malý). Let X be a locally compact Hausdorff space, let Π : O(X) [0, ] be a set function and µ be a finite Radon measure on X satisfying i) Π() Π(V ) Π( \ W ) for all, V, W O(X) such that W V ; ii) Given O(X), for all ε > 0 there exists ε O(X) such that ε and Π( \ ε ) ε. iii) Π(X) µ(x). iv) Π() µ() for all O(X). Then, Π = µ O(X) BV functions. We start by recalling some facts on functions of bounded variation which will be used in the sequel. We refer to [4] and the references therein for a detailed theory on this subject. A function u L (Ω; R d ) is said to be of bounded variation, and we write u BV (Ω; R d ), if all of its first distributional derivatives D j u i M(Ω) for i =,..., d and j =,..., N. The matrix-valued measure whose entries are D j u i is denoted by Du. The space BV (Ω; R d ) is a Banach space when endowed with the norm u BV := u L Du (Ω). By the Lebesgue Decomposition theorem Du can be split into the sum of two mutually singular measures D a u and D s u (the absolutely continuous part and singular part, respectively, of Du with respect to the Lebesgue measure L N ). By u we denote the Radon-Nikodým derivative of D a u with respect to L N, so that we can write Du = u L N Ω D s u. Let Ω u be the set of points where the approximate its of u exists and S(u) the jump set of this function, i.e., the set of points x Ω \ Ω u for which there exists a, b R N and a unit vector ν S N, normal to S(u) at x, such that a b and ε 0 ε N y ν(x,ε):(y x) ν>0 u(y) a dy = 0 (2.) and ε 0 ε N u(y) b dy = 0. (2.2) y ν(x,ε):(y x) ν<0 The triple (a, b, ν) is uniquely determined by (2.) and (2.2) up to permutation of (a, b), and a change of sign of ν and is denoted by (u (x), u (x), ν(u)(x)). If u BV (Ω; R d ) it is well known that S(u) is countably (N )-rectifiable, i.e. S(u) = K n N, n= where H N (N ) = 0 and K n are compact subsets of C hypersurfaces. Furthermore, H N ((Ω\Ω u )\S(u)) = 0 and the following decomposition holds Du = u L N [u] ν(u) H N S(u) D c u, where [u] := u u and D c u is the Cantor part of the measure Du, i.e., D c u = D s u (Ω u ). The space of special functions of bounded variation, SBV (Ω; R d ), introduced by De Giorgi and Ambrosio in [0] to study free discontinuity problems, is the space of functions u BV (Ω; R d ) such that D c u = 0, i.e. for which Du = u L N [u] ν(u) H N S(u). Proposition 2.2. If w BV (Ω; R d ) then

5 OPTIMAL DESIGN OF FRACTRED MEDIA WITH PRESCRIBED MACROSCOPIC STRAIN 5 i) for L N -a.e. x Ω ε 0 ε ε N (x,ε) w(y) w(x) w (x) (y x) N N N N dy = 0; ii) for H N -a.e. x S(w) there exist w (x), w (x) R d, and ν(x) S N normal to S(w) at x, such that w (y) w ε 0 ε N (x) dy = 0, w (y) w ν (x,ε) ε 0 ε N (x) dy = 0, ν (x,ε) where ν (x, ε) := y ν (x, ε) : (y x) ν > 0 and ν (x, ε) := y ν (x, ε) : (y x) ν < 0; iii) for H N -a.e. x Ω \ S(w) ε 0 ε N w(y) w (x) dy = 0. (x,ε) We next recall some properties of BV functions used in the sequel. We start with the following lemma whose proof can be found in [9]: Lemma 2.3. Let u BV (Ω; R d ). Then there exist piecewise constant functions u n SBV (Ω; R d ) such that u n u in L (Ω; R d ) and Du (Ω) = Du n (Ω) = [u n ](x) dh N. The next result is a Lusin-type theorem for gradients due to Alberti [2] and is essential to our arguments. Theorem 2.4. Let f L (Ω; R d N ). Then there exist u SBV (Ω; R d ) and a Borel function g : : Ω R d N such that Du = f L N g H N S(u), g dh N C f L (Ω;R ). d N S(u) S(u n) Remark 2.5. From the proof of Theorem 2.4 it also follows that u L (Ω;R d ) 2C f L (Ω;R d N ). Lemma 2.6 ([6, Lemma 2.6]). Let w BV (Ω; R d ), for H N -a.e. x in S(w), ε 0 ε N w (y) w (y) dh N = w (x) w (x). J w ν(x) (x,ε) 2.4. Sets of finite perimeter. In the following we give some preinary notions related with sets of finite perimeter. For a detailed treatment we refer to [4, 25]. Definition 2.7. Let E be an L N -measurable subset of R N. For any open set Ω R N the perimeter of E in Ω, denoted by P (E; Ω), is the variation of its characteristic function χ in Ω, i.e. P (E; Ω) := sup div ϕ dx : ϕ Cc (Ω; R d ), ϕ L. We say that E is a set of finite perimeter in Ω if P (E; Ω) <. E If L N (E Ω) is finite, then, denoting by χ its characteristic function, we have that χ L (Ω) (see [4, Proposition 3.6]). It follows that E has finite perimeter in Ω if and only if χ BV (Ω) and P (E; Ω) coincides with Dχ (Ω), the total variation in Ω of the distributional derivative of χ. The following approximation result can be found in [6] Lemma 2.8. Let E be a set of finite perimeter in Ω. Then, there exist a sequence of polyhedra E n, with characteristic functions χ n such that χ n χ in L (Ω) and P (E n ; Ω) P (E; Ω).

