Partial regularity for the crack set minimizing the two-dimensional Griffith energy

In this paper we prove a $\mathcal C^{1,\alpha}$ regularity result for minimizers of the planar Griffith functional arising from a variational model of brittle fracture. We prove that any isolated connected component of the crack, the singular set of a minimizer, is locally a $\mathcal C^{1,\alpha}$ curve outside a set of zero Hausdorff measure.

Following the original Griffith theory of brittle fracture [26], the variational approach introduced in [23] rests on the competition between a bulk energy, the elastic energy stored in the material, and a dissipation energy which is propositional to the area (the length in 2D) of the crack. In a planar elasticity setting, the Griffith energy is defined by where Ω ⊂ R 2 , which is bounded and open, stands for the reference configuration of a linearized elastic body, and A is a suitable elasticity tensor. Here, e(u) = (∇u+∇u T )/2 is the elastic strain, the symmetric most countable, but it seems difficult to exclude the possibility of uncountably many negligible connected components that accumulate to form a set with positive H 1 -measure. We could also imagine many small connected components of positive measure that accumulate near a given bigger component. The assumption to consider an isolated connected component in our main theorem rules out these pathological situations. The precise role of this hypothesis will be explained later.
Finally, our main result is stated on an isolated connected component of a general minimizer. An alternative could be to minimize the Griffith energy under a connectedness constraint, or under a uniform bound on the number of the connected components. Existence and Ahlfors regularity of a minimizer in this class are much easier to obtain, due to Blashke and Go lab Theorems, and our result in this case would imply that the singular set is C 1,α regular H 1 -almost everywhere. Indeed, a careful inspection to our proof reveals that all the competitors that we use preserve the topology of K, thus they can still be used under connectedness constraints on the singular set, leading to the same estimates. Moreover, it is quite probable that most of the results contained in this paper could be applied to almost minimizers instead of minimizers (i.e. pairs that minimize the Griffith energy in all balls of radius r with its own boundary datum, up to an error excess controlled by some Cr 1+α term). For sake of simplicity we decided to treat in this paper minimizers of the global functional only.
Comments about the proof. The Griffith energy is similar to the classical Mumford-Shah energy for some aspects, but it actually necessitates the introduction of new ideas and new techniques. For the classical Mumford-Shah problem, there are two main approaches. The first one, in dimension 2, see [7] (or [17], written a bit differently in the monograph [18]), is of pure variational nature. It was extended in higher dimensions in [30] with a more complicated geometrical stopping time argument. Alternatively, there is a PDE approach [5,3] (see also [4]), valid in any dimension, which consists in working on the Euler-Lagrange equation. However, none of the aforementioned approaches can be directly applied to the Griffith energy.
More precisely, while trying to perform the regularity theory for the Griffith energy, one has to face the following main obstacles: (i) No Korn inequality. The well-known Korn inequality in elasticity theory enables one to control the full gradient ∇u by the symmetric part of the gradient e(u). Unfortunately, it is not valid in the cracked domain Ω \ K, due to the possible lack of regularity of K (see [13,24]). Therefore, one has to keep working with the symmetric gradient in all the estimates.
(ii) No Euler-Lagrange equation. A consequence of the failure of the Korn inequality is the lack of the Euler-Lagrange equation. Indeed, while computing the derivative of the Griffith energy with respect to inner variations, i.e. by a perturbation of u of the type u • Φ t (x) where Φ t = id + tΦ, some mixtures of derivatives of u appear and these are not controlled by the symmetric gradient e(u). Therefore, the so-called "tilt-estimate", which is one of the key ingredients of the method in [5,3] cannot be used.
(iii) No coarea formula. A fondamental tool in calculus of variations and in geometric measure theory is the so-called coarea formula, which enables one to reconstruct the total variation of a scalar function by integrating the perimeter of its level sets. In our setting, on the one hand the displacement u is a vector field, and on the other hand even for each coordinate of u there would be no analogue of this formula with e(u) replacing ∇u. In the approach of [18] or [30], the coarea formula is a crucial ingredient which ensures that, provided the energy of u is very small in some ball, one can use a suitable level set of u to "fill the holes" of K, with very small length. It permits to reduce to the case where the crack K "separates" the ball in two connected components. This is essentially the reason why our regularity result only holds on (isolated) connected components of K.
