A Unified Model for Stress-Driven Rearrangement Instabilities

A variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wetting, and brittle fractures, is introduced. The model is characterized by an energy displaying both elastic and surface terms, and allows for a unified treatment of a wide range of settings, from epitaxially-strained thin films to crystalline cavities, and from capillarity problems to fracture models. The existence of minimizing configurations is established by adopting the direct method of the Calculus of Variations. The compactness of energy-equibounded sequences and energy lower semicontinuity are shown with respect to a proper selected topology in a class of admissible configurations that extends the classes previously considered in the literature. In particular, graph-like constraints previously considered for the setting of thin films and crystalline cavities are substituted by the more general assumption that the free crystalline interface is the boundary, consisting of an at most fixed finite number m of connected components, of sets of finite perimeter. Finally, it is shown that, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\rightarrow \infty $$\end{document}m→∞, the energy of minimal admissible configurations tends to the minimum energy in the general class of configurations without the bound on the number of connected components for the free interface.


Introduction
Morphological destabilizations of crystalline interfaces are often referred to as Stress-Driven Rearrangement Instabilities (SDRI), from the seminal paper [40] (see also Asaro-Grinfeld-Tiller instabilities [4,21]). SDRI consist in various mechanisms of mass rearrangements that take place at crystalline boundaries because of the strong stresses originated by the mismatch between the parameters of adjacent crystalline lattices. Atoms move from their crystalline order and different modes of stress relief may co-occur, such as deformations of the bulk materials with storage of elastic energy, and boundary instabilities that contribute to the surface energy.
In this paper we introduce a variational model displaying both elastic and surface energy that simultaneously takes into account the various possible SDRI, such as boundary discontinuities, internal cracks, external filaments, wetting and edge delamination with respect to a substrate, and brittle fractures. In particular, the model provides a unified mathematical treatment of epitaxially-strained thin films [22,31,33,42,48], crystal cavities [30,47,49], capillary droplets [11,24,26], as well as Griffith and failure models [9,13,14,39,50], which were previously treated separately in the literature. Furthermore, the possibility of delamination and debonding, i.e., crack-like modes of interface failure at the interface with the substrate [27,41], is treated in accordance with the models in [5,43,44], that were introduced by revisiting in the variational perspective of fracture mechanics the model first described in [50]. Notice that as a consequence the surface energy depends on the admissible deformations and cannot be decoupled from the elastic energy. As a byproduct of our analysis, we extend previous results for the existence of minimal configurations to anisotropic surface and elastic energies, and we relax constraints previously assumed on admissible configurations in the thin-film and crystal-cavity settings. For thin films we avoid the reduction considered in [22,23,31] to only film profiles parametrizable by thickness functions, and for crystal cavities the restriction in [30] to cavity sets consisting of only one connected starshaped void.
The class of interfaces that we consider is given by all the boundaries, that consists of connected components whose number is arbitrarily large but not exceeding a fixed number m, of sets of finite perimeter A. We refer to the class of sets of finite perimeter associated to the free interfaces as free crystals and we notice that free crystals A may present an infinite number of components. The assumption on the number of components for the boundaries of free crystals is needed to apply an adaptation to our setting of the generalization of Golab's Theorem proven in [36] that allows one to establish in dimension 2, to which we restrict, compactness with respect to a proper selected topology. To the best of our knowledge, no variational framework able to guarantee the existence of minimizers in dimension 3 in the settings of thin films and crystal cavities is available in the literature.
Furthermore, the class of admissible deformations is enlarged with respect to [22,23,30,31] to allow debonding and edge delamination to occur along the contact J u Σ S Ω A Fig. 1. An admissible free (disconnected) crystal A is displayed in light blue in the container , while the substrate S is represented in dark blue. The boundary of A (with the cracks) is depicted in black, the container boundary in green, the contact surface in red (thicker line) while the delamination region J u with a white dashed line surface := ∂ S ∩ ∂ between the fixed substrate S and the fixed bounded region containing the admissible free crystals (see Fig. 1). In what follows we refer to as the container in analogy with capillarity problems. Notice that the obtained results can be easily applied also for unbounded containers in the setting of thin films with the graph constraint (see Section 2.2). Mathematically this is modeled by considering admissible deformations u that are Sobolev functions only in the interior of the free crystals A and the substrate S, and G S B D, i.e., generalized special functions of bounded deformation (see [19] for more details), on A ∪ S ∪ . Thus, jumps J u that represent edge delamination can develop at the contact surface , i.e., J u ⊂ . The energy F that characterizes our model is defined for every admissible configuration (A, u) in the configurational space C m of free crystals and deformations by with respect to a positivedefinite elasticity tensor C and a mismatch strain E 0 . The mismatch strain is intro-duced to represent the fact that the lattice of the free crystal generally does not match the substrate lattice. We notice that the tensor C is assumed to be only L ∞ ( ∪ S), therefore not only allowing for different elastic properties between the material of the free crystals in and the one of the substrate, but also for non-constant properties in each material extending previous results. The surface energy S is defined as with surface tension ψ defined by where ϕ ∈ C( × R d ; [0, +∞)) is a Finsler norm representing the material anisotropy with c 1 |ξ | ϕ(x, ξ) c 2 |ξ | for some c 1 , c 2 > 0, β ∈ L ∞ ( ) is the relative adhesion coefficient on with |β(z)| ϕ(z, ν S (z)) (1.2) for z ∈ , ν is the exterior normal on the reduced boundary ∂ * A, and A (δ) denotes the set of points of A with density δ ∈ [0, 1]. Notice that the anisotropy ϕ is counted double on the sets A (1) ∩ ∂ A ∩ and A (0) ∩ ∂ A ∩ , that represent the set of cracks and the set of external filaments, respectively. On the free profile ∂ * A the anisotropy is weighted the same as on the delamination region J u , since delamination involves debonding between the adjacent materials by definition. Furthermore, the adhesion coefficient β is considered on the contact surface , alone on the reduced boundary ∩ ∂ * A\J u and together with ϕ on those external filaments A (0) ∩ ∂ A ∩ , to which we refer as wetting layer.
We refer the Reader to Section 2.3 for the rigorous mathematical setting and the main results of the paper, among which we recall here the following existence result: Main Theorem If v ∈ (0, | |) or S = ∅, then for every m 1 the volumeconstrained minimum problem This existence result is accomplished in Theorem 2.6, where we also solve the related unconstraint problem with energy F λ given by F plus a volume penalization depending on the parameter λ > 0.
The proof is based on the direct method of the Calculus of Variations, i.e., it consists in determining a suitable topology τ C in C m sufficiently weak to establish the compactness of energy-equibounded sequences in Theorem 2.7 and strong enough to prove that the energy is lower semicontinuous in Theorem 2.8. We notice here that Theorems 2.7 and 2.8 can also be seen as an extension, under the condition on the maximum admissible number m of connected components for the boundary, of the compactness and lower semicontinuity results in [15] to anisotropic surface tensions and to the other SDRI settings.
The topology τ C selected in C corresponds, under the uniform bound on the length of the free-crystal boundaries, to the convergence of both the free crystals and the free-crystal complementary sets with respect to the Kuratowski convergence and to the pointwise convergence of the displacements. In [22,23,31] the weaker convergence τ C consisting of only the Kuratowski convergence of complementary sets of free-crystals (together with the S) was considered, which in our setting without graph-like assumptions on the free boundary is not enough because not closed in C m . Working with the topology τ C also allows as to keep track or the surface energy of the possible external filaments of the admissible free crystals, which were in previous results not considered. However, to establish compactness with respect to τ C the Blaschke Selection Theorem employed in [22,23,30,31] is not enough, and a version for the signed distance functions from the free boundaries is obtained (see Proposition 3.1). Furthermore, in order to take in consideration the situation in which connected components of A k separates in the limit in multiple connected components of A, e.g., in the case of neckpinches, we need to introduce extra boundary in A k in order to divide their components accordingly (see Proposition 3.6). Otherwise, adding to u k different rigid displacements with respect to the components in A (which are needed for compactness of u k ) would results in jumps for the displacements in A k , which are not allowed in our setting with H 1 locdisplacements. Therefore, we pass from the sequence A k to a sequence D k with such extra boundary for which we can prove compactness. Passing to D k is not a problem in the existence in view of property (2.9) that relates the limin f of the energy with respect to A k to the one with respect to D k . However, in case S = ∅, in order to prove (2.9), we need to further modify the sequence D k from the original A k by cutting out the portion converging to delamination regions (e.g., portion containing accumulating cracks and voids at the boundary with S) using Proposition 3.9, and, in order to maintain the volume constraint, by replacing them with an extra set that does not contribute to the overall elastic energy.
The lower semicontinuity of the energy with respect to τ C is established for the elastic energy as in [31] by convexity, and for the surface energy in Proposition 4.1 in several steps by adopting a blow-up method (see, e.g., [1,8]). More precisely, given a sequence of configurations (A k , u k ) ∈ C m converging to (A, u) ∈ C m we consider a converging subsequence of the Radon measures μ k associated to the surface energy and (A k , u k ), and we estimate from below the Radon-Nikodym derivative of their limit denoted by μ 0 with respect to the Hausdorff measure restricted to the 5 portions of ∂ A that appear in the definition of the surface anisotropy ψ in (1.1). We overcome the fact that in general μ 0 is not a non-negative measure due to the presence of the contact term in the energy with β, by adding to μ k and μ 0 the positive measure defined for every Borel set B ⊂ R 2 and using (1.2). The estimates for the Radon-Nikodym derivative related to the free boundary ∩ ∂ * A and the contact region ( ∩ ∂ * A)\J u k follow from [1,Lemma 3.8]. For the estimates related to exterior filaments and interior cracks we first separately reduce to the case of flat filaments and cracks, and then we adapt some arguments from [36]. Extra care is needed to treat the exterior filament lying on to which we refer as wetting layer in analogy to the thin-film setting. The estimate related to the delamination region on follows by blow-up under condition (4.2) that ensures that the delamination regions between the limiting free crystal A and the substrate S can be originated from delamination regions between A k and S and from portions of free boundaries ∂ * A k or interior cracks collapsing on , as well as from accumulation of interior cracks starting from ( ∩ ∂ * A k )\J u k .
A challenging point is to prove that condition (4.2) is satisfied by (A k , u k ). In order to do this, in Theorem 2.8 we first extend the displacements u k to the set \(A k ∪ S) using Lemma 4.8. The extension of the u k is performed without creating extra jump at the interface on the exposed surface of the substrate, i.e., the jump set of the extensions is approximately J u k ∪ ( ∩ ∂ A k ). We point out that as a consequence we obtain also in Proposition 4.9 the lower semicontinuity, with respect to the topology τ C , of a version of our energy without exterior filaments (but with wetting layer) extending the lower semicontinuity results of [22,30,31].
Finally, we prove (1.3), which entails the existence of a minimizing sequence (A m , u m ) ∈ C m for the minimum problem of F in C. This is obtained by considering a minimizing sequence (A ε , u ε ) ∈ C for F λ , and then by modifying it into a new minimizing sequence (E ε,λ , v ε,λ ) ∈ C m such that F λ (A ε , u ε ) + δ ε F λ (E ε,λ , v ε,λ ) for some δ ε → 0 as ε → 0. The construction of (E ε,λ , v ε,λ ) ∈ C m requires 2 steps. In the first step we eliminate the external filaments, we remove sufficiently small connected components of A ε , and we fill in sufficiently small holes till we reach a finite number of connected components with a finite number of holes (see Fig. 2). In the second step we redefine the deformations in the free crystal by employing [14,Theorem 1.1] in order to obtain a deformation with jump set consisting of at most finitely many components, and such that the difference in the elastic energy and the length of the jump sets with respect to u ε remains small.
The paper is organized as follows: in Section 2 we introduce the model and the topology τ C , we refer to various SDRI settings from the literature that are included in our analysis, and we state the main results. In Section 3 we prove sequential compactness for the free crystals with the bound m on the boundary components in Proposition 3.3 and for C m in Theorem 2.7. In Section 4 we prove the lower semicontinuity of the energy (Theorem 2.8) by first considering only the surface energy S under the condition (4.2) (see Proposition 4.1), and we conclude the section by showing the lower semicontinuity of the energy without the external filament and wetting-layer terms with respect to the topology τ C (see Proposition 4.9). In Section 5 we prove the existence results (Theorems 2.6 and 2.9) and property (1.3). The paper is concluded with an Appendix where results related to rectifiable sets and Kuratowski convergence are recalled for reader's convenience.