6 6 JOSÉ MATIAS, MARCO MORANDOTTI, AND ELVIRA ZAPPALE In the sequel we denote by χ = the set of points in Ω with density, and by χ = 0, the set of points in Ω with density 0. We recall (see [4])that for every t [0, ], the set E t (set of all points where E has density t) is defined by E t := x R N L N (E B ϱ (x)) : ϱ 0 L N = t. (B ϱ (x)) The essential boundary of E is defined by E := R N \ (E 0 E ), and Dχ (Ω) = P (E; Ω) = H N (Ω E) = H N (Ω E 2 ). 3. Statement of the problem and main results Let Ω be a bounded open subset of R N and consider continuous functions W i : R d N [0, [, g i : R d S N [0, [, i 0,, and g 2 : 0, 2 (R d ) 2 S N [0, [ satisfying (H ) there exists c, C > 0 such that for i 0, and ζ, ζ 2 R d N, W i (ζ ) W i (ζ 2 ) C ζ ζ 2 ( ζ p ζ 2 p ), c ζ p W i (ζ ), for some p > ; (H 2 ) there exist c, C > 0 such that for all λ R d, ν S N, and for i 0,, c λ g i (λ, ν) C λ ; (H 3 ) (positive homogeneity of degree one in the first variable) for all t > 0, λ R d, ν S N, and for i 0,, g i (tλ, ν) = tg i (λ, ν); (H 4 ) (subadditivity) for all λ, λ 2 R d, ν S N, and for i 0,, g i (λ λ 2, ν) g i (λ, ν) g i (λ 2, ν); (H 5 ) there exists C > 0 such that for all a, b 0,, c, d R d, and ν S N, 0 g 2 (a, b, c, d, ν) C( a b c d ); (H 6 ) (mechanical consistency of the surface energy density) for all a, b 0,, c, d R d, and ν S N, g 2 (a, b, c, d, ν) = g 2 (b, a, d, c, ν); (H 7 ) there exists C > 0 such that for all a, b, 0,, c i, d i R d, i =, 2, and ν S N, g 2 (a, b, c, d, ν) g 2 (a, b, c 2, d 2, ν) C c c 2 d d 2 C (c d ) (c 2 d 2 ). Some comments on the hypotheses are in order. Observe that if g 2 (a, b, c, d, ν) = g 2 (b a, d c, ν), for some function g 2, then (H 7 ) corresponds to imposing Lipschitz continuity in the second variable for g 2. In particular, this model includes densities of the type g 2 (a, b, c, d, ν) = d c. In the sequel we will use the following notation, for the sake of simplicity, f(χ, u) := ( χ)w 0 ( u) χw ( u), (3.) and letting g (i, λ, ν) := g i (λ, ν) for every λ R d, ν S N, and for i 0,, we can include all the surface energy densities in one single function g : 0, 2 (R d ) 2 S N [0, [ by requiring In what follows, we assume further that g(0, 0, u, u, ν) = g (0, [u], ν) = g 0 ([u], ν), g(,, u, u, ν) = g (, [u], ν) = g ([u], ν), g(χ, χ, u, u, ν) = g 2 (χ, χ, u, u, ν), for χ χ. which is not a restriction, since g 2 is the density defined in S(χ) S(u). Remark 3.. We remark the following facts: (3.2) g 2 (,, c, c, ) = g 2 (a, a,,, ) = 0, (3.3)

7 OPTIMAL DESIGN OF FRACTRED MEDIA WITH PRESCRIBED MACROSCOPIC STRAIN 7 From condition (H ), it easily follows that there exists C > 0 such that, for all ζ R d N and i 0,, W i (ζ) C( ζ p ). The coercivity condition on the energies W i is not physically meaningful, since the Helmholtz free energy associated with crystals may have potential wells (at matrices where the energy vanishes). However, it can be dropped following arguments in [9, Proof of Prop. 2.22, Step 2] that are now standard: one considers a sequence of energies W i ε(ζ) := W i (ζ) ε ζ p and recovers the results for W i by letting ε 0. Conditions (H 2 ) and (H 3 ) may also rule out some important physical settings, but they can be relaxed following arguments in [9]: the coercivity condition (H 2 ) can be weakened by asking that the admissible sequences have bounded total variation (see [9, pages 76 and 77 ]); the homogeneity condition (H 3 ) can be relaxed to sublinearity g i (tλ, ν) tg i (λ, ν) (see [9, last paragraph of Section 3, on page 78]). We will extend by homogeneity the functions g i, i 0, to all of R N in the second variable. Let ξ R N, then g i (, ξ) := ξ g i (, ξ/ ξ ); We could replace the subadditivity assumption (H 4 ) by assuming Lipschitz continuity in the first variable. Without any extra difficulty one could replace f in (3.) by f : T R d N [0, [, where T is a set of finite cardinality of R m. We also believe that a similar analysis to the one presented below, can be performed when the range of χ is countable. Such a case is considered, e.g., in [8]; The following remark motivates the convergences in the definition of F od-sd. Remark 3.2 (Compactness). Assume that we have a sequence (χ n, u n ) BV (Ω; 0, ) SBV (Ω; R d ) such that u n is bounded in L and the energies F od-sd (χ n, u n ) are bounded. Then the growth assumptions (H ), (H 2 ), and (H 5 ) entail that u n Lp (Ω;R d N ) C, Du n (Ω) C and so there exist χ BV (Ω; 0, ), u SBV (Ω; R d ), and G L (Ω; R d N ) such that χ n χ in BV (Ω; 0, ), u n u in L (Ω; R d ), u n G in L p (Ω; R d N ). In order to prove our main result using the blow-up method from [6], we address the problem of finding an integral representation for the localized functional. Given O(Ω) and for f and g satisfying (3.), (3.2), and (3.3), define F od-sd (χ, u, G; ) := inf (u n,χ n) S(u n,χ n) inf f(χ n, u n ) dx n g(χ n, χ n, u n, u n, ν(χ n, u n )) dh N Dχ n () : χ n BV (; 0, ), u n SBV (; R d ), χ n χ in BV (; 0, ), u n u in L (; R d ), u n G in L p (; R d N ). Let F od-sd (χ, u, G) denote F od-sd (χ, u, G; Ω). Then, our main theorem reads as follows Theorem 3.3. Let χ BV (Ω; 0, ), u SBV (Ω; R d ), and G L (Ω; R d N ). Let F od-sd be defined by (.2) for functions W i, g, i i 0, and g 2 satisfying (H )-(H 7 ), for some p >. Then F od-sd (χ, u, G) admits an integral representation of the form F od-sd (χ, u, G) = H(χ, u, G) dx γ(χ, χ, u, u, ν) dh N (3.4) Ω Ω S(χ,u)

