AN EXTENSION THEOREM IN SBV AND AN APPLICATION TO THE HOMOGENIZATION OF THE MUMFORD-SHAH FUNCTIONAL IN PERFORATED DOMAINS

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1 AN EXTENSION THEOREM IN SBV AND AN APPLICATION TO THE HOMOGENIZATION OF THE MUMFORD-SHAH FUNCTIONAL IN PERFORATED DOMAINS FILIPPO CAGNETTI AND LUCIA SCARDIA Abstract. The aim of this paper is to prove the existence of extension operators for SBV functions from periodically perforated domains. This result will be the fundamental tool to prove the compactness in a non coercive homogenization problem. Keywords: extension theorem, homogenization, Γ-convergence, integral representation, brittle fracture, Mumford-Shah functional 2000 Mathematics Subject Classification: 74Q99, 74R05, 74R10 1. Introduction In this paper we show the existence of an extension operator for special functions with bounded variation with a careful energy estimate. Our motivation comes from [1], where the same problem is addressed in the context of Sobolev spaces (see also [19]. The main achievement in the quoted paper is the existence of a suitable extension operator in periodic domains, with extension constants invariant under homothety. This result turns out to be the fundamental tool for the analysis of the asymptotic behaviour of integral functionals on perforated domains. It seems very natural to look for an extension of the homogenization results in [1] to non coercive functionals consisting of a volume and a surface integral, as those occurring in computer vision and in the mathematical theory of elasticity for brittle materials. More precisely, we are interested in the study of the asymptotic behaviour of the Mumford-Shah functional on a periodically perforated domain, as the size of the holes and the periodicity parameter of the structure tend to zero. We recall that for every pair (u, U with U R n open and bounded and u SBV 2 (U, the Mumford-Shah functional (see [20] is defined as MS(u, U := u 2 dx + H n 1 (S u U, U where u and S u are the absolutely continuous gradient and the jump set of u, respectively. Therefore, we are led to consider energies of the form F ε MS(u, Ω(ε if u L (Ω(ε SBV 2 (Ω(ε, (u, Ω := + otherwise in L 2 (1.1 (Ω, where Ω R n is open and bounded, and Ω(ε is obtained intersecting the domain Ω with the ε-homothetic of a periodic connected open set E with Lipschitz boundary. More precisely, we define Ω(ε := Ω E ε, where E ε := εe. In order to analyze the behaviour of the family (F ε as ε 0, we need the analogue in the SBV framework of the extension estimates obtained in [1]. To this aim, the fundamental tool is provided by the following theorem, that is the main result of the paper. 1

2 2 EXTENSION THEOREMS AND HOMOGENIZATION Theorem 1.1. Let D, A be open subsets of R n. Assume that A is bounded and that D is connected and has Lipschitz boundary. Then there exists an extension operator L : SBV 2 (D L (D SBV 2 (A L (A and a constant c = c(n, D, A > 0 such that (i Lu = u a.e. in A D, (ii Lu L (A = u L (D, (iii MS(Lu, A c MS(u, D, (1.2 for every u SBV 2 (D L (D. The constant c is invariant under translations and homotheties. In the case of Sobolev spaces, the analogue of condition (iii is given by (Lu 2 dx c u 2. (1.3 A The classical argument to prove last relation is firstly due to Tartar (see [21] and [9]. In the quoted papers, the proof of (1.3 relies on the classical Poincaré-Wirtinger inequality, which cannot be obtained in the SBV case. Indeed, it is possible to construct non constant SBV functions whose absolutely continuous gradient is zero almost everywhere. On the other hand, the available extension of the Poincaré-Wirtinger inequality (see [15] does not lead directly to (iii. For this reason, we decided to follow a different approach. For the sake of simplicity, let us explain the main idea of the present work in the following simplified version of Theorem 1.1. Theorem 1.2. Let D, A R n be bounded open sets with Lipschitz boundary and assume that D is connected, D A and A \ D A. Then there exists an extension operator L : SBV 2 (D L (D SBV 2 (A L (A and a constant c = c(n, D, A > 0 such that D (i Lu = u a.e. in D, (ii Lu L (A = u L (D, (iii MS(Lu, A c MS(u, D, (1.4 for every u SBV 2 (D L (D. The constant c is invariant under translations and homotheties. To prove Theorem 1.2, we first consider a local minimizer of MS, that is a solution ˆv of the following problem: min MS(w, D W : w SBV 2 (D W, w = u a.e. in D }, where W A is a sufficiently small neighbourhood of D A (see Fig. 1. Figure 1. The set A; the set D; the neighbourhood W. Then, we carry out a delicate analysis of the behaviour of the function ˆv in the set W. More precisely, we define the extension Lu in A\ D modifying the function ˆv in different ways, according to the measure of the set Sˆv (W \ D. If this measure is large enough, then we consider Lu defined as ˆv in D W and zero in the remaining part of A. In this way we have essentially increased the energy in the surface term

3 EXTENSION THEOREMS AND HOMOGENIZATION 3 only, of an amount that is comparable to the measure of S u D. This guarantees that properties (i (iii are satisfied in this case. On the other hand, if H n 1( Sˆv (W \ D is small, then we may use the elimination property proved in [15, 13] to detect a subset of W \ D where the function ˆv has no jump (see also Theorem 2.5. This allows us to apply the extension property proved in the Sobolev setting in each connected component of. As already mentioned, the previous result finds an application in the study of the asymptotic behaviour of the functionals F ε defined in (1.1, as made precise by the following theorem. Theorem 1.3. Let E be a periodic, connected, open subset of R n, with Lipschitz boundary, let ε > 0, and set E ε := ε E. Given a bounded open set Ω R n, set Ω(ε := Ω E ε. Then, there exists an extension operator T ε : SBV 2 (Ω(ε L (Ω(ε SBV 2 (Ω L (Ω and a constant k 0 > 0, depending on E and n, but not on ε and Ω, such that T ε u = u a.e. in Ω(ε, for every u SBV 2 (Ω(ε L (Ω(ε. T ε u L (Ω = u L (Ω(ε, MS(T ε u, Ω k 0 ( MS(u, Ω(ε + H n 1 ( Ω This means that we can fill the ( holes of Ω(ε by means of an extension of u, whose Mumford- Shah energy is kept bounded by k 0 MS(u, Ω(ε + H n 1 ( Ω, where the constant k 0 = k 0 (n, E depends on n, and E, but is independent of Ω, ε and u. This is the key estimate to prove compactness of minimizing sequences for (F ε and thus, to identify a class of functions where the Γ-limit is finite. Within this class, we give a more explicit expression for the Γ-limit, characterizing the volume and the surface densities by means of two separate homogenization formulas (see Theorem 7.2. For completeness we mention that a previous work (see [16] shows that a very different situation occurs when the homogeneous Neumann boundary conditions on Ω(ε are replaced by homogeneous Dirichlet boundary conditions. In particular, in this case an extension theorem is not needed, since every function u SBV 2 (Ω(ε L (Ω(ε admits a natural extension by zero to the whole Ω. We finally remark that the same homogenization result has been independently obtained in the recent paper [17], where the lack of coerciveness is overcome by means of a localized Poincaré-type inequality. In dimension n = 2, this allows to construct a function satisfying the required energy estimate, and conciding with the original function in a set of positive measure. Then, the general n-dimensional case is recovered by a slicing argument. The plan of the paper is as follows. In Section 2 we recall the basic properties of special functions with bounded variation and the extension results available in the Sobolev setting. In order to simplify the exposition, in Sections 3 and 4 we focus on the case in which the set A \ D, where the extension has to be performed, is compactly contained in A. More precisely, in Section 3 we prove Theorem 1.2, while Section 4 is devoted to the corresponding simplified version of Theorem 1.3 (see Theorem 4.1. Then, we face the general case, proving Theorem 1.1 and Theorem 1.3 in Sections 5 and 6, respectively. In Section 7 we study the Γ-limit of the sequence of functionals (1.1. Finally, we postpone some technical lemmas in the Appendix. 2. Preliminaries Let us give some definitions and results that will be widely used throughout the paper. We denote with Q the unit cube in R n, i.e. Q = ( n, 1 2, 2 1 while (ei i=1,...,n stands for the canonical basis of R n. We use the following compact notation for the opposite hyperfaces of the cube: Q ±,i := Q x i = ± 1 } i = 1,..., n. 2 We say that a set E R n is periodic if E + e i = E for every i = 1,..., n.