Mathematical Setting
We start by introducing some notation. Since our model is two-dimensional, unless otherwise stated, all sets we consider are subsets of R 2 . We choose the standard basis {e 1 = (1, 0), e 2 = (0, 1)} in R 2 and denote the coordinates of x ∈ R 2 with respect to this basis by (x 1 , x 2 ). We denote by Int(A) the interior of A ⊂ R 2 . Given a Lebesgue measurable set E, we denote by χ E its characteristic function and by |E| its Lebesgue measure. The set where B r (x) denotes the ball in R 2 centered at x of radius r > 0, is called the set of points of density α of E. Clearly, is the topological boundary. The set E (1) is the Lebesgue set of E and |E (1) E| = 0. We denote by ∂ * E the reduced boundary of a finite perimeter set E [3,37], i.e., The vector ν E (x) is called the measure-theoretic normal to ∂ E.
The symbol H s , s 0, stands for the s-dimensional Hausdorff measure. An By [29,Theorem 2.3] any  (1/2) ) for any Borel set B.
The notation dist(·, E) stands for the distance function from the set E ⊂ R 2 with the convention that dist(·, ∅) ≡ +∞. Given a set A ⊂ R 2 , we consider also signed distance function from ∂ A, negative inside, defined as Remark 2.2. The following assertions are equivalent: [18,Chapter 4].
Moreover, either assumption implies ∂ E k K → ∂ E.

The Model
Given two open sets ⊂ R 2 and S ⊂ R 2 \ , we define the family of admissible regions for the free crystal and the space of admissible configurations by is the collection of all generalized special functions of bounded deformation [15,19]. Given a displace- we denote by e(u(·)) the density of e(u) = (Du+(Du) T )/2 with respect to Lebesgue measure L 2 and by J u the jump set of u. Recall that e(u) ∈ L 2 (A ∪ S) and J u is H 1 -rectifiable. Notice also that assumption u ∈ H 1 loc (Int(A) ∪ S; R 2 ) implies J u ⊂ ∩ ∂ * A. We denote the boundary trace of a function u : A → R n by tr A (if exists).
We endow C with the following notion of convergence: The energy of admissible configurations is given by the functional F : C → [−∞, +∞], where S and W are the surface and elastic energies of the configuration, respectively. The surface energy of (A, u) ∈ C is defined as where ϕ : × S 1 → [0, +∞) and β : → R are Borel functions denoting the anisotropy of crystal and the relative adhesion coefficient of the substrate, respectively, and ν := ν S . In the following we refer to the first term in (2.2) as the free-boundary energy, to the second as the energy of internal cracks and external filaments, to the third as the wetting-layer energy, to the fourth as the contact energy, and to the last as the delamination energy. In applications instead of ϕ(x, ·) it is more convenient to use its positively one-homogeneous extension |ξ |ϕ(x, ξ/|ξ |). With a slight abuse of notation we denote this extension also by ϕ.
The elastic energy of (A, u) ∈ C is defined as where the elastic density W is determined as the quadratic form sym is given by Given m 1, let A m be a collection of all subsets A of such that ∂ A has at most m connected components. Recall that since ∂ A is closed, it is H 1 -measurable. By Proposition A.2, ∂ A is H 1 -rectifiable so that A m ⊂ A. We call the set the set of constrained admissible configurations. We also consider a volume constraint with respect to v ∈ (0, | |], i.e.,

Applications
The model introduced in this paper includes the settings of various free boundary problems, some of which are outlined below.
-Epitaxially-strained thin films [10,22,23,31,35] (see also [6,38]). Notice that the container is not bounded, however, we can reduce to the situation of bounded containers where we can apply Theorem 2.9 since every energy equibounded sequence in A subgraph is contained in an auxiliary bounded set (see also Remark 2.10). -Crystal cavities [30,34,47,49]: ⊂ R 2 smooth set containing the origin, S := R 2 \ , free crystals in the subfamily See Remark 2.10.