(iv) No monotonicity formula for the elastic energy. One of the main ingredients to control the energy in [7] and [18] (in dimension 2), is the so-called monotonicity formula, which essentially says that a suitable renormalization of the bulk energy localized in a ball of radius r is a nondecreasing function of r. This is not known for the elastic energy, i.e. while replacing ∇u by e(u).
(v) No good extension techniques. To prove any kind of regularity result, one has to create convenient competitors, and the main competitor in dimension 2 is obtained by replacing K in some ball B where it is sufficiently flat, by a segment S which is nearly a diameter. While doing so, and in order to use the minimality of (u, K), one has to define a new function v which coincides with u outside B, which belongs to LD(B \ S), and whose elastic energy is controlled by that of u. Denoting by C ± both connected components of ∂B \ S, the way this is achieved in the standard Mumford-Shah theory (see [18] or [32]) consists in introducing the harmonic extensions of u| C ± to B using the Poisson kernel. This provides two new functions u ± ∈ H 1 (B), whose Dirichlet energies in the ball B are controlled by that of u on the boundary ∂B \ K. For the Griffith energy, the same argument cannot be used since there is no natural "boundary" elastic energy on ∂B \ K.
Let us now explain the novelty of the paper and how we obtain a regularity result, in spite of the aforementioned problems. We do not have any hope to solve directly the general problem (i), which would probably be a way to solve all the other ones. We follow mainly the two-dimensional approach of [18], for which one has to face the main obstacles (iii)-(v) described above.
Due to the absence of the coarea formula, we cannot control the size of the holes in K at small scales, when K is very flat, as done in [18]. This is a first reason why our theorem restricts to a connected component of K only. There is a second reason related to the decay of the normalized energy by use of a compactness argument (in the spirit of [29] or [5]) in absence of a monotonicity formula for the energy. In this argument, one of our main tool is the so-called Airy function w associated to a minimizer u, which can be constructed only in the two-dimensional case. This function has been already used in [6] to prove compactness and Γ-convergence results related to the elastic energy, and it is defined through the harmonic conjugate (see Proposition 5.2). The main property of w is that it is a scalar biharmonic function in Ω \ K, which satisfies |D 2 w| = |Ae(u)|. What is important is the fact that D 2 w is a full gradient, while e(u) is only a symmetric gradient. The other interesting fact in terms of boundary conditions, at least under connected assumptions, is the transformation of a Neumann type problem on the displacement u into a Dirichlet problem on the Airy function w, which is usually easier to handle.
We then obtain that, provided K is sufficiently flat in some ball B(x 0 , r) and the normalized energy is sufficiently small, we can control the decay of the energy r → ω(x 0 , r) as r → 0 (see Proposition 3.3). This first decay estimate is proved by contradiction, using a compactness and Γ-convergence argument on the elastic energy. In this argument, it is crucial the starting point x 0 to belong to an isolated connected component of K.
The second part of the proof is a decay estimate on the flatness, namely the quantity where the infimum is taken over all affine lines L passing through x, measuring how far is K from a reference line in B(x 0 , r). This quantity is particularly useful since a decay estimate of the type β(x 0 , r) ≤ Cr α leads to a C 1,α regularity result on K (see Lemma 6.4). The excess of density, namely controls the quantity β(x 0 , r) 2 , as a consequence of the Pythagoras inequality (see Lemma 6.3). In order to estimate the excess of density, the standard technique consists in comparing K, where K is already known to be very flat in a ball B(x 0 , r) (i.e. β(x 0 , r) ≤ ε), with the competitor given by the replacement of K by a segment S in B(x 0 , r). While doing this, one has to define a suitable admissible function v in B(x 0 , r) associated to the competitor S, that coincides with u outside B(x 0 , r) and has an elastic energy controlled by that of u. This is where we have to face the problem (v) mentioned earlier. The way we overcome this difficulty is a technical extension result (see Lemma 4.5). Whenever β(x 0 , r) + ω(x 0 , r) ≤ ε for ε sufficiently small (depending only on the Ahlfors regularity constant θ 0 ), one can find a rectangle U , such that B(x 0 , r/5) ⊂ U ⊂ B(x 0 , r), and a "wall set" Σ ⊂ ∂U , such that: where η is small. Moreover, if K ′ is a competitor for K in U (which "separates"), then there exists a The main point being that the set Σ ⊂ U where the values of u and v do not match, has very small length, essentially of order η > 0, that can be taken arbitrarily small. The price to pay is a diverging factor as η → 0 in the right-hand side of the previous inequality. A similar statement with H 1 (Σ) ≤ rβ(x 0 , r) is much easier to prove, and is actually used before as a preliminary construction (see Lemma 4.2). We believe Lemma 4.5 to be one of the most original part of the proof of Theorem 1.1. With this extension result at hand, estimating the flatness through the excess of density as described before, and choosing η of order ω(x 0 , r) 1/6 , enables one to obtain a decay estimate for the flatness of the type (see Proposition 3.2), The previous decay estimate together with the decay of the renormalized energy constitute the main ingredients which lead to the C 1,α regularity result.