4 4 EXTENSION THEOREMS AND HOMOGENIZATION Moreover, we say that an open set E R n has a Lipschitz boundary at a point x E (or equivalently, that E is locally Lipschitz at x if there exist an orthogonal coordinate system (y 1,..., y n, a coordinate rectangle R = (a 1, b 1... (a n, b n containing x, and a Lipschitz function Ψ : (a 1, b 1... (a n 1, b n 1 (a n, b n such that E R = y R : y n < Ψ(y 1,..., y n 1 }. If this property holds true for every x E with the same Lipschitz constant, we say that E has Lipschitz boundary (or equivalently, that E is Lipschitz. We will denote with M n the set of all the n n matrices with real entries. For the identity map we use the notation Id, i.e., Id(x = x for every x R n. For an open set A, C 0 (A denotes the class of C functions with compact support in A. Finally, inta is the interior of a set A R n. We recall now some properties of rectifiable sets and of the space SBV of special functions with bounded variation. We refer the reader to [5] for a complete treatment of these subjects. A set Γ R n is rectifiable if there exist N 0 Γ with H n 1 (N 0 = 0, and a sequence (M i i N of C 1 -submanifolds of R n such that Γ \ N 0 i N M i. Let x Γ \ N 0, and let i N such that x M i. We define the normal to Γ at x as the normal ν Mi (x to M i at x. It turns out that the normal is well defined (up to the sign for H n 1 -a.e. x Γ. Let U R n be an open bounded set with Lipschitz boundary. We define SBV (U as the set of functions u L 1 (U such that the distributional derivative Du is a Radon measure which, for every open set A U, can be represented as Du(A = u dx + [u](x ν u (x dh n 1 (x, A A S u where u is the approximate differential of u, S u is the set of jump of u (which is a rectifiable set, ν u (x is the normal to S u at x, and [u](x is the jump of u at x. For every p (1, + we set SBV p (U = u SBV (U : u L p (U; R n, H n 1 (S u < + }. If u SBV (U and Γ U is rectifiable and oriented by a normal vector field ν, then we can define the traces u + and u of u SBV (U on Γ, which are characterized by the relations 1 lim r 0 r n u(y u ± (x dy = 0 for H n 1 a.e. x Γ, U B ± r (x where B ± r (x := y B r (x : (y x ν 0}. The following extension theorems are the Sobolev versions of Theorem 1.1 and Theorem 1.3, respectively (see [1, Lemma 2.6] and [1, Theorem 2.1]. Theorem 2.1. Let D, A be open subsets of R n. Assume that A is bounded and that D is connected and has Lipschitz boundary at each point of D Ā. Then, there exists a linear and continuous operator τ : H 1 (D H 1 (A such that, for every u H 1 (D A τu = u a.e. in A D, τu 2 dx k 1 u 2 dx, A D (τu 2 dx k 2 u 2 dx, (2.1 where k 1 = k 1 (n, D, A and k 2 = k 2 (n, D, A are positive constants depending only on n, D, and A. Theorem 2.2. Let E be a periodic, connected, open subset of R n, with Lipschitz boundary, let ε > 0, and set E ε := εe. Given a bounded open set Ω R n, set Ω(ε := Ω E ε. Then, there D

5 EXTENSION THEOREMS AND HOMOGENIZATION 5 exists a linear and continuous extension operator τ ε : H 1 (Ω(ε Hloc 1 (Ω and three constants k 3, k 4, k 5 > 0 depending on E and n, but not on ε and Ω, such that Ω(εk 3 Ω(εk 3 τ ε u = u a.e. in Ω(ε, τ ε u 2 dx k 4 u 2 dx, (τ ε u 2 dx k 5 Ω(ε Ω(ε u 2 dx, for every u H 1 (Ω(ε. Here we used the notation Ω(εk 3 := x Ω : dist(x, Ω > εk 3 }. We give now the definition of a local minimizer for the Mumford-Shah functional. We recall that for an open set U R n and for w SBV 2 (U MS(w, U = w 2 dx + H n 1 (S w U. (2.2 U Definition 2.3. Let Ω R n be open. We say that w SBV 2 (Ω is a local minimizer for the functional MS(, Ω if MS(w, A MS(v, A for every open set A Ω, whenever v SBV 2 (Ω and v w} A Ω. Next theorem provides an estimate of the measure of the jump set for a local minimizer of the Mumford-Shah functional (see [15]. Theorem 2.4 (Density lower bound. There exists a strictly positive dimensional constant ϑ 0 = ϑ 0 (n with the property that if u SBV 2 (Ω is a local minimizer for the functional MS(, Ω defined in (2.2 for an open set Ω R n, n 2, then H n 1 (S u B ϱ (x > ϑ 0 ϱ n 1 for every ball B ϱ (x Ω with centre x S u and radius ϱ > 0. An equivalent but more appealing formulation of the previous theorem is the following elimination property (see [13]. Theorem 2.5 (Elimination property. Let Ω R n be open. There exists a constant β > 0 independent of Ω such that, if u SBV 2 (Ω is a local minimizer for the functional MS(, Ω defined in (2.2 and B ϱ (x 0 Ω is any ball with centre x 0 Ω with then S u B ϱ/2 (x 0 =. H n 1 (S u B ϱ (x 0 < βϱ n 1, We state now a theorem which provides an approximation result for SBV functions, with the property that the value of the Mumford-Shah functional along the approximating sequence converges to the value of the Mumford-Shah functional on the limit function. For the proof we refer to [10]. Theorem 2.6. Let Ω R n be open. Assume that Ω is locally Lipschitz and let u SBV 2 (Ω. Then there exists a sequence (u h SBV 2 (Ω such that for every h N (i S uh (ii S uh is essentially closed; is a polyhedral set; (iii u h W k, (Ω \ S uh for every k N; and such that (u h approximates u in the following sense: (iv u h u strongly in L 2 (Ω, (v u h u strongly in L 2 (Ω, (vi H n 1 (S uh H n 1 (S u.