Main Results
In this subsection we state the main results of the paper. Let us formulate our main hypotheses: (H1) ϕ ∈ C( × R 2 ; [0, +∞)) and is a Finsler norm, i.e., there exist c 2 c 1 > 0 such that for every x ∈ , ϕ(x, ·) is a norm in R 2 satisfying c 1 |ξ | ϕ(x, ξ) c 2 |ξ | for any x ∈ and ξ ∈ R 2 ; (2.4) (H2) β ∈ L ∞ ( ) and satisfies have a solution, where F λ : C m → R is defined as Furthermore, there exists λ 0 > 0 such that for every v ∈ (0, | |] and λ > λ 0 , (2.7) We notice that for λ > λ 0 solutions of (CP) and (UP) coincide (see the proof of Theorem 2.6) for any v ∈ (0, | |] and m 1. Moreover, (2.7) shows that a minimizing sequence for F in C can be chosen among the sets whose boundary have finitely many connected components.
The proof of the existence part of Theorem 2.6 is given mainly by the following two results in which we show that C m is τ C -compact and F is τ C -lower semicontinuous. Recall that an (infinitesimal) rigid displacement in R n is an affine for every k 1. Then there exist (A, u) ∈ C m of finite energy, a subsequence {(A k n , u k n )} and a sequence {(D n , v n )} ⊂ C m with v n := (u k n + a n )χ D n ∩A kn + u 0 χ D n \A kn for some piecewise rigid displacements a n associated to D n , such that A k n As a byproduct of our methods we obtain the following existence result in a subspace of C m with respect to a weaker topology previously used in [22,30,31] for thin films and crystal cavities.
admits a minimizer (A, u) in every τ C -closed subset of Remark 2. 10. The sets C subgraph and C starshaped defined in Section 2.2 are τ C -closed in C m (see e.g., [31,Proposition 2.2]). In the thin-film setting, we define ϕ and β as ϕ := γ f and where γ f , γ s , and γ f s denote the surface tensions of the film-vapor, substrate-vapor, and film-substrate interfaces, respectively. The energy F coincides (apart from the presence of delamination) with the thin-film energy in [22,23] in the case γ f , γ s , γ f s are constants, γ s − γ f s 0, γ s > 0, and γ f > 0. Therefore, Theorem 2.9 extends the existence results in [22,31] to all values of γ s and γ s − γ f s , as well as to anisotropic surface tensions and anisotropic elastic densities. Remark 2.11. All the results contained in this subsection hold true with essentially the same proofs by replacing (H3) with the more general assumption for some p 2, c c > 0 and f ∈ L 1 ( ∪ S).

Compactness
In this section we prove Theorem 2.7. Convergence of sets with respect to the signed distance functions has the following compactness property.
Proof. Without loss of generality we suppose A k / ∈ {R 2 , ∅}. By the Blaschke selection principle [3, Theorem 6.1], there exists a not relabelled subsequence {A k } and a closed set K ⊂ R 2 such that ∂ A k converges to K in the Kuratowski sense as k → ∞. Notice that by Proposition A.1, is 1-Lipschitz, by the Arzela-Ascoli Theorem, passing to a further not relabelled subsequence one can find f : Recall that K may have nonempty interior. Fix a countable set Q ⊂ Int(K ) dense in Int(K ), and define In general, the collection A is not closed under τ A -convergence. Indeed, let E := {x k } be a countable dense set in B 1 (0) and E k := {x 1 , . . . , x k } ∈ A. Then However, A m is closed with respect to the τ A -convergence.
Finally, by (3.2), |∂ A| = 0, and therefore, Now (b) follows from the uniform boundedness of {A k } and the Dominated Convergence Theorem.
Furthermore, sequences {A k } ⊂ A m with equibounded boundary lengths are compact with respect to the τ A -convergence.