Organization of the paper. The paper is organized as follows. In Section 2, we introduce the main notation used throughout the paper, and we precisely define the variational problem of fracture mechanics we are interested in. In Section 3, we prove our main result, Theorem 1.1, concerning the partial C 1,αregularity of the isolated connected components of the crack. The proof relies on two fundamental results. The first one, Proposition 3.2 is a flatness estimate in terms of the renormalized bulk energy which is established in Section 4. The second one, Proposition 3.3, is a bulk energy decay which is proved in Section 5. Eventually, we gather in the Appendix of Section 6 several technical results.
2. Statement of the problem 2.1. Notation. The Lebesgue measure in R n is denoted by L n , and the k-dimensional Hausdorff measure by H k . If E is a measurable set, we will sometimes write |E| instead of L n (E). If a and b ∈ R n , we write a · b = n i=1 a i b i for the Euclidean scalar product, and we denote the norm by |a| = √ a · a. The open (resp. closed) ball of center x and radius r is denoted by B(x, r) (resp. B(x, r)).
We write M n×n for the set of real n × n matrices, and M n×n sym for that of all real symmetric n × n matrices. Given a matrix A ∈ M n×n , we let |A| := tr(AA T ) (A T is the transpose of A, and trA is its trace) which defines the usual Frobenius norm over M n×n .
Given an open subset U of R n , we denote by M(U ) the space of all real valued Radon measures with finite total variation. We use standard notation for Lebesgue spaces L p (U ) and Sobolev spaces W k,p (U ) or H k (U ) := W k,2 (U ). If K is a closed subset of R n , we denote by H k 0,K (U ) the closure of C ∞ c (U \ K) in H k (U ). In particular, if K = ∂U , then H k 0,∂U (U ) = H k 0 (U ). Functions of Lebesgue deformation. Given a vector field (distribution) u : U → R n , the symmetrized gradient of u is denoted by In linearized elasticity, u stands for the displacement, while e(u) is the elastic strain. The elastic energy of a body is given by a quadratic form of e(u), so that it is natural to consider displacements such that e(u) ∈ L 2 (U ; M n×n sym ). If U has Lipschitz boundary, it is well known that u actually belongs to H 1 (U ; R n ) as a consequence of the Korn inequality. However, when U is not smooth, we can only assert that u ∈ L 2 loc (U ; R n ). This motivates the following definition of the space of Lebesgue deformation: (2.1) LD(U ) := {u ∈ L 2 loc (U ; R n ) : e(u) ∈ L 2 (U ; M n×n sym )}. If U is connected and u is a distribution with e(u) = 0, then necessarily it is a rigid movement, i.e. u(x) = Ax + b for all x ∈ U , for some skew-symmetric matrix A ∈ M n×n and some vector b ∈ R n . If, in addition, U has Lipschitz boundary, the following Poincaré-Korn inequality holds: there exists a constant c U > 0 and a rigid movement r U such that According to [2, Theorem 5.2, Example 5.3], it is possible to make r U more explicit in the following way: consider a measurable subset E of U with |E| > 0, then one can take provided the constant c U in (2.2) also depends on E.
Hausdorff convergence of compact sets. Let K 1 and K 2 be compact subsets of a common compact set K ⊂ R n . The Hausdorff distance between K 1 and K 2 is given by We say that a sequence (K n ) of compact subsets of K converges in the Hausdorff distance to the compact set K ∞ if d H (K n , K ∞ ) → 0. Finally let us recall Blaschke's selection principle which asserts that from any sequence (K n ) n∈N of compact subsets of K, one can extract a subsequence converging in the Hausdorff distance.