6 6 EXTENSION THEOREMS AND HOMOGENIZATION 3. Compactly contained hole: extension for a fixed domain In this section we prove Theorem 1.2. This is a simplified version of Theorem 1.1, under the additional assumption that the set A \ D, where the extension has to be performed, is compactly contained in A (see Fig. 1a and 1b. In this way, it will be possible to highlight the main ideas of the present work, without facing the further difficulties of the general case, that will be treated in Section 5. In order to prove the extension result we need to define, for every open set, a reflection map with respect to bounded Lipschitz subsets of the boundary, as made clear in the following theorem. Theorem 3.1. Let D R n be an open set, and assume that Λ D is a bounded, relatively open, nonempty Lipschitz set, with Λ x D : D has Lipschitz boundary at x}. Then, there exists a bounded open set W R n with Lipschitz boundary, such that Λ = W D, and a bilipschitz map φ : W W with φ Λ = Id and φ(w ± = W, where W + := W D and W := W (R n \ D. A pictorial idea of the previously stated reflection result is illustrated in Fig 2. Figure 2. The sets W +, W, Λ and the bilipschitz map φ. Proof. Since Λ is Lipschitz and compact, we can find a finite open cover U 1,..., U m of Λ such that we can associate to every U j a vector u 0 j Rn and a parameter η j (0, 1] with the following property. If x Λ U j for some j, then for every t (0, 1] and for every u j R n such that u j u 0 j < η j it turns out that x + t u j D and x t u j R n \ D. Set η := min j η j. Now, for every index j we fix an open set V j U j such that V 1,..., V m is still a covering of Λ. Let (ψ j j=1,...,m be a partition of unity for Λ subordinate to (V j j=1,...,m, i.e., ψ j C 0 (R n, supp ψ j V j, 0 ψ j 1 in R n, m ψ j = 1 on Λ. Let us fix α 0 > 0 so that for every collection of vectors u 1,..., u m } satisfying u i u 0 i < η for every i, we have m α 0 i=1 j=1 u i < dist(v j, U j for j = 1,..., m. Let us define Bη m (u 0 := u = (u 1,..., u m (R n m : u i u 0 i < η for every i}. For every α [ α 0, α 0 ] and for every u Bη m (u 0, we define the C function ru α : R n R n as r α u (x := x + α m ψ j (x u j. It turns out that, by construction, ru α Id has compact support and ru α Id 0 in C0 (R n ; R n as α 0. Let us set Ψ u (x := m j=1 ψ j(x u j and Ψ 0 (x := m j=1 ψ j(x u 0 j. Following the argument used in [14, Proposition 1.2], it is possible to show that, for every x Λ, we have that for every u Bη m (u 0, x + α Ψ u (x D if 0 < α α 0 and x + α Ψ u (x R n \ D if α 0 α < 0. j=1

7 EXTENSION THEOREMS AND HOMOGENIZATION 7 We claim that there exists η 0 (0, η] such that for every x Λ we have the following property: v Ψ 0 (x < η 0 x + α v D if 0 < α α0, x + α v R n \ D if α 0 α < 0. (3.1 We notice that in order to obtain (3.1 it is sufficient to prove that if v satisfies v Ψ 0 (x < η 0, then v = Ψ u (x for some u B m η (u 0. (3.2 Let us show (3.2. Let us fix x Λ; we define the linear map I x : ( R n m R n as u = (u 1,..., u m I x (u := Ψ u (x = m ψ j (x u j. Since x Λ, we have that j ψ j(x = 1. Hence, there exists ī 1,..., m} such that ψī(x 1 m. We claim that I x (B m η (u 0 contains a neighbourhood of I x (u 0. First of all, let us notice that I x (B m η (u 0 = I x( B η (u 0 1 B η (u 0 m A, (3.3 where A := I x( u 0 1} u 0 ī 1} B η (u 0 ī u 0 ī+1} u 0 m}. Easy computations show that y I x (u 0 : y A } = B η ψī(x(0. Therefore we can rewrite A as A = I x (u 0 + B η ψī(x(0 = B η ψī(x(i x (u 0 B η m (Ix (u 0. (3.4 The same argument can be repeated for every x Λ. Let us now show that (3.2 holds true with η 0 := η m. Let x Λ and v Rn such that v Ψ 0 (x < η 0, i.e., v B η0 (Ψ 0 (x = B η0 (I x (u 0. From (3.3 and (3.4 we have that v A I x (Bη m (u 0, hence there exists u Bη m (u 0 such that v = I x (u = Ψ u (x. This proves (3.2. For every x 0 Λ let us consider the following Cauchy problem: ẋ(t = Ψ 0 (x(t, (3.5 x(0 = x 0. We denote by (x 0, t Φ(x 0, t the flow associated to (3.5. Using (3.1 and the compactness of Λ, we have that there exists t 0 > 0, independent of x 0 Λ, such that Φ(x 0, t : t (0, t 0 } D and Φ(x 0, t : t (0, t 0 } R n \ D. Clearly, the restriction Φ Λ ( t0,t 0 is bijective. In particular we have that Φ(x 0, 0 : x 0 Λ} = Λ. Now we define W, W +, W as W := Φ(x 0, t : (x 0, t Λ ( t 0, t 0 } (3.6 W + := W D = Φ(x 0, t : (x 0, t Λ (0, t 0 }, (3.7 W := W (R n \ D = Φ(x 0, t : (x 0, t Λ ( t 0, 0 }. (3.8 Using classical properties of the flow, and the fact that Λ is Lipschitz, it is possible to show that the map Φ Λ ( t0,t 0 : Λ ( t 0, t 0 W is bilipschitz. We define φ : W W in the following way. Let x W. Then, by definition of W, there exists a pair (x 0, t Λ ( t 0, t 0 such that x = Φ(x 0, t. We set φ(x := Φ(x 0, t. This map is bijective and bilipschitz, and satisfies the required properties. Hence the theorem is proved. In the periodic case, the previous theorem can be modified in the following way. Corollary 3.2. Let E R n be a periodic open connected set with Lipschitz boundary. Then, there exists a periodic neighbourhood W of E with Lipschitz boundary, and a bilipschitz periodic map φ : W W such that φ E = Id and φ(w ± = W, where W + := W E and W := W (R n \E. j=1

8 8 EXTENSION THEOREMS AND HOMOGENIZATION Proof. We repeat the proof of Theorem 3.1, with D = E and Λ = E Q. We observe that, due to the periodicity of E, the open covering U 1,..., U m of Λ and the vectors u 0 1,..., u 0 m can be chosen to be periodic, in the following sense. If j 1,..., m} and i 1,..., n} are such that U j Q +,i, then there exists k 1,..., m} such that U k = U j + e i and u 0 j = u0 k. Similarly, If j 1,..., m} and i 1,..., n} are such that U j Q,i, then there exists k 1,..., m} such that U k = U j e i and u 0 j = u0 k. By the previous argument, proceeding as in the proof of Theorem 3.1, it is possible to construct a vector field Ψ 0 : R n R n such that Ψ 0 Q +,i = Ψ 0 Q,i for every i = 1,..., n. Therefore without loss of generality we can assume that Ψ 0 is Lipschitz and periodic. Indeed, we can otherwise replace it with the Lipschitz periodic extension of Ψ 0 Q. As in the proof of Theorem 3.1, for every x 0 E Q we consider the Cauchy problem (3.5 and we denote with Φ(x 0, t the associate flow. Then we define a positive real number t 0, the set K := Φ(x 0, t : (x 0, t E Q ( t 0, t 0 }, and a periodic bilipschitz map ˆφ : K K such that ˆφ E Q = Id. Moreover, K and ˆφ can be extended by periodicity, so that the set ( W := int (K + h h Z n and the function φ : W W defined as φ(x := h + ˆφ(x h for x (K + h and h Z n have the required properties. We can now prove Theorem 1.2. Proof of Theorem 1.2. Let u SBV 2 (D L (D. By Theorem 3.1 applied to Λ = D A, we can find an open set W containing D A, and a bilipschitz map φ : W W such that φ D A = Id and φ(w ± = W, where W + = W D and W = W (R n \ D. Without loss of generality, we can assume W A. We define v : D W R as u(x if x D, v(x := u(φ(x if x W. It turns out that v SBV 2 (D W and that the following estimate holds true: MS(v, D W (1 + C 1 MS(u, D, (3.9 where, setting ψ := φ 1, the constant C 1 = C 1 (n, D, A is given by C 1 := det ψ ( ψ T L (W ;M n. (3.10 For the rigorous proof of (3.9 we refer to Theorem 8.1 in the Appendix. Now, let us consider a solution ˆv of the following problem: } min w 2 dx + H n 1 (S w : w SBV 2 (D W, w = u a.e. in D. D W By definition of ˆv and using (3.9, we have that ˆv = u a.e. in D and MS(ˆv, D W MS(v, D W (1 + C 1 MS(u, D. (3.11 By a truncation argument, it follows that ˆv L (D W = u L (D. Let us analyze more carefully the structure of W. By (3.6, (3.7 and (3.8, we have W = Φ(x 0, t : (x 0, t ( D A ( t 0, t 0 }, W + = W D = Φ(x 0, t : (x 0, t ( D A (0, t 0 }, W = W (R n \ D = Φ(x 0, t : (x 0, t ( D A ( t 0, 0 },