Proposition 3.3. (Compactness of
Then there exists a subsequence {A k l } and A ∈ A m such that H 1 (∂ A) < ∞ and sdist(·, ∂ A k l ) → sdist(·, ∂ A) locally uniformly in R 2 as l → ∞.
Proof. By Proposition 3.1 there exists a not relabelled subsequence {A k } and a set A such that ∂ A k K → ∂ A and sdist(·, ∂ A k ) → sdist(·, ∂ A) locally uniformly in R 2 as k → ∞. By Lemma 3.2, A ∈ A m and H 1 (∂ A) < ∞.  We postpone the proof until after the next lemma. Before this, we need to introduce some notation. Let n 0 > 1 be such that E h ∩ {dist(·, ∂ A) > 1 n } = ∅ for every h ∈ I 1 ∪ I 2 and n > n 0 . Given h ∈ I 1 ∪ I 2 , let {E n h } n>n 0 be an increasing sequence of connected open sets satisfying By the sdist-convergence and the finiteness of I 1 ∪ I 2 , for any n n 0 there exist k 0 Note that 0 < d n < 1 2n . The idea of the proof of Proposition 3.4 is to "partition" the connecting components of Int(A k ) which in the limit break down into connected components {E h } h∈I of Int(A) such that I ∩ I 1 = ∅ and I ∩ I 2 = ∅, for example in the case of neckpinches. More precisely, we cut out at most m-circles from Int(A k ) such that for any n > n 0 , for all sufficiently large k (depending only on n), intersects at least one of these circles. The following lemma consists in performing this argument for fixed i ∈ I 1 and j ∈ I 2 : Lemma 3.5. Under the assumptions of Proposition 3.4, let i ∈ I 1 , j ∈ I 2 , and n > n 0 be such that the set is infinite. Then, there exists k i j n > k 0 n such that for any k ∈ Y with k > k i j n there exists a collection {B r l k (z l k )} l of at most m balls contained in A k such that r l k < d n and any curve γ ⊂⊂ Int(A k ), connecting a point of E n i to a point of E n j , intersects at least one of B r l k (z l k ).
Proof. We divide the proof into four steps.
Step 1: for any k ∈ Y, let C k ⊂⊂ Int(A k ) be any closed connected set intersecting both E n 0 i and E n 0 j . Then lim By contradiction, assume that there exists > 0 such that for infinitely many k ∈ Y. By the Kuratowski-compactness of closed sets there exist a closed connected set C and a not relabelled subsequence {C k } k∈Y satisfying Then by the definition of the Kuratowski convergence, there exist sequences x k ∈ C k and y k ∈ ∂ A k such that x k → x and y k → y. Since Thus, C ⊂⊂ Int(A). In particular, (3.7) implies that the non-empty connected open set {dist(·, C) < 4 } is compactly contained in Int(A) and intersects both E n 0 But this is a contradiction since E n i and E n j belong to different connected components of Int(A).
where sup is taken over all closed connected sets D ⊂⊂ Int(A k ), intersecting both E n 0 i and E n 0 j (such sets exist by definition of Y ). Moreover, there exists k 1 n > 0 such that L k contains E n i ∪ E n j and δ k < d n for any k > k 1 n . Indeed, in view of the Kuratowski-compactness of closed sets and from the Kuratowski-continuity of dist(·, ∂ A k ), (3.8) has a maximizer L k . Applying Step for all k >k 1 n . By construction, dist(L k , ∂ A k ) = δ k , and since δ k → 0, there exists k 1 n >k 1 n such that δ k < d n for any k k 1 n . Note that by (3.9) for such k we have also E n i ∪ E n j ⊂ L k . Let us show that L k is also path-connected. Indeed, given Thus, L k is locally path-connected. Now the compactness and the connectedness of L k imply its path-connectedness.
Step 3: given x ∈ E n 0 i and y ∈ E n 0 j , let γ k ⊂ L k be a curve connecting x to y. Then for any k > k 0 there exists Indeed, otherwise slightly perturbing the curve γ k around the points of the compact set γ : Step 4: now we prove the lemma. Applying Steps 1-3 with A k , we find an integer k 1 where sup is taken over all connected and closed D ⊂⊂ Int(A k ) intersecting both E n 0 i and E n 0 j , and a ball B r 1 where sup is taken over all connected and closed D ⊂⊂ Int(A 1 k ) intersecting both E n 0 i and E n 0 j , and a ball B r 2 and consider the set Y 2 of all k ∈ Y 1 for which there exists a closed connected Proof of Proposition 3.4. Given i ∈ I 1 , j ∈ I 2 and n > n 0 , let Y n i j be given by (3.5). If Y n i j is infinite, let k i j n be given by Lemma 3.5, otherwise set k i j } is the collection of balls given by Lemma 3.5. Without loss of generality we assume that {k n } n is strictly increasing and set Being a union of at most N 1 N 2 m circles, γ n is H 1 -rectifiable; here N i is the cardinality of I i . By Lemma 3.5, Then lim n→∞ H 1 (γ n ) = 0 and therefore, γ n converges in the Kuratowski sense to at We claim that the sequences {A k n } and {γ n } satisfy assertions (a)-(c). Indeed, by (3.11), {γ n } satisfy (a). Since γ n converges to at most N 1 N 2 m points on ∂ A in the Kuratowski sense, (b) follows. To prove (c) , we take any connected open sets D ⊂⊂ E and D ⊂⊂ F. By connectedness and the definitions of E h and E n h , there exist i ∈ I 1 and j ∈ I 2 andn > n 0 such that D ⊂⊂ E n i and D ⊂⊂ E n j for all n >n. By the construction of γ n , the sets E n i and E n j (and hence, D and D ) belong to different connected components of Int(A k n )\γ n for all n >n.
By inductively applying Proposition 3.4 and by means of a diagonal argument we modify a sequence {A k } τ A -converging to a set A into a sequence {B k } with same τ A -limit and whose (open) connected components "vanish" or "converge to the corresponding" connected components of A. This construction will be used in Step 1 of the proof of Theorem 2.7. We notice here that if S = ∅, then the sequence {D n } from Theorem 2.7 coincides with the sequence {B n }. Actually, if S = ∅, it would be enough to take D n = B n , where B n is constructed in the Step 1 of the proof of the next proposition, since in this case we do not need properties (e) and (f) of the statement of the next proposition. Proof. Given N , n 1, we define the index set I N n by We notice that I N n is finite since A is bounded.
. This is done by using Proposition 3.4 iteratively in N ∈ N and a diagonal argument.
Substep 1: Base of iteration. By Proposition 3.4 applied with {A k } k∈Y 0 with Y 0 := N, I 1 = {1}, and I 2 = I 1 n inductively with respect to n ∈ N, we find a decreasing sequence Y 0 ⊃ Y 1 ⊃ . . . of infinite subsets of N such that for the subsequence {A k } k∈Y n there exists a sequence {γ n k } k∈Y n of H 1 -rectifiable sets such that for any n 1 : for any k ∈ Y n and lim k∈Y n , k→∞ H 1 (γ n k ) = 0; -for any connected open sets D ⊂⊂ E 1 and D ⊂⊂ ∪ j∈I 1 n E j there exists k > 0 such that D and D belong to different connected components of A k \γ n k for any k ∈ Y n with k > k ; Then by a diagonal argument, we choose an increasing sequence n ∈ N → k 1 n ∈ Y n such that n } n such that for any N 1 : By condition b N n in Substep 2 and by the uniform boundedness of { B N ,n }, there exists an increasing sequence N ∈ N → n N ∈ N such that the sequence Step 2: By the Area Formula applied with dist(·, E i ) we have for any i and a.e. t > 0. Thus, we can choose t s 0 for which for any s ∈ N and i, and thus, by a diagonal argument we find a further subsequence for any i and s. Let and since ∂ A k Ns n Ns Step 3: so that, passing to further not relabelled subsequence if necessary, we can choose Proof. Indeed, suppose that there exists a ball B ⊂⊂ Q, a measurable function u : B → R n and a not relabelled subsequence {u k } such that u k → u a.e. in some subset E of B with positive measure. Since u k ∈ H 1 (B ; R n ), by the Poincaré-Korn inequality, there exists a rigid displacement a k : R n → R n such that has a finite perimeter in and u ∈ G S B D 2 (Q; R n ) such that u k → u a.e. in Q\F and

be an increasing sequence of connected Lipschitz open sets such that
where c j is independent on k. Then by the Rellich-Kondrachov Theorem, every subsequence {u k l } admits further not relabelled subsequence such that u k l +a Since a.e.-convergence of linear functions implies the local strong H 1 -convergence, The following corollary of Proposition 3.7 is used in the proof of Theorem 2.7: Corollary 3.8. Let P, P k ⊂ R n be connected bounded open sets such that for any G ⊂⊂ P there exists k G such that G ⊂⊂ P k for all k > k G , and let u k ∈ H 1 loc (P k ; R n ) be such that Proof. Let B ⊂⊂ P be any ball. By assumption, B ⊂⊂ P k for all large k. By the Poincaré-Korn inequality, for all such k there exists a rigid displacement b k such that This, (3.18) and the Rellich-Kondrachov Theorem imply that there exist a not relabelled subsequence Proposition 3.9. Assume (H1)-(H2) and let x 0 ∈ , δ ∈ (0, 1 2 ) and r ∈ (0, 1) be such that ν 0 := ν (x 0 ) exists. Then for any k > k δ .
We postpone the proof unfil after the following lemma: and A k τ A → A and u k → u a.e. in U r ∩ Int(A) and |u k | → +∞ a.e. in S ∩ U r . Then for every > 0 there exists k > 0 such that for any k > k , [3, pp. 80] there exists at most countably many C 1 -curves { k i } i 1 such that Selecting closed arcs inside curves if necessary, we suppose that k i ⊂ U r and where dist H is the Hausdorff distance (see e.g., (A.1) for the definition). Let V k 0 := U r \Int(A k ) ∪ S be the "voids". By the definition of In particular, by (3.22), sup and |w k | → +∞ a.e. in U r ∩ S. We show that Under the notation of [15], given ξ ∈ S 1 let π ξ be the orthogonal projection onto the line ξ := {η ∈ R 2 : ξ · η = 0}, perpendicular to ξ ; given a Borel set F ⊂ R 2 and y ∈ ξ , let F ξ y := {t ∈ R : y + tξ ∈ F} be the one-dimensional slice of F, and given u ∈ G S B D(U r ; R 2 ) and y ∈ ξ , let u ξ y (t) = u(y + tξ) · ξ be the one-dimensional slice of u. Since w h → w a.e. in U r \S, by [15,Eq. 3.23], for any > 0 and Borel set F ⊂ U r , for a.e. ξ ∈ S 1 and a.e. y ∈ ξ , where the integral of f ξ y (w k ) over ξ is uniformly bounded independent on ξ and k (see also (4.21)