8 8 JOSÉ MATIAS, MARCO MORANDOTTI, AND ELVIRA ZAPPALE where, for i 0, and A, B R d N, H(i, A, B) = inf W i ( u) dx g([u], i ν(u)) dh N : u S(u) u SBV (; R d ), u L p (), u = Ax, u dx = B, and, for a, b 0,, c, d R d, ν S N, γ(a, b, c, d, ν) := inf g([u], 0 ν(u)) dh N g([u], ν(u)) dh N ν χ=0 S(u) ν χ= S(u) g 2 (χ, χ, u, u, ν(u)) dh N Dχ ( ν ) : ν S(χ) S(u) (χ, u) A od-sd (a, b, c, d, ν) (3.5) (3.6) where with A od-sd (a, b, c, d, ν) := (χ, u) BV ( ν ; 0, ) SBV ( ν ; R d ) : χ a,b,ν (x) := a if x ν > 0, b if x ν 0, χ ν = χ a,b,ν, u ν = u c,d,ν, u = 0 L N -a.e. and u c,d,ν (x) := c if x ν > 0, d if x ν 0. Remark 3.4. The formula for H only sees each component separately, so for i 0, and A, B R d N, H(i, A, B) = Hsd i (A, B), where the latter is the bulk energy density given by formula (2.6) in [9]. In particular, arguing as in [9] (see formula (4.22) therein), for i 0, and A, B R d N, H(i, A, B) C( A B p ), (3.7) which will be used in the sequel. The formula for γ can be specialized giving rise to 3 cases: in S(χ) S(u) we have in fact the formula in its full generality, and it could be denoted as γ od-sd as it fully reflects both the optimal design and structured deformation effects. In particular (H 2 ) and (H 5 ) entail that for a, b 0,, c, d, R d, ν S N, γ(a, b, c, d, ν) C( d c ); (3.8) in S(u) \ S(χ) the formula for γ reads as γ sd (i, λ, ν) := inf g([v], i ν(v)) dh N, v A sd (λ, ν) ν S(v) for i 0,, λ R d and ν S N, with and A sd (λ, ν) := v SBV ( ν ; R d ) : v ν = v λ,ν, v = 0 L N -a.e. v λ,ν := λ if x ν > 0, 0 if x ν 0. The symbol γ sd is adopted to underline that it is similar to the formula for the interfacial energy density in [9] and is due only to structured deformations. In fact, for every a, b 0,, c, d R d, ν S N, and for i 0,, we have γ(a, b, c, d, ν) inf g([u], i ν(u)) dh N : (χ, u) A od-sd (a, b, c, d, ν) ν χ=i S(u) = inf g([u], i ν(u)) dh N : u A sd (λ, ν) ν S(u) where c d = λ. In order to prove the latter equality one can argue as in the proof of (5.8) below, where we show that the contribution of g i on ν χ = j S(u) with j 0,, j i is negligible. (3.9)

9 OPTIMAL DESIGN OF FRACTRED MEDIA WITH PRESCRIBED MACROSCOPIC STRAIN 9 On the other hand, when a = b = i, the function χ a is admissible for the definition of γ, hence we can conclude that γ(i, i, c, d, ν) = γ sd (i, λ, ν). On the other hand (H 2 ) and (H 5 ) entail that there exists a constant C > 0 such that, for every λ R d, ν S N, γ sd (λ, ν) C λ. (3.0) in S(χ) \ S(u) the formula for γ reads γ od (a, b, ν) := inf Dχ ( ν ), χ A od (a, b, ν) = (b a) ν = for a, b 0, and ν S N, with A od (a, b, ν) := χ BV ( ν ; 0, ) : χ ν = χ a,b,ν =. We use the symbol γ od in order to emphasize that it only reflects the optimal design setting ([8]). In fact, for every c, d R d we have γ(a, b, c, d, ν) inf Dχ ( ν ), χ A od (a, b, ν) (b a) ν. On the other hand, if c = d, then the function u c is admissible for the definition of γ(a, b, c, d, ν) and we have γ(a, b, c, c, ν) = γ od (a, b, ν) = (b a) ν, thus we can conclude that on S(χ) \ S(u), the surface term reduces to the perimeter of S(χ). 4. Auxiliary Results The following result can be proven following arguments analogous to [9, Proposition 3.] and [9, Lemma 2.20]. Lemma 4.. Let i 0, and A, B R d N, and define H(i, A, B) = inf inf W i ( u n ) dx g u n ([u i n ], ν(u n )) dh N : n S(u n) u n SBV (; R d ), u n L p (), u n Ax in L, u n B in L p (; R d ). (4.) nder the assumptions (H ) (H 4 ) it results that H = H, where H is the density defined in (3.5). In fact, by [9, Lemma 2.20], (H ), and (4.), we have that H(i, A, B) = inf inf W i ( u n )dx g u n ([u i n ], ν(u n ))dh N : S(u) u n SBV (; R d ), u n Ax in L (; R d ), sup u n L (;R d ) <, u n L p (; R d ), u n B in L p (; R d ), for i 0,, A, B R d N. On the other hand, exploiting the same arguments in [9, Proposition 3.] we have H(i, A, B) = inf W i ( u) dx g([u], i ν(u)) dh N : u S(u) u SBV (; R d ) L (; R d ), u L p (), u = Ax, for i 0,, A, B R d N. Analogously, we can prove that u = B,