9 EXTENSION THEOREMS AND HOMOGENIZATION 9 where the function (x 0, t Φ(x 0, t is the flux associated to problem (3.5. Now we set Γ := Φ(x 0, t 0 /2 : x 0 D A} W. For every z Γ let ϱ(z be defined as ϱ(z := sup ϱ > 0 : B ϱ (z W }, and let γ be the positive constant given by The situation is shown in Fig. 3. γ := 1 2 inf z Γ ϱ(z. Figure 3. A point z Γ and the ball B γ (z. Here A, D and W are those shown in Figure 1. Let ω > 0 be defined as ω := β γ n 1, where β > 0 is the constant given by the Elimination Theorem 2.5. In order to construct the required extension, we need to distinguish two cases, that will be treated in a different way. First case: large jump set We assume that H n 1 (Sˆv W ω. Let us define Lu as ˆv(x if x D W, (Lu(x := ( if x A \ (D W. It turns out that Lu SBV 2 (A. Moreover, using (3.11 we have MS(Lu, A MS(ˆv, D W + H n 1 ( W \ D = MS(ˆv, D W + C 2 ω MS(ˆv, D W + C 2 H n 1 (Sˆv W max1, C 2 }MS(ˆv, D W where C 2 = C 2 (n, D, A is the positive constant given by (1 + C 1 max1, C 2 }MS(u, D, (3.13 C 2 := Hn 1 ( W \ D. ω Second case: small jump set We assume that H n 1 (Sˆv W < ω. Let us fix z Γ and let us consider the ball B γ (z W. Clearly, H n 1 (Sˆv B γ (z H n 1 (Sˆv W < ω. By our definition of ω, this implies that H n 1 (Sˆv B γ (z < β γ n 1. Hence, by Theorem 2.5 we have that Sˆv B γ/2 (z = (see Fig. 4. The same argument can be repeated for every z Γ. Therefore, we deduce that the set W defined as := z Γ B γ/2 (z does not intersect the jump set of ˆv (see Fig. 5.

10 10 EXTENSION THEOREMS AND HOMOGENIZATION Figure 4. The ball B γ 2 (z. Figure 5. The sets, U and U. Without loss of generality, we can assume connected, otherwise the same argument can be repeated for every connected component of. We observe that disconnects A \ D. Indeed, we can write (A \ D \ = U U, where U and U are open, connected disjoint sets, with D U. Now, let us define Lu as ˆv(x if x A \ ( U (Lu(x :=, (3.14 (τ ˆv(x if x ( U, where τ denotes the extension operator from H 1 ( to H 1 ( U provided by Theorem 2.1. By relation (2.1, we have that (τ ˆv 2 dx k 2 ˆv 2 dx, (3.15 U where k 2 = k 2 (n,, U. Furthermore, up to truncation, we can always assume that the L bound is preserved. Then, it turns out that Lu SBV 2 (A, Lu = u a.e. in D and Lu L (A = u L (D. By (3.15, we have MS(Lu, A = MS(ˆv, D U + (τ ˆv 2 dx U MS(ˆv, D W + k 2 MS(ˆv, where in the last inequality we used (3.11. max1, k 2 }MS(ˆv, D W (1 + C 1 max1, k 2 }MS(u, D, (3.16 Estimate in the general case. The function Lu defined in (3.12 and (3.14 respectively clearly satisfies properties (i and (ii of Theorem 1.2. By (3.13 and (3.16, estimate (1.4 holds true setting c(n, D, A := (1 + C 1 max1, C 2, k 2 }. The arguments used in the proof are clearly invariant under translations. Thus, it remains to prove that the constant c is invariant under homotheties. Let w SBV 2 (D L (D and let λ > 0. We define the function w λ : λd R as w λ (x := λ w( x λ for every x λd. Let Lw SBV 2 (A L (A denote the extension provided by the theorem just proved. Then, we can define an extension operator L λ from SBV 2 (λd L (λd to SBV 2 (λa L (λa as (L λ w λ (x := ( x λ (Lw for every x λa. λ This concludes the proof, since MS(L λ w λ, λa = λ n 1 MS(Lw, A c λ n 1 MS(w, D = c MS(w λ, λd.

11 EXTENSION THEOREMS AND HOMOGENIZATION Compactly contained hole: ε- periodic extension In this section we prove a simplified version of Theorem 1.3. We will consider the case in which the set E is obtained removing from the unit square a compactly contained hole and repeating this construction by periodicity (see Fig. 6. More precisely, let F Q be an open Lipschitz set. We assume that E is given by E := h Z n ((Q \ F + h. (4.1 Figure 6. The set Ω(ε. Here, for simplicity, F is a cube, and F and Q are concentric. We state now the main result of this section. Theorem 4.1. Fix ε > 0. Let Ω R n be a bounded open set with Lipschitz boundary, and let E R n the periodic set defined as in (4.1. Set E ε := εe and Ω(ε := Ω E ε. Then there exists an extension operator T ε : SBV 2 (Ω(ε L (Ω(ε SBV 2 (Ω L (Ω and a constant k 0 > 0 depending on E and n, but not on ε and Ω, such that for every u SBV 2 (Ω(ε. T ε u = u a.e. in Ω(ε, T ε u L (Ω = u L (Ω(ε, MS(T ε u, Ω k 0 ( MS(u, Ω(ε + H n 1 ( Ω Proof. Let u SBV 2 (Ω(ε L (Ω(ε. Let Z ε be the set of vectors h Z n such that the ε-homothetic of the domain h + Q has a nonempty intersection with Ω, and let us introduce an ordering of its elements. More precisely, we set Z ε := h Z n : ε(h + Q Ω } = h 1, h 2,..., h N(ε }, (4.2 where with N(ε N we denoted the cardinality of Z ε. For shortening the notation, we set and We define ũ : E ε R as Q k := h k + Q, Q ε k := ε Q k k = 1,..., N(ε, (4.3 Ω Q (ε := int ũ := ( N(ε Q ε k u in Ω(ε 0 in E ε \ Ω(ε.. (4.4 Clearly, the function ũ satisfies ũ = u a.e. in Ω(ε, ũ L (E ε = u L (Ω(ε, and Notice that we can write MS(ũ, E ε = MS(ũ, E ε MS(u, Ω(ε + H n 1 ( Ω. (4.5 N(ε MS(ũ, Q ε k E ε + H n 1 (Sũ E ε ( N(ε Q ε k. (4.6