below). Let
Then F := U r ∩ π −1 ξ ( A) is Borel and, thus, integrating (3.26) over A and using the definition of A and Fatou's Lemma we get Note that by construction, J w k is a union of open sets, thus, for a.e. y ∈ π ξ (J w k ), the line π −1 ξ (y) passing through y and parallel to ξ crosses J w k at least at two points. Thus, From (3.25) and (3.24) it follows that there exists k > 0 such that for any k > k . Now (3.23) follows from (3.29).
We anticipate here that in Lemma 4.7, below, we establish a similar result.
Proof of Proposition 3.9. For simplicity, assume that x 0 = 0, ν = e 2 and φ(ξ ) = ϕ(0, ξ). Denote the left-hand side of (3.21) by α k . By (3.19) and (2.5), where By Lemma 3.10 applied with φ and := δ U r ∩ ϕ(x, ν )dH 1 , there exists k δ such that and therefore, Applying (2.5) in the first integral we get and hence, (3.31) and the definition of imply for any k 1. In view of (3.32) and Proposition 3.3, there exists A ∈ A m with H 1 (∂ A) < ∞ and a not relabelled subsequence {A k } such that sdist(·, ∂ A k ) → sdist(·, ∂ A) locally uniformly in R 2 . Now we construct the sequence {(B n , v n )} in three steps. In the first step we apply Proposition 3.6 and Corollary 3.8 to obtain a (not relabelled) subsequence and to construct a sequence In the second step we take care of the fact that adding different rigid motions in B k and in S can create extra jump at making difficult to satisfy (2.9). More precisely, by Proposition 3.9 we modify {B k } and {u k } so that the modified sequence Step  Notice that by condition (a1), Thus, (3.34) Now we define the piecewise rigid displacements a k associated to B k . Let {S j } j∈Y be the set of connected components of S for some index set Y. We define the index sets I n ⊂ I and Y n ⊂ Y inductively on n in such a way that Corollary 3.8 holds with P k = E k i and P = E i and also with P k = P = S j for every i ∈ I n and j ∈ Y n with the same rigid displacements a n k independent of i and j. More precisely, let I 0 := Y 0 := ∅, and given the sets I 1 , . . . , I n−1 and Y 1 , . . . , Y n−1 for n 1, we define I n and Y n as follows. By Corollary 3.8 applied with P k = P = S j n with j n the smallest element of Y \ n−1 l=1 Y l , we find a not relabelled subsequence {(B k , u k )}, a sequence {a n k } of rigid displacements and w n ∈ H 1 loc (S j n ; R 2 ) such that u k + a n k → w n a.e. in S j n . Let I n and Y n be the sets such that there exists a not relabelled subsequence {(B k , u k )} such that the sequence (u k + a n k )χ E k i converges a.e. in E i for i ∈ I n and the sequence (u k + a n k )χ S j converges a.e. in S j for j ∈ Y n . Recall that j n ∈ Y n . Let By the definition of I n and Y n , and by diagonalization the sequence (u k + a n k )χ F k n converges as k → ∞ a.e. in F n to some function in H 1 loc (F n ; R 2 ), which we still denote by w n .
Note that for large n, Y n is empty since Y is finite by assumption. Notice also that by definition of I n and Y n , and Proposition 3.7 applied in connected open sets P ⊂⊂ E i ∪ S j , we have |u k + a n k | → +∞ a.e in E i ∪ S j for every i ∈ I \I n and j ∈ Y \Y n , We now define the rigid displacements in E k i for i ∈ I \ n I n . By a diagonal argument and by Corollary 3.8 applied with P k = E k i and P = E i for any i ∈ I \ n I n , we find a further not relabelled sequence {B k , u k }, sequence { a i k } of rigid displacements and Finally, we define rigid displacements in connected components C k i of B k \ i E k i whose interior in the limit becomes empty, i.e., C k i turns into an external filament. Recall that By construction, a k is a piecewise rigid displacement associated to B k , u ∈ H 1 loc (Int(A) ∪ S; R 2 ) and u k + a k → u a.e. in Int(A) ∪ S. Note that e(u k + a k ) = e (u k  for all large k (depending only D). Since u k + a k → u a.e. in D, by the Poincaré-Korn inequality, e(u k +a k ) e(u) weakly in L 2 (D; M 2×2 sym ). Then by the convexity We observe that if S = ∅, the terms of the surface energy S(A k , u k ) related to disappears, and hence, using e(u k + a k ) = e(u k ) and property (a1), where o(1) → 0 as n → ∞, and so we can define D n = B n .
Step 2: Further modification of {B k }. Without loss of generality we assume v < | |. It remains to control J u k +a k at since, as mentioned above, adding different rigid displacements to u k in connected components of the substrate and the free crystal whose closures intersect can result in a larger jump set J u k +a k than J u k . Recall that by condition (a4) and (a6), Hence, we need only to control J u k ∩ ∂ * A. The idea here is to remove a "small" subset R k of B k containing almost all points x ∈ ∩ ∂ * A ∩ (∂ * A k \J u k ) which in the limit becomes jump for u. In order to keep volume constraint, we will insert a square U k of volume |R k | in \A k . This is possible since |R k | → 0 and |A k | v for all k.
More precisely, we prove that for any δ ∈ (0, 1/16), there exist k δ > 0 and for any k > k δ . We divide the proof into four steps. Substep 2.1. By assumptions of the theorem, conditions (a2) and (a5) and Lemma 3.2 (b), |A| v. Hence, we can choose a square U ⊂⊂ \A. By (a2) and the definition of τ A -convergence, there is no loss of generality in assuming U ⊂⊂ \B k for any k. Let Without loss of generality, we assume 0 ∈ (0, 1 2 ). First we observe that for any δ ∈ (0, 1) : (b1) since is Lipschitz, for H 1 -a.e. x ∈ , there exist a unit normal ν (x) to and r x > 0 such that for any r ∈ (0, r x ), U r,ν (x) (x) ∩ can be represented as a graph of a Lipschitz function over tangent line U r,ν (x) (x) ∩ T x at x in the direction ν (x); (b2) since ϕ is uniformly continuous, for any x ∈ there exists r δ x > 0 such that for any y ∈ U r δ x ,ν (x) (x) and ξ ∈ S 1 , |ϕ(y, ξ) − ϕ(x, ξ)| < δ; (b3) since H 1 -a.e. x ∈ is the Lebesgue point of β, there exists r δ x > 0 such that for any r ∈ (0, r δ x ), thus, there exists r x > 0 such that for any r ∈ (0, r x ), for any k >k δ . Fix δ ∈ (0, 1 16 ) and let t δ > 0 be such that where 0 is given in (3.37). We now consider connected components E i and S j such that the associated rigid displacements are different. Let I be the set of all i ∈ I such that H 1 (∂ * E i ∩∂ S) > 0 and lim inf k→∞ H 1 (∂ E k i ∩ ∂ S) > 0, and there exists a connected component S j of S such that u k + b i k → u a.e. in E i and |u k + b i k | → +∞ a.e. in S j for the associated sequence {b i k } of rigid displacements in E i .
Note that L ⊂ and by (b1)-(b5), for a.e. x ∈ L for which ν (x) and ν A (x) exist and ν (x) = ν A (x) there is such that properties (b1)-(b5) holds with x and r = r x . Note that for any such x : x ; (c2) by Proposition 3.9 applied with u k + a i k , there exists k δ x >k δ such that and Q x ⊂ ∩ be the open set whose boundary consists of 1 := U r ∩ , two segments 2 , 3 ⊂ ∂U r of length at most 2δr, parallel to ν (x), and the segment Indeed, without loss of generality, we assume that x = 0 and ν (x) = e 2 . By the anisotropic minimality of segments, Denoting by α k the left-hand side of (3.41), by condition (a1) and the definition of In the last inequality using 1 1+ δ c 2 1 − δ c 2 and inequality (2.4) once more we deduce Now condition (b4) and (2.4) and the inequality H 1 ( 1 ) 2r imply Inserting this in (3.48) we get (3.42). Substep 2.3. Now we choose finitely many points x 1 . . . , x N ∈ L with corresponding r 1 , . . . , r N satisfying (b1)-(b5) and (c1)-(c2) such that the squares {U r j ,ν (x j ) (x j )} N j=1 are pairwise disjoint and Recalling the definition of k x δ in condition (c2) and the definitionk δ in (3.38), let k δ := max{k δ , k x 1 δ , . . . , k x N δ } and let Q x j ⊂ ∩ U r j ,ν (x j ) (x j ) be as in Substep 2.2. Set Then, as in the proof of (3.42), so that by the pairwise disjointness of {U r j ,ν (x j ) (x j )}, Recalling (3.49) and (3.38), and using (2.4), we estimate hence, from (3.50), we get On the other hand, since Int( B δ k ) ⊂ Int(B k ) and e(u k ) = e(u k + a k ), However, by construction, In order not to increase the number of connected components of B δ k , we translate U k in \B k until it touches to ∂ B δ k . Define Then {(B δ k , u δ k )} ⊂ C m and for any k > k δ by (3.51) and (3.52) By the choice of U k , its sidelength is less that δ 0 , hence, using 0 < 1 2 and (2.4), S(U k , u 0 ) 2c 2 δ so that Step 3: Construction of (D n , v n ). Notice that the sequence {(B δ k , u δ k )} in general does not need to satisfy B δ k τ A → A, since we removed "something" from B k and added a square U k . To overcome this problem, we take δ = δ n := 1 16n and (D n , v n ) := (B δ n k n , u δ n k n ), where k n := k δ n + 1, and there is no loss of generality in assuming n → k n is increasing. Denote r n j := r δ n x j , where the latter is defined in Substep 2.3 and notice that by (3.39) and (3.40) r n j → 0 as n → ∞ In particular, thus (2.9) follows from (3.53) and (3.34).