10 0 JOSÉ MATIAS, MARCO MORANDOTTI, AND ELVIRA ZAPPALE Lemma 4.2. Assume that (H 2 ) and (H 4 ) (H 7 ) hold. Then, for every a, b 0,, c, d, R d and ν S N, it results where γ(a, b, c, d, ν) = inf v n inf γ(a, b, c, d, ν) := γ(a, b, c, d, ν), ν S(χ n,v n) g(χ n, χ n, v n, v n, ν(χ n, v n )) dh N Dχ n ( ν ) : χ n χa,b,ν in BV ( ν ; 0, ), v n u c,d,ν in L ( ν ; R d ), v n 0 in L p. Proof. Trivially we have that γ(a, b, c, d, ν) γ(a, b, c, d, ν). Indeed it suffices to observe that any function u SBV ( ν ; R d ) such that u = u c,d,ν on ν and u = 0, and χ BV ( ν ; 0, ) such that χ = χ a,b,ν on ν are constant sequences admissible for defining γ. In order to prove the opposite inequality consider ν S N, u n SBV ( ν ; R d ) and χ n BV ( ν ; 0, ) such that u n u c,d,ν in L, with u n 0 in L p strongly, and χ n χa,b,ν, with Dχ n ( ν ) Dχ ( ν ), i.e. χ n χ strictly. By Theorem 2.4 we can take a sequence v n SBV ( ν ; R d ) such that u n = v n L N -a.e. and Dv n ( ν ) C u n L ( ν;r N). d Then by Lemma 2.3 there exist piecewise constant functions w n,m such that w n,m v n as m and Dw n,m ( ν ) Dv n ( ν ). Define z n,m := u n v n w n,m. It results z n,m = 0 L N -a.e. Furthermore n,m z n,m u c,d,ν L = 0. Moreover, using the fact that D s v n ( ν ) D s w n,m ( ν ) C u n dx 0 ν as n and exploiting (H 2 ) and (H 7 ), we have that g(χ n, χ n, z n,m, zn,m, ν(χ n, z n,m ))dh N k ν S(χ k,z k ) n,m ν S(χ n,z n,m) ν S(χ n,u n) g(χ n, χ n, u n, u n, ν(χ n, u n ))dh N. Extract a diagonal sequence in n and m, say (χ k, z k ), such that z k u c,d,ν in L ( ν ) with z k = 0 L N -a.e. and χ k χ a,b,ν strictly in BV ( ν, 0, ), are such that g(χ k, χ k, z k, z k, ν(χ k, z k ))dh N g(χ n, χ n, u n, u n, ν(χ n, u n ))dh N. ν S(χ n,u n) Finally we modify the sequences z k and χ k near the boundary of ν. Applying Fubini s theorem we can find r k such that, up to a subsequence, trχ k χ a,b,ν dh N 0, trz k u c,d,ν dh N, as k. Define and k ν S( χ k, z k ) ν(0, r k ) χ k (x) := z k (x) := ν(0, r k ) χ k if x ν (0, r k ), χ a,b,ν if x ν (0, ) \ ν (0, r k ), z k if x ν (0, r k ), u c,d,ν if x ν (0, ) \ ν (0, r k ). Clearly z k = 0 L N -a.e. and (H 2 ), (H 4 ), (H 5 ), (H 7 ), and the above convergences entail that g( χ k, χ k, z k, z k, ν( χ k, z k ))dh N g(χ n, χ n, u n, u n, ν(χ n, u n ))dh N, which concludes the proof. ν S(χ n,u n)

11 OPTIMAL DESIGN OF FRACTRED MEDIA WITH PRESCRIBED MACROSCOPIC STRAIN Remark 4.3. A similar argument leads to the following sequential characterization of γ sd (see Remark 3.4 and Proposition 4. in [9]): γ sd (i, λ, ν) = γ sd (i, λ, ν), for every i 0,, λ R d and ν S N, where γ sd (i, λ, ν) = inf inf g v n ([v i n ], ν(v n )) dh N : v n u c,d,ν in L ( ν ; R d ), v n 0 in L. p ν S(v n) Lemma 4.4. Let g satisfy (H 2 ), (H 4 ), and (H 7 ). Then γ(a, b, c, d, ν) γ(a, b, c, d, ν) C( c c d d ) (4.2) for every a, b 0,, c, c, d, d R d, ν S N. Moreover, γ is upper semicontinuous with respect to ν. Proof. We start by proving (4.2). By Lemma 4.2, for any given ε > 0 there exist sequences χ n BV ( ν ; 0, ) such that χ n χa,b,ν, and v n SBV ( ν ; R d ) such that v n u c,d,ν in L ( ν ; R d ), v n 0 in L p ( ν ; R d ) and ε γ(a, b, c, d, ν) g(χ n, χ n, v n, vn, ν(χ n, v n )) dh N Dχ n ( ν ). ν S(χ n,v n) By Lemma 2.3 there exists a sequence of piecewise constant functions u n such that u n u c,d,ν u c,d,ν, Du n ( ν ) D(u c,d,ν u c,d,ν) ( ν ) = (c c ) (d d ). By Lemma 4.2 we have that γ(a, b, c, d, ν) inf inf ν S(χ n,w n) ν S(χ n,v n) g(χ n, χ n, w n, w n, ν(χ n, w n )) dh N Dχ n ( ν ) g(χ n, χ n, v n, v n, ν(χ n, v n )) dh N Dχ n ( ν ) ε γ(a, b, c, d, ν) ε C (c c ) (d d ) ε, where (H 2 ), (H 4 ), and (H 7 ) have been exploited. It suffices to send ε 0 to achieve one of the inequalities in (4.2). The reverse inequality can be proven in the same way. In order to prove the upper semicontinuity of γ od-sd in the last variable, we observe that for every ε > 0 there exists χ ε BV (; 0, ), χ ε = χ a,b,ν on ν and u ε SBV ( ν ; R d ), u ε = u c,d,ν on ν, with u ε = 0 L N -a.e. in ν and such that γ(a, b, c, d, ν) g(χ ε, χ ε, u ε, u ε, ν(χ ε, u ε )) dh N < ε. (4.3) ν S(χ ε,u ε) For every sequence ν n ν we can take a family of rotations R n, such that R n ν = ν n and it results clearly that R n converges to the identity. Then γ(a, b, c, d, ν n ) ν S(χ ε,u ε) g(χ ε, χ ε, u ε, u ε, ν(χ ε, u ε ))dh N, which in turn, by virtue of (4.3), provides The proof is concluded by sending ε 0. sup γ(a, b, c, d, ν n ) γ(a, b, c, d, ν) ε. Lemma 2.20 in [9] holds in our context leading to the following result. Lemma 4.5. Let u SBV (Ω; R d ) L (Ω; R d ). Assume that (H ) (H 7 ) hold. Then F od-sd (χ, u, G; ) = F od-sd(χ, u, G; ),