12 12 EXTENSION THEOREMS AND HOMOGENIZATION Let us denote with L ε k : SBV 2 (Q ε k Eε L (Q ε k Eε SBV 2 (Q ε k L (Q ε k the extension operator provided by Theorem 1.2, with k = 1,..., N(ε, and we define v ε SBV 2 (Ω Q (ε L (Ω Q (ε as v ε (x := (L ε kũ(x if x Q ε k, k 1,..., N(ε}. We have that for every k = 1,..., N(ε v ε = ũ a.e. in Q ε k E ε, v ε L (Q ε k = ũ L (Q ε k Eε, MS(v ε, Q ε k c MS(ũ, Q ε k E ε. (4.7 Since the constant provided by the theorem is invariant under translations and homotheties, c = c(n, F, Q is independent of k and ε. Then, using (4.6 and (4.7, we get ( N(ε N(ε MS(v ε, Ω Q (ε = MS(v ε, Q ε k + H (S n 1 v ε E ε N(ε c MS(ũ, Q ε k E ε + H n 1 (Sũ E ε k 0 MS(ũ, E ε, Q ε k ( N(ε where k 0 := maxc, 1}. Combining the previous expression with (4.5 we have therefore the claim follows defining T ε u := v ε Ω. MS(v ε, Ω Q (ε k 0 ( MS(u, Ω(ε + H n 1 ( Ω, Q ε k 5. General connected sets: extension for a fixed domain In this section we prove Theorem 1.1. First of all we state two lemmas that follow, up to some slight variation, from [1, Lemma 2.2] and from [1, Lemma 2.3]. Lemma 5.1. Let P, ω, ω be open subsets of R n. Assume that ω, ω are bounded, with ω ω and that P has Lipschitz boundary at each point of P ω. Then the number of connected components of P ω that intersect P ω is finite. Lemma 5.2. Let D be a connected open subset of R n, with Lipschitz boundary, and let A R n be open and bounded, with A D. Then, there exists k N, k 2, such that A kq and A D is contained in a single connected component of kq D. Proof of Theorem 1.1. Let k be given by Lemma 5.2; applying Lemma 5.1 with P = R n \D, ω = A and ω = (k + 1Q, we have that the number of connected components of (R n \ D ((k + 1Q that intersect A is finite, say M N. Let us denote these connected components by U 1,..., U M. The situation is described in Figure 7. Notice that, since A kq, then δ := dist(a, (kq > 0. We want to extend the function u to the sets U 1 A,..., U M A, in such a way that conditions (i, (ii and (iii of Theorem 1.1 are satisfied. Let W, W ±, Φ, φ and t 0 be those defined in the proof of Lemma 3.1 with M Λ := Λ i, where Λ i := U i D, for i = 1,..., M. Let us define the sets where, for i = 1,..., M, i=1 M M Θ 1 := Θ i 1,, Θ 2 := Θ i 2, i=1 Θ i 1 := Λ i (kq, Possible restriction of the interval [ t 0, t 0 ]. i=1 Θ i 2 := Λ i ((k + 1Q.

13 EXTENSION THEOREMS AND HOMOGENIZATION 13 Figure 7. The set D; the set A (here k = 2; the sets U i s (notice that M = 4. In the case Θ 2 we restrict the interval [ t 0, t 0 ] to some [ η, η], with η (0, t 0 ], to guarantee that the image under the map Φ of an η-neighbourhood of Θ 1 is well separated by A. More precisely, we proceed in the following way. If Θ 2 = we just set η := t 0. If instead Θ 2, we have also Θ 1. Then, for every x 0 Θ 1, x 0 = Φ(x 0, 0 and dist(x 0, A δ. Since (x, t Φ(x, t is uniformly continuous in the compact set Θ 1 [ t 0, t 0 ], and x dist(x, A is continuous (in fact Lipschitz, there exists η (0, t 0 ] such that dist(φ(x 0, t, A > δ/2 for every (x 0, t Θ 1 [ η, η], i.e., dist ( Φ ( Θ 1 [ η, η], A > δ/2. Notice that, since Θ 1 Θ 2 =, the sets Θ 1 [ η, η] and Θ 2 [ η, η] are mapped by the flow Φ into two parallel (in the sense of the flow Φ disjoint sets Φ(Θ 1 [ η, η] and Φ(Θ 2 [ η, η]. Thus, dist ( Φ(Θ 1 [ η, η], Φ(Θ 2 [ η, η] > 0. (5.1 Definition of an auxiliary minimum problem. We define the following subsets of W W := Φ(Λ ( η, η, W + := Φ(Λ (0, η, W := Φ(Λ ( η, 0. Without loss of generality, we assume that W (k + 2Q. This will be useful in order to prove Theorem 1.3. Notice that W W, and that φ(w = W +. Now we define v : D W R as u(x if x D, v(x = u(φ(x if x W. It turns out that v SBV 2 (D W and MS(v, D W (1 + C 1 MS(u, D, (5.2 where, setting ψ := φ 1, C 1 = C 1 (n, D, A is given by (see Theorem 8.1 C 1 := det ψ ( ψ T L (W ;M n. (5.3 Let us consider a solution ˆv of the following problem: } min w 2 dx + H n 1 (S w : w SBV 2 (D W, w = u a.e. in D. D W By definition of ˆv and using (5.2, we have ˆv = u a.e. in D and MS(ˆv, D W MS(v, D W (1 + C 1 MS(u, D. (5.4 A truncation argument shows that ˆv L (D W = u L (D.

14 14 EXTENSION THEOREMS AND HOMOGENIZATION Now, for every i = 1,..., M, we denote with Γ i the connected component of Φ(Λ i, η/2 kq intersecting A. We notice that Γ i W, for i = 1,..., M. Then we set Γ := M Γ i W. (5.5 For every z Γ, let ϱ(z be defined as ϱ(z := sup ϱ (0, µ : B ϱ (z W }, where δ if Θ 2, µ := 2 + otherwise, i=1 and let γ be given by γ := 1 2 infϱ(z : z Γ}. Thanks to (5.1, we have γ > 0. In the case Θ 2 we require γ < δ/2 since, as will be clear in the sequel, we need to disconnect the sets U i kq (for i = 1,..., M, by covering Γ with balls of radius γ 2. Let ω > 0 be defined as ω := β γ n 1, where β > 0 is the constant given by the Elimination Theorem 2.5. In order to construct the required extension, we need to distinguish two cases, that will be treated in a different way. First case: large jump set We assume that H n 1 (Sˆv W ω. Let us define Lu as ˆv(x if x A (D W, (Lu(x := 0 if x A \ (D W. (5.6 It turns out that Lu SBV 2 (A. Moreover, using relations (5.4 and (5.9, where we set MS(Lu, A MS(ˆv, A (D W + H n 1 ( W \ D MS(ˆv, D W + C 2 ω MS(ˆv, D W + C 2 H n 1 (Sˆv W max1, C 2 }MS(ˆv, D W (1 + C 1 max1, C 2 }MS(u, D, (5.7 C 2 := Hn 1 ( W \ D. ω Second case: small jump set We assume that H n 1 (Sˆv W < ω. Let us fix i 1,..., M} and z Γ i, and let us consider the ball B γ (z W. Clearly, H n 1 (Sˆv B γ (z H n 1 (Sˆv W < ω. By our definition of ω, this implies that H n 1 (Sˆv B γ (z < β γ n 1. Hence, by Theorem (2.5 we have that Sˆv B γ/2 (z =. The same argument can be repeated for every z Γ i. Therefore we deduce that, for every i = 1,..., M, the set i defined as i := z Γ i B γ/2 (z W does not intersect the jump set of ˆv. Moreover, by definition, i is Lipschitz, connected, and disconnects the set U i kq. Indeed, for every i = 1,..., M, we can write (U i kq\ i = U i U i, where U i and U i are open, disjoint, and D U i. The situation is illustrated in Figure 8, where for simplicity we focused on the set U 4 of the previous Figure 7. Then, we define M := i. i=1