Lower Semicontinuity
In this section we consider more general surface energies. For every A ∈ A and J A ∈ J A , where The main result of this section is the following.
relatively open subset L k of with H 1 ( We prove Proposition 4.1 using a blow-up around the points of the boundary of A. Given y o ∈ R 2 and ρ > 0, the blow-up map σ ρ,y o : R 2 → R 2 is defined as is an open square of sidelength 2ρ > 0 centered at x whose sides are either perpendicular or parallel to ν; if ν = e 2 and x = 0, we write U ρ,ν (0) and σ ρ,x (U ρ,ν (x)) = U 1,ν (0). We denote by π the projection onto x 1 -axis i.e., π(x) = (x 1 , 0). (4.5) The following auxiliary results will be used in the proof of Proposition 4.1:

Lemma 4.2. Let U be any open square, K ⊂ U be a nonempty closed set and
Proof. We prove only the first assertion, the second being the same. If x k ∈ E k is such that x k → x, then by assumption, On the other hand, given x ∈ K suppose that there exists r > 0 such that B r (x) ∩ E k = ∅ for infinitely many k. Then for such k, sdist(x, ∂ E k ) = dist(x, E k ) r > 0, which contradicts to the assumption.
In the next lemma we observe that the endpoints of every curve contained in the boundary of any bounded set A with connected boundary are still arcwise connected if we remove the boundary of Int(A) belonging to .