12 2 JOSÉ MATIAS, MARCO MORANDOTTI, AND ELVIRA ZAPPALE for every χ BV (; 0, ), G L p (; R d ), and O(Ω), where F od-sd(χ, u, G; ) := inf (u n,χ n) S(u n,χ n) inf f(χ n, u n ) dx g(χ n, χ n, u n, u n, ν(χ n, u n )) dh N Dχ n () : χ n BV (; 0, ), u n SBV (; R d ), χ n χ in BV (; 0, ), u n u in L (; R d ), u n L < C, u n G in L p (; R d N ). Proof. In order to prove this result it clearly suffices to show that given χ n BV (; 0, ) such that χ n χ in BV, and un SBV (; R d ) such that u n w 0 in L, with w 0 L (; R d ) and u n G, in L p (; R d N ), u n L p (), there exist w n SBV (; R d ) L (; R d ) such that w n w 0 in L, w n L < and inf f(χ n, u n ) dx g(χ n, χ n, u n, u n, ν(χ n, u n )) dh N S(u n,χ n) sup f(χ n, w n ) dx g(χ n, χ n, w n, wn, ν(χ n, w n )) dh. N S(w n,χ n) Let φ i C 0 (R d ; R d ) be such that φ i (s) := s if s < e i, 0 if s e i. (4.4) and φ i L. Since w 0 L, there exists i 0, such that for i i 0, we have that w 0 L e i and φ i (w 0 ) = w 0 L N -a.e. Let i i 0 and define wn(x) i := φ i (u n (x)), where u n w 0 in L and u n G in L p. Clearly wn i L e i, S(wn) i S(u n ), and by the chain rule formula wn i = φ i (u n ) u n L N -a.e. Furthermore, arguing as in [9, Lemma 2.20], and wn i G in L p as n. Estimating the energies we have f(χ n, wn) i dx g(χ S(χ n,wn i ) n, χ n, wn i = f(χ n, u n ) dx x: u n e i x: u n e i S(χ n,u n) x:e i u n e i S(χ n,u n) w i n w 0 L (;R d ) u n (x) w 0 (x) L (;R d ), x:e i u n e i, w i n, ν(χn, w i n)) dh N g(χ n, χ n, u n, u n, ν(χ n, u n )) dh N f(χ n, φ i (u n ) u n ) dx f(χ n, 0) dx x: u n >e i g(χ n, χ n, wn i, w i n, ν(χn, wn)) i dh N f(χ n, u n ) dx g(χ n, χ n, u n, u n, ν(u n, χ n )) dh N S(χ n,u n) C u n L e i C ( u n p )dx C x:e i u n e i C ( [u n ] )dh N, x:e i u n e i S(u n) S(χ n) x:e i u n e i (S(u n)\s(χ n)) [u n ] dh N

13 OPTIMAL DESIGN OF FRACTRED MEDIA WITH PRESCRIBED MACROSCOPIC STRAIN 3 where we have used (H ), (H 2 ), and the fact that L N (x : u n > e i ) e (i) u n L. Then for M > i 0, M i 0 M i=i 0 f(χ n, wn) i dx g(χ S(χ n,wn i ) n, χ n, wn i, w i n, ν(w i n, χ n )) dh N f(χ n, u n ) dx g(χ n, χ n, u n, u n, ν(χ n, u n )) dh N S(χ n,u n) C M M i 0 e i ( u n p )dx [u n ] dh N Dχ n (). i=i S(u 0 n) The three terms in the last line above are uniformly bounded independently on n, thus we may take M so large that their sum is less than ε. Hence there exists some i i 0,..., M such that f(χ n, wn) i dx g(χ S(χ n,wn i ) n, χ n, wn i, w i n, ν(χn, wn)) i dh N f(χ n, u n ) dx g(χ n, χ n, u n, u n, ν(χ n, u n )) dh N ε. S(χ n,u n) Thus it suffices to diagonalize first and then to send ε to zero to obtain the result. 5. Proof of the main result This section is devoted to the proof of Theorem 3.3 and is divided in four subsections. First we prove that the functional F od-sd (χ, u, G; ), in (.2), is the restriction of a suitable Radon measure to open subsets of Ω, then we prove a lower bound and an upper bound estimate in terms of its integral representation when the target field u is in L (Ω; R d ) and finally we prove the general case via a truncature argument. 5.. Localization. This subsection is devoted to show that F od-sd (χ, u, G; ), O(Ω), is the trace of a Radon measure absolutely continuous with respect to L N H N S(χ,u). Proposition 5.. Assume that (H ), (H 2 ), and (H 5 ) hold and let u SBV (Ω; R d ). Then F(χ, u, G; ) is the trace on O(Ω) of a finite Radon measure on B(Ω). Proof. The proof relies on Lemma 2.. First we prove that, for every χ BV (Ω; 0, ), u SBV (Ω; R d ), and G L p (Ω; R d N ), F(χ, u, G; ) Dχ () L N () Du () G L p (;R d N ). We observe that by Theorem 2.4 there exists h SBV (; R d ) such that h = G L N -a.e. in and Dh () C G L (;R d N ). By Lemma 2.3 there exists a sequence of piecewise constant functions ū n such that ū n u h in L, Dū n () Du Dh (). Define now u n := ū n h. Clearly u n (x) = G(x) for L N -a.e. x and u n u in L.

14 4 JOSÉ MATIAS, MARCO MORANDOTTI, AND ELVIRA ZAPPALE Thus, the definition of F od-sd (χ, u, G; ), (H ), (H 2 ), and (H 5 ) entail that F od-sd (χ, u, G; ) inf f(χ, u n )dx g(χ, χ, u n, u n, ν(χ, u n ))dh N Dχ () S(χ,u n) inf f(χ, G)dx C u n u n dh N C Dχ () S(χ,u n) inf f(χ, G)dx C Du n () C Dχ () inf f(χ, G)dx C Dū n () C G L (;R d N ) C Dχ () C f(χ, G)dx Du Dh () G L (;R d N ) Dχ () C f(χ, G)dx Du () G L (;R d N ) Dχ () C L N () G L p (;R d N ) Du () Dχ (). We start proving (iv) in Lemma 2.. We know that (H ) and the lower semicontinuity of total variation entail the existence of a sequence (χ n, u n ) BV (Ω; 0, ) SBV (Ω; R d ) such that χ n χ in BV and un u in L (Ω; R d ), u n G in L p (Ω; R d N ) and F od-sd (χ, u, G; Ω) = f(χ n, u n )dx Ω Ω S(χ n,u n) p to the extraction of a further subsequence we know that as n, and (5.) g(χ n, χ n, u n, u n, ν(χ n, u n ))dh N Dχ n (Ω) f(χ n, u n )dx g(χ n, χ n, u n, u n, ν(χ n, u n ))dh N S(χn,u n) Dχ n ( ) µ in M(Ω), µ(ω) = F od-sd (χ, u, G; Ω). (5.2) For every O(Ω) we can say that F od-sd (χ, u, G; ) inf f(χ n, u n )dx g(χ n, χ n, u n, u n, ν(χ n, u n ))dh N Dχ n () S(χ n,u n) µ(). Next we prove that (i) in Lemma 2.. Consider, V, W O(Ω) such that V W. Fix ε > 0 and consider (χ n, u n ) BV (V ; 0, ) SBV (V ; R d ) and (χ n, v n ) BV (W \ ; 0, ) SBV (W \ ; R d ) almost minimizing sequences for F od-sd, i.e. V f(χ n, u n )dx g(χ n, χ n, u n, u n, ν(χ n, u n ))dh N Dχ n (V ) V S(χ n,u n) ε F od-sd (χ, u, G; V ), f(χ n, v n )dx (W \) ε F od-sd (χ, u, G; W \ ), (W \) S(χ n,vn) g(χ n, χ n, v n, v n, ν(χ n, v n ))dh N Dχ n (W \ ) with χ n χ in V, χ n χ in W \, un u in L (V ; R d ), v n u in L (W \ ; R d ), u n G in L p (V ; R d N ) and v n G in L p (W \ ; R d N ). (5.3).