15 EXTENSION THEOREMS AND HOMOGENIZATION 15 Figure 8. In this figure we present a step-by-step construction of the set 4. Notice that, by construction, 1,..., M are the connected components of the set. We underline that this fact is crucial in order to get the desired extension, since we are going to apply M times Theorem 2.1, by extending the function u from the sets i. Now, let us define the function Lu as ˆv(x if x A \ U i for i = 1,..., M, (Lu(x := (τ iˆv(x if x A U i (5.8 for i = 1,..., M, where, for every i = 1,..., M, τ i denotes the extension operator provided by Theorem 2.1 from H 1 ( i to H 1 ( i ( i U i U i. By (2.1, we have that for every i = 1,..., M (τ iˆv 2 dx K 2 ˆv 2 dx, (5.9 i where we set i ( i U i U i K 2 := max k 2(n, i, i ( i U i U i }. (5.10 i=1,...,m Furthermore, up to truncation, we can always assume that the L bound is preserved. Then, Lu SBV 2 (A, Lu = u a.e. on A D and Lu L (A = u L (D. To conclude the proof of the theorem in the case of a small jump set it remains to estimate the Mumford-Shah functional of the extended function Lu on A in terms of the function u on D. By (5.4 and (5.9, ( M M MS(Lu, A = MS ˆv, (A \ U (τ iˆv 2 dx i=1 i + i=1 A U i M MS(ˆv, D W + K 2 MS(ˆv, i i=1 max1, K 2 }MS(ˆv, D W (1 + C 1 max1, K 2 }MS(u, D. (5.11 Estimate in the general case. The function Lu defined in (5.6 and (5.8 clearly satisfies properties (i and (ii of Theorem 1.1. By (5.7 and (5.11, we obtain (1.2 setting c(n, D, A := (1 + C 1 max1, C 2, K 2 }. The invariance of the constant c under translations and homotheties follows as in the proof of Theorem 1.2.

16 16 EXTENSION THEOREMS AND HOMOGENIZATION 6. General connected domains: ε-periodic extension We now prove Theorem 1.3, stated in the Introduction. For a pictorial idea of the set E, see Figure 9. Proof. Following closely the proof of Theorem 4.1, we define Z ε, Q k, Q ε k and Ω Q(ε as in (4.2, (4.3 and (4.4, respectively. From now on, we will consider the positive constants M, k, ω, C 1, K 2, C 2, the sets W, W +, W, Γ,, 1,..., M, U 1,..., U M, and the bilipschitz function φ : W W defined in the proof of Theorem 1.1, with D = E and A = Q. We introduce also the sets N(ε N(ε W := (h k + W, W ± := (h k + W ±, and the function φ ε : εw εw given by ( y φ ε εhk (y := εφ + εh k y ε(h k + W, k = 1,... N(ε. ε By Corollary 3.2, the sets W, W ± are Lipschitz, and the function φ ε is well defined and bilipschitz. Setting ψ := φ 1 and ψ ε := (φ ε 1, we have ( z ψ ε εhk (z = εψ + εh k ε for every z ε(h k + W. Notice that ( z ( z ψ ε εhk (z = ( ψ ε z ε(h k + W, k = 1,... N(ε, (6.1 where z denotes the gradient with respect to the variable z. Let ũ : E ε R be defined as u in Ω(ε, ũ := 0 in E ε \ Ω(ε. Clearly the function ũ satisfies ũ = u in Ω(ε, ũ L (E ε = u L (Ω(ε, and MS(ũ, E ε MS(u, Ω(ε + H n 1 ( Ω. (6.2 We define the extension v : E ε εw R as ũ(x if x E ε, v(x := ũ ( φ ε (x if x εw. Figure 9. A periodic connected set with its periodicity cell. Notice that E Q is not connected and that, in this case, k = 2.

17 EXTENSION THEOREMS AND HOMOGENIZATION 17 It turns out that v SBV 2 (E ε εw ; moreover, by Theorem 8.1 and using (6.2, MS(v, E ε εw (1 + C 1 MS(ũ, E ε (1 + C 1 ( MS(u, Ω(ε + H n 1 ( Ω, (6.3 where, thanks to (6.1, the constant C 1 is independent of ε and is given by C 1 = det ψ ( ψ T L (W ;M n = det( z ψ ε ( z ψ ε T L (εw ;M n. Consider now a solution ˆv ε to the minimum problem min w 2 dx + H n 1 (S w : w SBV 2 (E ε εw, w = ũ a.e. in E ε}. E ε εw As in the proof of Theorem 1.1, we get that ˆv ε L (E ε εw = u L (Ω(ε. Moreover, since v is a competitor for the minimum problem defining ˆv ε, using (6.3 we have MS(v, E ε εw MS(ˆv ε, E ε εw (1 + C 1 MS(ũ, E ε (1 + C 1 ( MS(u, Ω(ε + H n 1 ( Ω. (6.4 In order to construct the required extension, we divide the cubes into two groups, that will be treated in a different way. More precisely, we enumerate the vectors h 1,..., h N(ε as follows: H n 1 ( ε(h k + W Sˆv ε ω ε n 1 k = 1,..., N 1 (ε, (6.5 where N 1 (ε 0, 1,..., N(ε}, and H n 1 ( ε(h k + W Sˆv ε < ω ε n 1 k = N 1 (ε + 1,..., N(ε. (6.6 First case: large jump set. We start proving a bound for the number N 1 (ε of cubes with large jump set, showing that they cannot be too many as ε approaches zero. Indeed, by (6.4 we have MS(ũ, E ε 1 (1 + C 1 MS(ˆvε, E ε 1 εw (1 + C 1 MS(ˆvε, εw 1 (1 + C 1 C 3 1 (1 + C 1 C 3 N(ε N 1(ε ω ε n 1 (1 + C 1 C 3, N 1(ε MS (ˆv ε, ε(h k + W Sˆv ε H n 1( ε(h k + W Sˆv ε where, recalling that W (k + 2Q, we denoted with C 3 = C 3 (n, k the smallest integer such that each point x R n is contained in at most C 3 different cubes of the form (h + (k + 2Q h Z n. From the previous estimate it follows that Second case: small jump set. Once again, following the proof of Theorem 1.1, and defining N 1 (ε (1 + C 1C 3 ω ε n 1 MS(ũ, E ε. (6.7 F := N(ε k=n 1(ε+1 (h k +, F ε := εf, we have that Sˆv ε F ε =. Notice that, arguing as it has been done to prove Corollary 3.2, one can show that F is Lipschitz. We also set G := N(ε k=n 1(ε+1 M (h k + U j, j=1 G ε := εg.