Capacities. In the sequel, we will use the notion of capacity for which we refer to [1,27]. We just recall the definition and several facts. The (k, 2)-capacity of a compact set K ⊂ R n is defined by One of the interests of capacity is that it enables one to give an accurate sense to the pointwise value of Sobolev functions. More precisely, every u ∈ H k (R n ) has a (k, 2)-quasicontinuous representativeũ, which means thatũ = u a.e. and that, for each ε > 0, there exists a closed set A ε ⊂ R n such that Cap k,2 (R n \ A ε ) < ε andũ| Aε is continuous on A ε (see [1, Section 6.1]). The (k, 2)-quasicontinuous representative is unique, in the sense that two (k, 2)-quasicontinuous representatives of the same function u ∈ H k (R n ) coincide Cap k,2 -quasieverywhere. In addition, if U is an open subset of R n , then u ∈ H k 0 (U ) if and only if for all multi-index α ∈ N n with length |α| ≤ k − 1, ∂ α u has a (k − |α|, 2)-quasicontinuous representative that vanishes Cap k−|α|,2 -quasi everywhere on ∂U , i.e. outside a set of zero Cap k−|α|,2capacity (see [1,Theorem 9.1.3]). In the sequel, we will only be interested in the cases k = 1 or k = 2 in dimension n = 2.

2.2.
Definition of the problem. We now describe the underlying fracture mechanics model and the related variational problem.
Reference configuration. Let us consider a homogeneous isotropic linearly elastic body occupying Ω ⊂ R 2 in its reference configuration. The Hooke law associated to this material is given by where λ and µ are the Lamé coefficients satisfying µ > 0 and λ + µ > 0. Note that this expression can be inverted into where E := 4µ(λ + µ)/(λ + 2µ) is the Young modulus and ν := λ/(λ + 2µ) is the Poisson ratio.
Griffith energy. For all (u, K) ∈ A(Ω), we define the Griffith energy functional by In this work, we are interested in (interior) regularity properties of the global minimizers of the Griffith energy under a Dirichlet boundary condition, i.e., solutions to the (strong) minimization problem where ψ ∈ W 1,∞ (R 2 ; R 2 ) is a prescribed boundary displacement. Note that, this formulation of the Dirichlet boundary condition permits to account for possible cracks on ∂Ω, where the displacement does not match the prescribed displacement ψ.
The question of the existence of solutions to (2.3) has been addressed in [12] (see also [11,25]), extending up to the boundary the regularity results [14,10]. For this, by analogy with the classical Mumford-Shah problem, it is convenient to introduce a weak formulation of (2.3) as follows with GSBD 2 a suitable subspace of that of generalized special functions of bounded deformation (see [16]) where the previous energy functional is well defined. According to [11,Theorem 4.1], if Ω has Lipschitz boundary, the previous minimization problem admits at least a solution, denoted by u. In addition, if Ω is of class C 1 , thanks to [12, Theorems 5.6 and 5.7], there exist θ 0 > 0 and R 0 > 0, only depending on A, such that the following property holds: for all x 0 ∈ J u and all r ∈ (0, R 0 ) such that B(x 0 , r) ⊂ Ω ′ , then The previous property of J u ensures that, setting K := J u , then H 1 (K \ J u ∩ Ω) = 0, so that the pair (u, K) ∈ A(Ω) is a solution of the strong problem (2.3). In addition, the crack set K is H 1 -rectifiable and Ahlfors regular: for all x 0 ∈ K and all r ∈ (0, R 0 ) such that B(x 0 , r) ⊂ Ω ′ , then where C is a constant depending only on Ω. The second inequality is obtained by comparing (u, K) with the most standard competitor (v, K ′ ) where v := u1 Ω ′ \(Ω∩B(x0,r)) and K ′ : Next, taking in particular K ′ = K and any v ∈ LD(Ω \ K) as competitor implies that u ∈ LD(Ω \ K) is also a solution of the minimization problem Note that u is unique up to an additive rigid movement in each connected component of Ω \ K disjoint from ∂Ω \ K. It turns out that u satisfies the following variational formulation: for all test functions ϕ ∈ H 1 (Ω \ K; R 2 ) with ϕ = 0 on ∂Ω \ K, In particular, u is a solution to the elliptic system −div(Ae(u)) = 0 in D ′ (Ω \ K; R 2 ), and, as a consequence, elliptic regularity shows that u ∈ C ∞ (Ω \ K; R 2 ).