15 OPTIMAL DESIGN OF FRACTRED MEDIA WITH PRESCRIBED MACROSCOPIC STRAIN 5 In order to connect the functions without adding more interfaces, we argue as in [7] (see also [8]). For > 0 small enough, consider := x V : dist(x, ) <. For x W, let d(x) := dist(x; ). Since the distance function to a fixed set is Lipschitz continuous (see [25, Exercise.]), we can apply the change of variables formula (see [3, Theorem 2, Section 3.4.3]), to obtain [ ] u n (x) v n (x) Jd(x)dx = u n (x) v n (x) dh N dy 0 d (y) \ and, as since Jd(x)is bounded and u n v n 0 in L (V (W \ ); R d ), it follows that for almost every ϱ [0; ] we have u n (x) v n (x) dh N (x) = d (ϱ) u n (x) v n (x) dh N = 0. ϱ (5.4) An argument entirely analogous guarantees that d (ϱ) χ n (x) χ n(x) dh N (x) = ϱ χ n (x) χ n(x) dh N = 0. (5.5) Fix ϱ 0 [0; ] such that (5.4) and (5.5) hold. We observe that ϱ0 is a set with locally Lipschitz boundary since it is a level set of a Lipschitz function (see e.g. Evans and Gariepy [3]). Hence we can consider χ n, χ n, u n, v n on ϱ0 in the sense of traces and define χ χ n in ϱ0 u n in ϱ0 n = χ w n = n in W \ ϱ0, v n in W \ ϱ0. By the choice of ϱ 0, χ n and w n are admissible for F od-sd (χ, u, G, W ). In particular Thus we have F od-sd (χ, u, G; W ) inf f(χ n, w n )dx W f(χ n, u n )dx inf W \ V χ n χ in BV (W ; 0, ), w n u in L (W ; R d ), w n G in L p (W ; R d N ). g(χ W S(χ n,wn) n, χ n, w n, wn, ν(χ n, w n ))dh N Dχ n (W ) V S(χ n,u n) g(χ n, χ n, u n, u n, ν(χ n, u n ))dh N Dχ n (V ) f(χ n, v n )dx g(χ V S(χ n,vn) n, χ n, v n, vn, ν(χ n, v n ))dh N Dχ n (W \ ) g(χ ϱ0 S(χ n,wn) n, χ n, w n, wn, ν(χ n, w n ))dh N Dχ n ((S(w n ) S(χ n)) ϱ0 ) F od-sd (χ, u, G; V ) F od-sd (χ, u, G; W \ ) 2ε inf g(χ ϱ0 S(χ n,wn) n, χ n, w n, wn, ν(χ n, w n ))dh N Dχ n ( ϱ0 ) S(χ n, w n ) Observing that, by (H 2 ), (H 5 ), and (5.4), the first integral converges to 0, while the convergence to 0 of the latter term is ensured by (5.5), the proof of (i) follows sending ε to 0..

16 6 JOSÉ MATIAS, MARCO MORANDOTTI, AND ELVIRA ZAPPALE It remains to prove (iii) and (ii) in Lemma 2.. To this end, fix ε > 0 and take W V such that µ(v \ W ) < ε. By (i), (5.2), and (5.3), it results µ(v ) µ(w ) ε = µ(ω) µ(ω \ W ) ε F od-sd (χ, u, G; Ω) F od-sd (χ, u, G; Ω \ W ) ε F od-sd (χ, u, G; V ) ε. Letting ε 0, we obtain µ(v ) F od-sd (χ, u, G; V ), which proves (iii). On the other hand, by (5.), we have F od-sd (χ, u, G; ) C( G p )L N Du Dχ. Next, denote by λ the Radon measure on the right-hand side, take K a compact set such that K V with λ(v \ K) < ε, and W an open set such that K W V. sing (i) and (5.3) we have F od-sd (χ, u, G; V ) F od-sd (u, χ, G; W ) F od-sd (χ, u, G; V \ K) µ(w ) λ(v \ K) µ(v ) ε, and this concludes the proof as ε Lower bound. This subsection is devoted to prove in (3.4) in two steps, first identifying a lower bound for the bulk density and then for the surface one Bulk. pon considering a sequence µ n of bounded Radon measures associated with a sequence (χ n, u n ) admissible for F od-sd (χ, u, G), and denoting by µ the weak-star it of (a subsequence of) µ n, we want to show that dµ dl N (x 0) H(χ(x 0 ), u(x 0 ), G(x 0 )), for L N -a.e. x 0 Ω. Let x 0 be a point of absolute continuity and approximate differentiability for χ and u, and a point of absolute continuity for G. Namely, assume that 0 N (x 0,) χ(y) χ(x 0 ) N N N N dy = 0, (5.6) and 0 N (x 0,) d Du dl N (x 0) = u(x 0 ), u(y) u(x 0 ) u (x 0 ) (y x 0 ) d Dχ dl N (x 0) = 0, (5.7) N N N N dy = 0, (5.8) 0 N G(x) G(x 0 ) u(x) u(x 0 ) dx = 0. (5.9) (x 0,) Observe that the above requirements are satisfied for L N -a.e. x 0 Ω. Without loss of generality suppose that χ(x 0 ) =, the other choice can be handled similarly.