18 18 EXTENSION THEOREMS AND HOMOGENIZATION Next lemma, whose proof is postponed to the Appendix, gives the correct estimate for the cubes with small jump set. Lemma 6.1. There exists an extension operator J ε : H 1 (F ε H 1 (F ε G ε and a constant C 4 = C 4 (n, E, independent of ε and Ω, such that, for every w H 1 (F ε, J ε w = w a.e. in F ε, J ε w L (F ε G ε = w L (F ε, the following estimate holds true: (J ε w 2 dx C 4 w 2 dx. F ε G ε F ε Estimate in the general case. Let us denote with L ε k : SBV 2 (Q ε k Eε L (Q ε k Eε SBV 2 (Q ε k L (Q ε k the extension operator provided by Theorem 1.1, with k = 1,..., N 1 (ε. We define the function v ε : Ω Q (ε R as (L ε kũ(x if x Qε k, k = 1,..., N 1(ε, v ε (x := (J εˆv ε (x if x F ε G ε, ˆv ε (x otherwise in Ω Q (ε. Notice that v ε = ˆv ε = ũ = u a.e. in Ω(ε and v ε L Ω Q (ε = u L (Ω(ε. Moreover, MS(v ε, Q ε k c MS(ũ, Q ε k E ε k = 1,..., N(ε. (6.8 Recalling that the constant provided by Theorem 1.1 is invariant under translations and homotheties, c = c(n, E, Q is independent of k and ε. We notice that the function v ε can possibly jump along the boundaries of the cubes Q ε k, for k = 1,..., N 1(ε, and this contribution is controlled by N 1 (εε n 1. Therefore we have, by (6.2, (6.4, (6.7 and (6.8, MS(v ε, Ω Q (ε N 1(ε N 1(ε c MS(ũ, Q ε k E ε + (1 + C 1C 3 ω MS(L ε kũ, Q ε k + N 1 (εε n 1 + MS(J εˆv ε, F ε G ε + MS(ˆv ε, E ε εw k 0 MS(ũ, E ε k 0 ( MS(u, Ω(ε + H n 1 ( Ω, where ( C3 k 0 := c + (1 + C 1 ω + C Therefore, the claim follows setting T ε u := v ε Ω. MS(ũ, E ε + C 4 MS(ˆv ε, F ε + MS(ˆv ε, E ε εw 7. Homogenization of Neumann problems In this section we consider an application of the extension property to a non coercive homogenization problem. The starting point is the energy associated to a function u SBV 2 (Ω, i.e., F ε (u := u 2 dx + H n 1 (Ω(ε S u, (7.1 Ω(ε where Ω(ε := Ω εe, and E is an open connected periodic subset of R n with Lipschitz boundary. Notice that we can rewrite the functional F ε as ( x x F ε (u = a u 2 dx + a( dh n 1 (x, Ω ε S u ε where a is a Q-periodic function given by 1 in E, a(y = 0 in R n \ E.

19 EXTENSION THEOREMS AND HOMOGENIZATION Compactness. In this subsection we prove a compactness result for a sequence having equibounded energy F ε. Theorem 7.1. Let (u ε SBV 2 (Ω L (Ω be a sequence satisfying the following bounds: u ε L (Ω(ε c and F ε (u ε c < +, where c > 0 is a constant independent of ε. Then there exist a sequence (ũ ε SBV 2 (Ω and a function u SBV 2 (Ω such that ũ ε = u ε a.e. in Ω(ε for every ε and (ũ ε converges to u weakly in BV (Ω. Proof. Let us define ũ ε := T ε u ε, where T ε is the extension operator defined in Theorem 1.3. Then, from the assumptions on the sequence (u ε and using the properties of T ε we obtain ũ ε L (Ω c and MS(ũ ε, Ω c < +. Hence, by Ambrosio s compactness Theorem we have directly the claim Integral representation. The present subsection is devoted to the identification of the Γ-limit of the sequence (F ε in SBV 2 (Ω, with respect to the strong convergence in L 2 (Ω. Let us define for u SBV 2 (Ω the functional F hom as F hom (u := f hom ( u dx + ϕ(ν u dh n 1. (7.2 Ω S u The limit densities f hom : R n [0, + ] and ϕ : S n 1 [0, + ] are characterized by means of the following homogenization formulas: } f hom (ξ := min a(y ξ + w(y 2 dy : w H#(Q 1, (7.3 Q where H# 1 (Q denotes the space of H1 (Q functions with periodic boundary values on Q, and } 1 ϕ(ν := lim inf a(y dh n 1 : w SBV (λq λ + λn 1 ν, w = 0 a.e., w = w 1,ν on λq ν, S w (7.4 where Q ν is any unit cube in R n with centre at the origin and one face orthogonal to ν, and 1 if x, ν 0, w 1,ν (x = 0 if x, ν < 0. For notational brevity we denote with P the class of admissible functions for the infimum in the definition of ϕ, that is, P := w SBV (λq ν : w = 0 a.e., w = w 1,ν on λq ν }. (7.5 Theorem 7.2. The family (F ε Γ-converges with respect to the strong topology of L 2 (Ω to the functional F hom introduced in (7.2. More precisely for every u SBV 2 (Ω the following properties are satisfied: (i for every (u ε SBV 2 (Ω converging to u strongly in L 2 (Ω F hom (u lim inf ε 0 F ε (u ε, (ii there exists a sequence (u ε SBV 2 (Ω converging to u strongly in L 2 (Ω such that F hom (u lim sup F ε (u ε. ε 0 For the proof of Theorem 7.2 we rely on [8, Theorem 2.3]. Due to the lack of coerciveness, we cannot apply the results in [8] directly to the functionals F ε. So we first modify the sequence to get the coerciveness we need, and then we obtain the stated Γ-convergence by approximation. Let us define for η > 0 the approximating functionals Fη ε : SBV 2 (Ω [0, + as x Fη(u ε = a η( Ω ε u 2 dx + S u a η( x ε dh n 1,

20 20 EXTENSION THEOREMS AND HOMOGENIZATION where a η is a Q-periodic function given by 1 if y E, a η (y = η if y R n \ E. Theorem 7.3. The family (Fη ε Γ-converges with respect to the strong topology of L 2 (Ω to the functional Fη hom : SBV 2 (Ω [0, + defined as Fη hom (u := fη hom ( u dx + ϕ η (ν u dh n 1. Ω S u The limit densities fη hom following homogenization formulas: : R n [0, + ] and ϕ η : S n 1 [0, + ] are identified by means of the fη hom (ξ := min ϕ η (ν := lim λ + Q } a η (y ξ + w(y 2 dy : w H#(Q 1, (7.6 1 inf λn 1 where H# 1 (Q and P are defined as before. S w a η (y dh n 1 : w P }, (7.7 Proof. The functionals Fη ε satisfy all the assumptions required in order to apply [8, Theorem 2.3] and hence the thesis follows directly. Now we are ready to give the proof of Theorem 7.2. Proof of Theorem 7.2. We split the proof into three steps. First step: approximation. It turns out that Indeed, since a η a pointwise as η 0, one has F hom = inf F η hom = lim Fη hom. (7.8 η η 0 f hom (ξ = inf η f η hom (ξ = lim fη hom (ξ. (7.9 η 0 For the surface integral one can proceed as follows. Since (ϕ η is decreasing and ϕ η ϕ for every η > 0, taking the limit as η goes to zero we have directly ϕ(ν inf ϕ η(ν = lim ϕ η (ν. η η 0 On the other hand, for every w P and for λ > 0, the following estimate holds true: 1 λ n 1 a η (y dh n 1 1 S w λq ν λ n 1 a(y dh n 1 + η S w λq ν λ n 1 Hn 1 (S w λq ν. (7.10 In particular for a minimizing sequence (w h P of the cell problem (7.4, by (7.10 we have 1 λ n 1 a η (y dh n 1 1 S wh λq ν λ n 1 a(y dh n 1 + η S wh λq ν λ n 1 Hn 1 (S wh λq ν. (7.11 Notice that, from the definition of the class P (see (7.5, we have S wh λq ν a(y dh n 1 = MS(w h, λq ν E, (7.12 Now we denote with w h the extension of w h to the whole domain λq ν obtained by applying Theorem 1.3 in the following way. We define the function λ wh λ(y := w h(λy. Clearly wh λ is defined in Q ν ( 1 λ E and by Theorem 1.3 it admits an extension T 1 λ wh λ to the whole Q ν satisfying ( ( MS (T 1 λ w λ h, Q ν k 0 MS w λ h, Q ν ( 1 λ E + H n 1 ( Q ν. (7.13 At this point, for x λq ν, we can define w h as w h (x := λ (T 1 λ w λ h ( x λ,