Lemma 4.3. Let A ⊂ R 2 be a bounded set such that ∂ A is connected and has finite H 1 measure. Suppose that x, y ∈ ∂ A are such that x = y and ⊂ ∂ A is a curve connecting x to y. Then there exists a curve ⊂ ∂ A\( ∩ ∂Int(A)) connecting x to y.
Proof. Without loss of generality we assume G := Int(A) = ∅, otherwise we simply take = . Note that Since connected compact sets of finite length are arcwise connected (see Proposition A.2), it suffices to show that x and y belong to the same connected component of ∂ A\( ∩ ∂G). Suppose that there exist two open sets P, Q ⊂ R 2 with disjoint closures such that where x ∈ P ∩ ∂ A\( ∩ ∂G) and y ∈ Q ∩ ∂ A\( ∩ ∂G). Then \P ∪ Q = ∅ and of \P ∪ Q connecting both P and Q is at most finite. Moreover, since has no self-intersections (see Section A.2 for the definition of the curve in our setting) and the endpoints of belong to P and Q, respectively, n must be odd. However, by (4.8) L i ⊂ ∂G, and hence, by (4.6), every neighborhood of L i contains points belonging to both Int(A) and R 2 \A. We reached a contradiction since in this case Int(A) would be unbounded. Then, for every δ ∈ (0, 1), there exists k δ > 1 such that for any k > k δ , (4.10) Proof. Let us denote the left hand side of (4.10) by α k . We may suppose sup k α k < ∞. By assumption (a), for every δ ∈ (0, 1) there exists k 1,δ > 1 such that Step 1. Assume that for some k > k 1,δ , ∂ E k has a connected component K 1 intersecting both {x 1 = 1} and {x 1 = − 1}. In this case by Lemma 4.3, ∂ E k \(K 1 ∩ ∂Int(A)) is also connected and contains a path K 2 connecting {x 1 = 1} to {x 1 = − 1}. Note that K 1 and K 2 may coincide on (E (1) i , i = 1, 2, be the segments along the vertical lines {x 1 = ±1} connecting the endpoints of K 1 and K 2 to (±1, 0), respectively. Since K 1 ∩ ∂ * E k and K 2 ∩ ∂ * E k are disjoint up to a H 1 -negligible set which implies (4.10).
Step 2. Assume now that every connected component of U 1 ∩ ∂ E k intersects at most one of {x 1 = 1} and {x 1 = − 1}. In this case, let K 1 , . . . , K m k stand for the connected components of ∂ E k lying strictly inside of U 1 (that is, not intersecting Hence there exists k 2,δ > k 1,δ such that (4.14) for any k > k 2,δ . Then repeating the proof of (4.13) with K j in (a j , b j ) × (−1, 1), for every j = 1, . . . , m k + 2 we find Therefore, by (4.14) and (2.4), Since m o 1, this implies (4.10). Then for every δ ∈ (0, 1) there exists k δ > 1 such that for any k > k δ , Proof. The assertion follows from applying Lemma 4.5 to U 1 \E k .
The following result extends the lower semicontinuity result of [15, Theorem 1.1] to the anisotropic case.
Proof. We divide the proof into two steps.
Step 1. First we prove the (4.17) assuming that φ is independent on x ∈ D, i.e., Let W = {φ • 1} be the Wulff shape of φ, i.e, the unit ball for the dual norm Note that φ •• = φ and by (4.15), where sup is taken over finite disjoint open sets {F n } N n=1 whose closures are contained in G. Now we prove (4.17). Under the notation of [7,15], for any ∈ (0, 1), open set F ⊂ D with F ⊂ D and for H 1 -a.e. ξ ∈ ∂ W we have where ξ := {y ∈ R d : y · ξ = 0}, is the hyperplane passing through the origin and orthogonal to ξ, given y ∈ R d , F ξ y := {t ∈ R : y + tξ ∈ F} is the section of the straight line passing through y ∈ R d and parallel to ξ, given u : Recall that by (4.19), and by (4.16), Now taking sup over {F n } and letting → 0 we obtain (4.17).
Step 2. Now we prove (4.17) in general case. Without loss of generality we suppose that the limin f in (4.17) is a finite limit. Consider the sequence {μ h } h 0 of positive Radon measures in D defined at Borel subsets of B ⊆ D as Since sup h μ h (D) < ∞, by compactness, there exist a positive Radon measure μ and a not relbelled subsequence {μ h } h 1 such that μ h * μ as h → ∞. We prove that μ μ 0 , (4.23) in particular from μ(D) μ 0 (D) (4.17) follows. Since μ 0 is absolutely continuous with respect to H d−1 J w , to prove (4.23) we need only to show For this aim fix ∈ (0, c 1 ). By the uniform continuity of φ, there exists r > 0 such that for any ν ∈ S d−1 and x, y ∈ D with |x − y| < r . In particular, given x ∈ J w and for a.e. r ∈ (0, r ), where in the equality we use the weak* convergence of {μ h } and in the inequality (4.25) with y = x 0 and x ∈ B r (x 0 ) ∩ J w h By Proposition 4.6 applied with φ(x 0 , ·), where in the second equality we again used (4.25). Moreover, by (4.15), Since and r ∈ (0, r ) are arbitrary, (4.24) follows from the Besicovitch Derivation Theorem.
Proof. Denote the left-hand-side of (4.26) by α k . We suppose that sup k |α k | < ∞ so that by (4.9) for some M > 0. Moreover, passing to a not relabelled subsequence if necessary, we assume that By assumption (b), k is "very close" I 2 , hence, by the area formula [3, Theorem 2.91] for any K k ⊂ k one has lim sup where in the last inequality and in the first equality we used that φ is a norm, and the last equality follows from |l k | 1 k . Hence, We divide the proof into two steps.
Step 1. For shortness, let J k := J E k and C k := k \∂ * E k . We claim that for any δ ∈ (0, 1) there exists k 1 δ > 0 such that for any k > k 1 δ , Indeed, by adding to both sides of (4.29) the quantity 2 U 1 ∩J k φ(ν k )dH 1 + (4.30) and hence, we will prove (4.30). Note that since J k ⊂ k is H 1 -rectifiable, given δ ∈ (0, 1) there exists a finite union R k of intervals of k such that where c 2 > 0 is given in (4.9). Possibly slightly modifying u k around the (approximate) continuity points of R k and around the boundary of the voids U 2 \E k we assume that J k := R k , L k = ∅ and . By relative openness of C k = k \∂ * E k and J k in k and assumption (d), K k is a union j K j k of at most countably many pairwise disjoint connected rectifiable sets K j k relatively closed in U 1 . Let co(K j k ) denote the closed convex hull of K j k . Observe that if K j k is not a segment, then the interior of co(K j k ) is non-empty and Note that j o ∈ D k i . As in (4.32) we observe that Then K k ⊂ i V k i and by (4.33) where T is the set of all indices i for which V k i is a line segment. For every i ∈ T we replace the segment V k i with a closed rectangle Q i containing V k i and not intersecting any  [46,Proposition 2.6]). Therefore, v k := u k χ , and by assumption (g) and inequalities (4.9), and (4.27), Repeating the same arguments of the proof of (3.25) we obtain Note that the direct application of Proposition 4.6 would not be enough since we would obtain the estimate: without coefficient 2 on the left. From (4.34) and (4.35) it follows that there exists k 1 δ > 0 such that for any k k 1 δ . By (4.28) we may suppose that for such k, Thus, in view of (4.31), from (4.36) we get (4.30).
Proof of Proposition 4.1. Without loss of generality, we suppose that the limit in the left-hand side of (4.3) is reached and finite. Define Then g + is Borel, g + (·, s) ∈ L 1 ( ) for s = 0, 1, and by (4.1), g + 0 and for H 1 -a.e. x ∈ . Consider the sequence μ k of Radon measures in R 2 , associated to S(A k , J A k ; ϕ, g), defined at Borel sets B ⊂ R 2 by Analogously, we define the positive Radon measure μ in R 2 associated to S(A, J A ; ϕ, g), writing A in place of A k in the definition of μ k . By (2.4), assumption A k τ A → A and the nonnegativity of g + , Thus, by compactness there exists a (not relabelled) subsequence {μ k } and a nonnegative bounded Radon measure μ 0 in R 2 such that μ k * μ 0 as k → ∞. We claim that μ 0 μ, (4.39) which implies the assertion of the proposition. In fact, (4.3) follows from (4.39), the weak*-convergence of μ k , and the equalities Therefore, by [32,Theorem 1.153] and assumption (a2), Proof of (4.40b). Consider points where T x is the approximate tangent line to ∂ A and σ ρ,x is given by (4.4); (b4) exists and finite.
By the H 1 -rectifiability of ∂ A, Proposition A.4 (applied with the closed connected component K of ∂ A containing x) and the Besicovitch Derivation Theorem, the set of points x ∈ A (0) ∩ ∂ A not satisfying these conditions is H 1 -negligible, hence we prove (4.40b) for x ∈ A (0) ∩ ∂ A satisfying (b1)-(b4). Without loss of generality we assume x = 0, ν A (x) = e 2 and T x = T 0 is the x 1 -axis.
Proof of (4.40c). We repeat the same arguments of the proof of (4.40b) using Proof of (4.40d). Given x ∈ \∂ A, there exists r x > 0 such that Thus, for any r ∈ (0, r x /2), for H 1 -a.e. Lebesgue points x ∈ \∂ A of g + . Proof of (4.40e). Consider points where T x is the approximate tangent line to ∂ A; (e4) x is a Lebesgue point of g + (·, 1), i.e., exists and is finite.
By the H 1 -rectifiability of ∂ A, the Lipschitz continuity of , the Borel regularity of ν (·), Proposition A.4 (applied with closed connected component K of ∂ A containing x), the continuity of ϕ, assumptions on g + and the Besicovitch Derivation Theorem, the set of x ∈ ∩ ∂ A not satisfying these conditions is H 1 -negligible. Hence, we prove (4.40e) for x satisfying (e1)-(e6). Without loss of generality we assume x = 0, ν (x) = ν A (x) = e 2 and T x = T 0 is the x 1 -axis. Let r n 0 be such that By the weak*-convergence, for any h 1 we have By Proposition A.5 (b), (e2) and (e3) imply sdist(·, σ r n (∂ A)) → dist(·, T 0 ) uniformly in U 1 . Since for any n, sdist(·, σ r n (∂ A k )) → sdist(·, σ r n (∂ A)) uniformly in U 1 as k → ∞, by a diagonal argument, we can find a subsequence {k n } and not relabelled subsequence {r n } such that μ k n (U r n ) μ 0 (U r n ) + r 2 n (4.48) for any n 1 and sdist(·, σ r n (A k )) → σ (·, T 0 ) uniformly in U 1 as k → ∞, thus, by Lemma 4.2, as n → ∞. Notice also that by (e2) and Proposition A.4 (applied with the closed connected component K of ), U 1 ∩ σ r n ( ) K → I 1 as n → ∞. By (4.38), ϕ(y, ν (x)) + g + (y, 0) g + (y, 1) for H 1 -a.e. on , in particular on J A k , hence, by Remark 2.3 and the definition of μ k , Adding and subtracting U rn ∩ ∩∂ * A kn φ(y, ν A kn )dH 1 to the right and using (4.38) once more in the integral over U r n ∩ \∂ A k n we get By the uniform continuity of ϕ, given ∈ (0, 1) there exists n > 0 such that |ϕ(y, ν) − ϕ(0, ν)| < for all y ∈ U r n , ν ∈ S 1 and n > n . We suppose also that Lemma 4.4 holds with n when δ = . Since the number of connected components of ∂ A k n lying strictly inside U r n is not greater than m, in view of (4.49) and the non-negativity of g + , as in (4.46) for all n > n we obtain By the non-negativity of g + , (2.4) and (4.48), thus again using (4.48), also (4.47), (4.50) and (4.51), as well as (e1) and (e3)-(e5) we establish Now letting → 0 and using ν (0) = e 2 we obtain (4.40e).
By the H 1 -rectifiability of J A , ∂ A and , assumption (b) of Proposition 4.1 (recall that J A ⊂ J w ), the definition of the jump set of G S B D-functions, (4.38), and the Besicovitch Derivation Theorem, the set of points x ∈ J A not satisfying these conditions is H 1 -negligible. Hence we prove (4.40g) for x ∈ J A satisfying (g1)-(g6). Without loss of generality, we assume x = 0 and ν (x) = e 2 . Let r 0 = r x and w k ∈ G S B D 2 (B r 0 (0); R 2 ) be given by assumption (b) of Proposition 4.1. Note that by the weak*-convergence of μ k , lim k→∞ μ k (U r ) = μ 0 (U r ).
Proof. By Proposition 3.7, u ∈ H 1 loc (P; R n )∩ G S B D 2 (P; R n ). By Poincaré-Korn inequality, there exist c P > 0 and a sequence {a k } of rigid displacements such that u k + a k H 1 (P) c P e(u k ) L 2 (P) (4.66) for any k. Since u k → u a.e. in P, reasoning as in the proof of Proposition 3.7 (with P in place of B ), up to a not relabelled subsequence, a k → a a.e. in R n for some rigid displacement a : R n → R n . In particular, H 1 (P; R n )-norm of a k is uniformly bounded independently of k, hence, The lower semicontinuity of the elastic-energy part can be shown by using convexity W (x, ·). Indeed, let D ⊂⊂ Int(A). Then by τ A -convergence of A k , D ⊂⊂ Int(A k ) for all large k. Since u k → u a.e. in A ∪ S, by (4.69) and the weak-compactness of L 2 (D ∪ S), e(u k ) e(u) in L 2 (D ∪ S). Therefore, from the convexity of W(D, ·) it follows that Since S(E, v) = S(E, J v ; ϕ, g) with J E = J v and g(x, s) = β(x)s, the lower semicontinuity of of the surface part, follows from Proposition 4.1 provided that for H 1 -a.e. x ∈ J u there exists r x > 0, w k ∈ G S B D(B r x (x); R 2 ) and relatively open sets L k of with H 1 (L k ) < 1/k such that (4.2) holds. Let so that B r x 0 (x) ⊂⊂ ∪ ∪ S, and choose r = r x ∈ (0, r x 0 ) such that (see [46,Proposition 2.6]) and B r (x) ∩ S is connected. We construct {w k } by extending {u k } in B r (x)\(A k ∪ S) without creating extra jumps at the interface on the exposed surface of the substrate. More precisely, we apply Lemma 4.8 with Q := B r (x), P := B r (x) ∩ S, and u k P . Since u k → u a.e. in P, by Lemma 4.8, there exist v ∈ H 1 (Q; R 2 ) and a not relabelled subsequence {u k } such that the Sobolev extension Eu k of u k P to Q converges to v a.e. in Q. Define Perturbing w k slightly if necessary, we can assume J w k = : up to a H 1 -negligible set. In fact, by [46, Proposition 2.6] there exist ξ ∈ R 2 with arbitrarily small |ξ | > 0 for which H 1 ({y ∈ : [u k ](y) = ξ }) = 0 (with [u k ](x) the size of the jump of u k ), and hence, we can perturb u k with a W 1,∞ (B r (x)\ )-function with arbitrarily small norm, which is equal to ξ on an arbitrarily large subset of . By construction, Thus w k and w satisfy (4.2).
We conclude this section by proving a lower semicontinuity property of F with respect to τ C . Observe that if (A k , u k ) τ C → (A, u), then A = Int(A) so that the weak convergence of u k to u in H 1 loc (A ∪ S; R 2 ) is well-defined. However, notice that C m is not closed with respect to τ C -convergence.  Proof. Consider the auxiliary functional F : C → R defined as Since F does not see wetting layer energy,