17 OPTIMAL DESIGN OF FRACTRED MEDIA WITH PRESCRIBED MACROSCOPIC STRAIN 7 We assume that the sequence is chosen in such a way that µ( (x 0, )) = 0, thus µ((x 0, )) N inf f(χ N n (x), u n (x))dx x 0 g(χ n, χ n, u n, u n, ν(χ n, u n ))dh N (x 0) S(χ n,u n) = f(χ n (x 0 y), u n (x 0 y))dy S(χn,un) x 0 g(χ n (x 0 y), χ n (x 0 y), u n (x 0 y), u n (x 0 y), ν(χ n, u n )(x 0 y))dh N Since we are estimating a lower bound, in the right hand side we can neglect the term g 2 in g, moreover, according to the notations established in (3.2), g (i,, ) will denote g i (, ), where i 0,. Defining one has and Analogously, by defining it easily follows that χ n, (y) := χ n(x 0 y) χ(x 0 ), χ n, L () =,n,n = 0 u n, w 0 L (;R d ) =,n 0 Moreover u n, (y) = u n (x 0 y). We have that = 0 N χ n (x 0 y) χ(x 0 ) dy χ(x 0 y) χ(x 0 ) dy χ(x) χ(x 0 ) dx = 0. x 0 u n, (y) := u n(x 0 y) u(x 0 ) w 0 (y) := u(x 0 )y, = 0 N u(x 0 y) u(x 0 ) u(x 0 )y dy u(x) u(x 0 ) u(x 0 )(x x 0 ) dx = 0. dµ dl N (x 0) = µ n k,n k N ((x 0, k )), for a sequence of sides lengths k 0 as k, and choose this sequence so that In fact, since (5.0) (5.) H N (S(χ n ) (x 0, k )) k,n k N = 0. (5.2) H N (S(χ n ) (x 0, k )) H N (S(χ n ) (x 0, k )) k,n k N k,n k N for (5.2) to hold, it is enough to choose k so that k Dχ ((x 0, k )) N k = k,n Dχ n ((x 0, k )) N k = k Dχ ((x 0, k )) N k Dχ ((x 0, k )) k k N, = d Dχ dl N (x 0) = 0,.

18 8 JOSÉ MATIAS, MARCO MORANDOTTI, AND ELVIRA ZAPPALE where the last equality holds since x 0 / S(χ). Consequently we may estimate from below dµ (x dl N 0 ) as inf f(χ(x 0 ) k χ n,k (y), u n,k (y))dy k,n g (χ(x 0 ) k χ n,k (y), [ k u n,k (y)], ν n,k (y))dh N k (S(u n,k )\S(χ n,k )) x 0 k = inf k,n I n,k In,k, 2 where we wrote for simplicity χ n,k := χ n,k and u n,k := u n,k, and ν n,k denotes the unit exterior normal to S(u n,k ). A diagonalization argument allows to define subsequences (not relabelled) k, χ k := χ nk, k, and u k := u nk, k, such that and χ k χ(x 0 ) L k 0 () = 0, Dχ k (x 0, k ) k 0 k N = 0, u k w 0 L (,R k 0 d ) = 0, dµ dl N (x 0) inf f(χ(x 0 ) k χ k (y), u k (y))dy k g (χ(x 0 ) k χ k (y), k [u k ](y), ν k (y))dh N k (S(u k )\S(χ k )) x 0 k = infik Ik, 2 k (5.3) where, as above, ν k denotes the unit normal to S(u k ), and Ik and I2 k denote I n k, k and In 2 k, k, respectively. Without loss of generality, up to subsequences if necessary, the above inf is a it, and, by (5.) and Lemma 4.2 applied to and to w 0 := u(x 0 ) y, we can assume that u k is uniformly bounded in L. We aim to fix χ(x 0 ) and to estimate k [Ik I2 k ] from below with a sequence that satisfies the conditions in the definition of H(χ(x0 ), u(x 0 ), G(x 0 )) (see Lemma 4.). For the sake of exposition, we control each term of the sum Ik I2 k separately and then add them. First we consider I k. Chacon biting Lemma ([4, Lemma 5.32]) guarantees the existence of a not relabelled subsequence u k and of a decreasing sequence of Borel sets E r, such that L N (E r ) 0, as r and the sequence u k p is equiintegrable in \ E r for any r N. Since f 0 and by (H ), f(χ(x 0 ) k χ k (y), u k (y)) dy f(χ(x 0 ) k χ k (y), u k (y)) dy k k \E r f(χ(x 0 ), u k (y)) dy χ k (x 0 k y) χ(x 0 ) C( u k (y) p ) dy k \E r \E r f(χ(x 0 ), u k (y)) dy sup C u k k \E p dy r k \E r where (5.0) has been used. In order to pass from \E r f(χ(x 0 ), u k (y))dy to f(χ(x 0), u k (y))dy, we extract a further subsequence. Indeed, we claim that for each j N there exists k = k(j) and r j N, such that f(χ(x 0 ), v j (y)) dy f(χ(x 0 ), v j (y)) dy C \E rj j, (5.4)

19 OPTIMAL DESIGN OF FRACTRED MEDIA WITH PRESCRIBED MACROSCOPIC STRAIN 9 where v j := u k(j). In light of (H ), in order to guarantee that (5.4) holds, we need to make sure that for each j, there exists k = k(j) and r(j), such that E rj ( u k(j) p ) dy j. Suppose not. Then, there exists j 0 such that, for all r and k, E r ( u k p ) dy > j 0. (5.5) For k fixed, and for r N noting that w r = u k is a constant sequence (and hence with p-equiintegrable gradients), letting r we get a contradiction from (5.5). Therefore, by (5.4), the sequence v j gives the right estimate from below for Ik(j), that is, up to the extraction of a further subsequence and denoting in what follows χ j := χ k(j), j := k(j) and E j := E rj, we have dµ dl N (x 0) j (S(v j)\s(χ k(j) ) f(χ(x 0 ), v j (y))dy = j f(χ(x 0 ), v j (y))dy sup j g (χ(x 0 ) k(j) χ j (y), [v j ](y), ν(χ j, v j )(y))dh N sup j C j v j p dy, \E j (S(v j)\s(χ j)) C j v j p dy \E j g (χ(x 0 ) j χ j (y), [v j ](y), ν(χ j, v j )(y))dh N where the positive -homogeneity of g in the first variables has been exploited. Recall also that, by the choice of the sizes of the cubes in (5.3), we are going to neglect the contribution supported in S(χ j ). This sequence still needs to be slightly changed in order to control the surface term Ik 2 and to comply with the conditions in Lemma 4.. Set now: F j := y : x 0 j y S(χ j ) and χ j (x 0 j y) χ(x 0 ) = 0 and note that Define: ṽ j := L N ( \ F j ) 0. (5.6) v j in F j u(x 0 )y in \ F j. Note that ṽ j is still uniformly bounded in L and it converges in L norm to w 0 (y) = u(x 0 )y. Moreover, since v j G(x 0 ) in L p, by (5.6) the same holds for ṽ j. Next we show that passing from v j to ṽ j there is no change in the control from below of Ij. By (H ) we have that: f(χ(x 0 ), v j (y)) dy f(χ(x 0 ), ṽ j (y)) dy C v j (y) p u(x 0 ) p dy. \F j Since, by (5.4) it results v j (y) p dy = v j (y) p dy v j (y) p dy \F j (\E j)\f j E j\f j v j (y) p dy v j (y) p dy \E j E j\f j v j (y) p dy \E j j,