21 EXTENSION THEOREMS AND HOMOGENIZATION 21 and from (7.13 we have directly the estimate This implies in particular that MS( w h, λq ν = λ n 1 MS(T 1 λ w λ h, Q ν λ n 1 ( ( k 0 MS w λ h, Q ν ( 1 λ E + H n 1 ( Q ν ( = k 0 MS(wh, λq ν E + H n 1 ( λq ν. H n 1 (S ewh λq ν k 0 S wh λq ν a(y dh n 1 + k 0 λ n 1, where we used (7.12. The minimality of w h ensures that there exists a constant c > 0 such that 1 λ n 1 Hn 1 (S ewh λq ν c. (7.14 Since w h = w h a.e. in λq ν E, we can assume without loss of generality that (7.14 holds for the sequence w h. Therefore in (7.11 we obtain 1 λ n 1 inf a η (y dh n 1 1 w P S w λq ν λ n 1 inf a(y dh n 1 + c η. w P S w λq ν If we let λ + and then η 0 we get ϕ(ν = inf ϕ η(ν = lim ϕ η (ν. (7.15 η η 0 Hence, from (7.9, (7.15 and monotone convergence we obtain (7.8. Second step: liminf inequality (i. It is immediate to remark that for every u SBV 2 (Ω F ε η(u F ε (u + η MS(u, Ω. (7.16 Let u SBV 2 (Ω and let (u ε SBV 2 (Ω be a sequence converging to u strongly in L 2 (Ω and having equibounded energy F ε (u ε. By (7.16 we have F ε η(u ε F ε (u ε + η MS(u ε, Ω. (7.17 Using Theorem 4.1 we can assume that the sequence (u ε has equibounded energy F ε (u ε + ηms(u ε, Ω. Hence, since in particular MS(u ε, Ω c, we get from (7.17 and from Theorem 7.3 F hom η (u lim inf ε 0 F ε η(u ε lim inf ε 0 F ε (u ε + η c, that holds true for every η > 0. If we now let η 0 in the previous expression we obtain the required bound F hom (u = lim Fη hom η 0 (u lim inf F ε (u ε. ε 0 Third step: limsup inequality (ii. In this case we simply use the trivial estimate F ε η F ε. (7.18 Indeed, let u SBV 2 (Ω and let (u ε SBV 2 (Ω be a recovery sequence for the functionals F ε η. Then F hom η This implies in particular that (u = lim sup ε 0 Fη(u ε ε lim sup F ε (u ε. ε 0 F hom (u = inf η and therefore the proof is concluded. F hom η (u lim sup F ε (u ε, ε 0

22 22 EXTENSION THEOREMS AND HOMOGENIZATION 7.3. Γ-convergence under Dirichlet conditions. This subsection is devoted to the proof of a result which is a version of Theorem 7.2 that takes into account boundary data. We need a preliminary observation concerning the approximating coercive functionals F ε η. First of all let us fix ψ H 1 (R n and a set Ω with Ω Ω. We define the densities fη ε : Ω R n [0, + ] and gη ε : Ω [0, + ] as ( x ( x a η ξ 2 if x Ω, a η if x Ω, fη(x, ε ξ := ε ξ 2 if x Ω gη(x ε := ε \ Ω, 2 if x Ω \ Ω. Therefore we define the sequence of functionals F η ε on SBV 2( Ω as F η(u ε := fη( u ε dx + gη(x ε dh n 1. eω S u Using Theorem 7.3, it is easy to verify that F ε η Γ- converges with respect to the strong L 2 topology to F η, where F η (u := f η (x, u dx + g η (x, ν u dh n 1, eω S u and the limit densities f η and g η satisfy the relations f η (x, ξ = f hom η (ξ if x Ω, ξ 2 if x Ω \ Ω, g η (x, ν = ϕ η (ν if x Ω, f hom η and ϕ η being defined in (7.6 and (7.7, respectively. Lemma 7.4. The functionals F η,ψ ε defined on SBV 2( Ω as F F η,ψ(u ε ε := η (u if u = ψ on Ω \ Ω, + otherwise Γ-converge with respect to the strong L 2 topology to the functional Fη,ψ given by Fη (u if u = ψ on F η,ψ (u := Ω \ Ω, + otherwise. Proof. We omit the proof of this Lemma, which can be directly obtained by [11, Lemma 7.1]. Using the same notation adopted so far, we can define the functionals F ε on SBV 2( Ω as F ε (u := f ε (x, u dx + g ε (x dh n 1, eω S u where f ε : Ω R n [0, + ], g ε : Ω [0, + ] are given by ( x ( x a ξ 2 if x Ω, a if x Ω, f ε (x, ξ := ε ξ 2 if x Ω g ε (x := ε \ Ω, 2 if x Ω \ Ω. We can finally state the Γ-convergence result for the functionals F ε under Dirichlet boundary conditions. Notice that F ε (, Ω = F ε (, Ω, F ε being defined in (7.1. Theorem 7.5. The functionals F ψ ε defined on SBV 2( Ω as F F ψ(u ε ε (u if u = ψ on := Ω \ Ω, + otherwise

23 EXTENSION THEOREMS AND HOMOGENIZATION 23 Γ-converge with respect to the strong L 2 topology to the functional Fψ given by F(u if u = ψ on Ω \ Ω, F ψ (u := + otherwise. The limit functional F is defined as F(u := f(x, u dx + g(x, ν u dh n 1, eω S u where the limit densities f : Ω R n [0, + ] and g : Ω S n 1 [0, + ] satisfy f hom (ξ if x Ω, f(x, ξ = ξ 2 if x Ω g(x, ν = ϕ(ν if x Ω, \ Ω, f hom and ϕ being defined in (7.3 and (7.4, respectively. Proof. The convergence is a direct consequence of Lemma 7.4 and Theorem Appendix In this last section we prove some technical results that have been used in the paper. First, we show in a rigorous way an integral estimate for the composition of an SBV function with a bilipschitz map. This provides a stability result for the Mumford-Shah functional under bilipschitz transformations of the domain. More precisely, we have the following theorem. Theorem 8.1. Let W, W be bounded open subsets of R n with Lipschitz boundary, let φ : W W be a bilipschitz function and let us set ψ := φ 1. For every u SBV 2 (W, let us define the function v : W R as v(x := u(φ(x. Then, for every u SBV 2 (W we have that v SBV 2 (W and ( v 2 dx + H n 1 (S v C 1 W u 2 dx + H n 1 (S u, (8.1 where W C 1 := det ψ ( ψ T L (W ;M n. (8.2 Proof. It is well known that the function v belongs to SBV (W (see for example [5]. In order to prove the estimate (8.1, we split the proof into two steps. First step: approximation of u. As first step we approximate u with more regular functions and we prove the claim for the approximating functions. More precisely, let (u h be the sequence provided by Theorem 2.6, and set v h := u h φ. We claim that relation (8.1 holds true for the functions v h, i.e. that ( v h 2 dy + H n 1 (S vh C 1 u h 2 dx + H n 1 (S uh h N, (8.3 W W where C 1 is defined in (8.2. Let us set ψ := φ 1. By property (iii of Theorem 2.6 we can apply the standard chain rule and we get v h = ( φ T ( u h φ L n -a.e. on W \ ψ( S uh, that is, since ψ maps L n -negligible sets into L n -negligible sets, v h = ( φ T ( u h φ L n -a.e. on W. (8.4 Notice that, from the fact that (φ ψ(x = x for every x W, we get ( φ ψ ψ = Id ( φ ψ = ( ψ 1.