Existence
In this section we prove Theorems 2.6 and 2.9.
Proof of Theorem 2.6. We start by showing the existence of solutions of problems (CP) and (UP). For the constrained minimum problem, let {(A k , u k )} ⊂ C m be arbitrary minimizing sequence such that so that (A, u) is a minimizer. The case of the unconstrained problem is analogous. Now we prove (2.7). Observe that in general  F(A, u) and |A|, we can choose a finite set I ⊂ I such that |A| j∈I |A j | + 8λ .
Thus, setting A := j∈I A j and u := u A we get that (A , u ) ∈ C and Hence, setting A := A ∪ j∈J \J F j and u := u χ A + u 0 χ j∈J \J F j we get that (A , u ) ∈ C and Notice that for j ∈ J \J , the set ∂ A ∩ ∂ F j becomes the internal crack for A , and there is no elastic energy contribution in F j ; see Fig. 2.
Hence, A is a union of finitely many connected open sets with finitely many "holes" inside so that ∂ A = ∂ * A consists of finitely many connected sets with finite length. Moreover, by (5.5), (5.6) and (5.7), inf F λ + 2 F λ (A , u ).  As J v is contained in at most finitely many closed C 1 -curves, we can find finitely many arcs of those curves whose union ⊂ B still contains J v and satisfies Combining this with (5.14) we obtain (5.9).
Proof of Theorem 2.9. In view of Proposition 4.9 the assertion follows from the direct methods of the Calculus of Variations.

Appendix A
In this section we recall some results from the literature for the reader's convenience.

A.1. Kuratowski Convergence
Let {E k } be a sequence of subsets of R 2 . A set E ⊂ R 2 is the K-lower limit of {E k } if for every x ∈ E and ρ > 0 there exists n > 0 such that B ρ (x) ∩ E k = ∅ for all k > n. A set E ⊂ R 2 is the K-upper limit of {E k } if for every x ∈ E and ρ > 0 and n ∈ N there exists k > n such that B ρ (x) ∩ E k = ∅. Observe that by the definition of K-convergence, E k and E k have the same K-upper and K-lower limits. Moreover, Kuratowski limit is always unique.
Proposition A.1. The following assertions are equivalent: (a) E k K → E; (b) if x k ∈ E k converges to some x ∈ R 2 , then x ∈ E, and for every x ∈ E there exists a subsequence x n k ∈ E n k converging to x; (c) dist(·, E k ) → dist(·, E) locally uniformly in R 2 ; 14] it is differentiable at a.e. ∈ (0, H 1 ( )) and |γ o ( )| 1. Hence has an (approximate) tangent line at H 1 -a.e. x ∈ and we can define the approximate unit normal ν (x) of at H 1 -a.e. x ∈ . We recall the following characteristics of compact connected H 1 -rectifiable sets (see [29,Theorem 3.14] and [36,Section 2] ((0, 1) Note that |E h | < v and since the bi-Lipschitz maps do not increase the number of connected components, (E h , v h ) ∈ C m . Let us estimate where θ F (x) is 1 for H 1 -a.e. on ∂ * F, is 2 for H 1 -a.e. on F (1) ∪ F (0) ∩∂ F and is 0 otherwise. By the choice of v h , I 2 0. Moreover, by (A.4) and the area formula as well as from (2.4), (A.2) and equality θ E h ( (y)) = θ A h (y) for H 1 -a.e. y ∈ ∂ A h , Moreover, by (2.4), thus, Finally, repeating the same arguments of Step 4 in the proof of [28, Theorem 1.1], we obtain (A.5) Since the dependence of the right-side of (A.5) on h is only through λ h , for sufficiently large , which contradicts to the minimality of (A h , u h ).