20 20 JOSÉ MATIAS, MARCO MORANDOTTI, AND ELVIRA ZAPPALE we obtain dµ dl N (x 0) f(χ(x 0 ), ṽ j (y))dy j sup j C j 2 v j p dy \E j = f(χ(x 0 ), ṽ j (y))dy j (S(v j)\s(χ(x 0) jχ j)) (S(v j)\s(χ(x 0) jχ j)) g (χ(x 0 ) j χ j (y), [v j ](y), ν(v j (y)))dh N g (χ(x 0 ) j χ j (y), [v j ], ν(v j (y)))dh N where the last equality follows by the equiintegrability of ṽ j p in \ E j. Now we control Ik 2, observing that inf g (χ(x 0 ) j χ j (y), [v j ](y), ν(v j (y))dh N j S(v j)\s(χ j) inf g (χ(x 0 ) j χ j (y), [ṽ j ](y), ν(ṽ j (y)))dh N j S(ṽ j)\s(χ j) inf g([ṽ j ](y), ν(ṽ j (y)))dh N j F j S(ṽ j) = inf g([ṽ j ](y), ν(ṽ j (y)))dh N. j S(ṽ j) This last equality comes from the fact that ṽ j has no jumps in y : χ j (y) χ(x 0 ) = and g j ([ṽ j ], ν(ṽ j )) dh N = 0. (5.7) S(ṽ j) S( χ j) In fact, by Lemma 4.5 and (H 2 ), we have: g j ([ṽ j ], ν(ṽ j )) dh N C H N (S(ṽ j ) S(χ j )) C H N (S(χ j )) 0, j j S(ṽ j) S(χ j) since x 0 / S(χ), by the appropriate choice of the sizes of the cubes j so that (5.2) holds. Thus dµ dl N (x 0) f(χ(x 0 ), ṽ j (y))dy g j ([ṽ j ], ν j (y))dh N. S(v j) Since ṽ j is admissible for the definition of H, the proof is concluded Interfacial. We want to show that df od-sd (χ, u, G) dh N S(χ, u) (x 0) γ(χ (x 0 ), χ (x 0 ), u (x 0 ), u (x 0 ), ν(χ, u)(x 0 )), namely taking into account Remark 3.4, for H N -a.e. x 0 S(u) \ S(χ), df od-sd (χ, u, G) dh N S(u) (x 0) γ sd (χ(x 0 ), [u](x 0 ), ν(u)(x 0 )), (5.8) df od-sd (χ, u, G) dh N S(χ, u) (x 0) γ(χ(x 0 ), [u](x 0 ), ν(χ, u)(x 0 )), (5.9)

21 OPTIMAL DESIGN OF FRACTRED MEDIA WITH PRESCRIBED MACROSCOPIC STRAIN 2 for H N -a.e. x 0 S(u) S(χ), and df od-sd (χ, u, G) dh N S(χ) (x 0) Dχ (x 0 ), (5.20) for H N -a.e. x 0 S(χ) \ S(u). Let O(Ω), open and let (χ n, u n ) be an admissible sequence for the definition of F od-sd (χ, u, G)(), such that, for η > 0 fixed, η F od-sd (χ, u, G)() f(χ n, u n ) dx n g(χ n, χ n, u n, u n, ν(χ n, u n )) dh N Dχ n () (5.2) S(χ n,u n) = n µ n (), where µ n is a bounded sequence of Radon measures, such that, upon a choice of a (non-relabelled) subsequence, µ n µ. We divide the proof in three parts according to the choice of x0. First consider x 0 (S(u) \ S(χ)). In this case, we prove (5.8), taking into account the sequential characterization of γ sd (see Remark 4.3). The desired lower bound follows from proving that dµ dh N (x 0) γ sd (χ(x 0 ), [u](x 0 ), ν(u)(x 0 )), for H N -a.e. x 0 S(u) \ S(χ) and by letting η 0. Choose a sequence of radii k 0 such that µ( ν (x 0, k )) = 0, k N. Then we have that dµ dh N (x 0) k,n N g (χ n, [u n ], ν(u n )) dh N k ν(x 0, k ) S(u n)\s(χ n) = g (χ n (x 0 k y), [u n ](x 0 k y), ν(u n (x 0 k y)) dh N, k,n ν S(un)\S(χn) x 0 k where ν := ν(u)(x 0 ). Define and The point x 0 S(u) \ S(χ) is to be chosen H N -a.e so that χ n,k (y) := χ n (x 0 k y), (5.22) u n,k (y) := u n (x 0 k y) u (x 0 ). (5.23) χ n,k (y) χ(x 0 ) L ( ν) = 0, (5.24) k,n and u k,n (y) u λ,ν L ( ν;r d ) = 0, (5.25) k,n where λ = [u] (x 0 ), ν is as defined above, and u λ,ν is defined according to (3.9). Following arguments in [9], we get a diagonalizing sequence v k := u n(k),k such that v k u λ,ν in L and v k 0 in L p. Let χ k := χ n(k),k. Next we change slightly this sequence in order to fix χ(x 0 ). For that, we need to prove that g ( χ k (y), [v k ](y), ν(v k )) dh N = 0, k y ν: χ k (y) χ(x 0) =0 S(v k )\S( χ k ) where g (i,, ) = g(, i ) is as in (3.2). By (H 2 ), it is enough to prove that [v k ] (y) dh N. k y ν: χ k (y) χ(x 0) =0 S(v k )\S( χ k ) Similarly to the lower bound bulk, this is achieved by changing v k.