3. The main quantities and proof of the C 1,α regularity We now introduce the main quantities that will be at the heart of our analysis.
3.1. The normalized energy. Let (u, K) ∈ A(Ω). Then for any x 0 and r > 0 such that B(x 0 , r) ⊂ Ω we define the normalized elastic energy by Sometimes we will write ω u (x 0 , r) to emphasize the underlying displacement u.
Remark 3.1. By definition of the normalized energy, for all 0 < t < r, we have 3.2. The flatness. Let K be a closed subset of R 2 . For any x 0 ∈ R 2 and r > 0, we define the flatness by where the infimum is taken over all affine lines L passing through x 0 . In other words Sometimes we will write β K (x 0 , r) to emphasize the underlying crack K.
Remark 3.2. By definition of the flatness, we always have that for all 0 < t < r, In the sequel, we will consider the situation where for ε > 0 small. This implies in particular that K ∩ B(x 0 , r) is contained in a narrow strip of thickness εr passing through the center of the ball. Let L(x 0 , r) be a line containing x 0 and satisfying We will often use a local basis (depending on x 0 and r) denoted by (e 1 , e 2 ), where e 1 is a tangent vector to the line L(x 0 , r), while e 2 is an orthogonal vector to L(x 0 , r). The coordinates of a point y in that basis will be denoted by (y 1 , y 2 ). Provided (3.3) is satisfied with ε ∈ (0, 1/2), we can define two discs D + (x 0 , r) and D − (x 0 , r) of radius r/4 and such that D ± (x 0 , r) ⊂ B(x 0 , r) \ K. Indeed, using the notation introduced above, setting x ± 0 := x 0 ± 3 4 re 2 , we can check that D ± (x 0 , r) := B(x ± 0 , r/4) satisfy the above requirements. A property that will be fundamental in our analysis is the separation in a closed ball.
The following lemma guarantees that when passing from a ball B(x 0 , r) to a smaller one B(x 0 , t), and provided that β K (x, r) is relatively small, the property of separating is preserved for t varying in a range depending on β K (x, r).
Proof. We will need the following elementary inequality resulting from the mean value Theorem Using the notations introduced above, considering the local basis {e 1 , e 2 } such that e 1 is a tangent vector to L(x 0 , r) and e 2 is a normal vector to L(x 0 , r), we have For all t ∈ (16τ r, r), we have We first note that, similarly to (3.7), we can estimate From (3.8) and (3.6) we deduce Denoting by α = arccos(ν(x 0 , t) · e 2 ) the angle between ν(x 0 , t) and e 2 , the previous inclusion implies where we have used (3.5), and that t > 16τ r. Let y 0 := x 0 + 3 4 ν(x 0 , t)t be the center of the disc D + (x 0 , t). We have that |(y 0 − x 0 ) 2 | = cos(α) 3 4 t. In particular, using the elementary inequality | cos(α) − 1| ≤ α and (3.9) we get hence, since t > 16τ r, we infer that for all y ∈ B(y 0 , t 4 ), All in all, we have proved that Arguing similarly for D − (x 0 , t), we get Since by (3.6) we have that K ∩ B(x 0 , t) ⊂ {y ∈ B(x 0 , t) : |y 2 | ≤ τ r}, we deduce that D ± (x 0 , t) must belong to two distinct connected components of B(x 0 , t)\K, and thus that K actually separates The following topological result is well-known.
Lemma 3.2. Let K ⊂ Ω be a relatively closed set, and let x 0 ∈ K, r > 0 be such that B(x 0 , r) ⊂ Ω.
Then, there exists an injective Lipschitz curve Γ ⊂ K that still separates D ± (x 0 , r) in B(x 0 , r).

3.3.
Initialization of the main quantities. We prove that, if (u, K) is a minimizer of the Griffith functional, one can find many balls B(x 0 , r) ⊂ Ω such that K separates D ± (x 0 , r) in B(x 0 , r) and such that β K (x 0 , r) and ω u (x 0 , r) are small for r > 0 small enough, and for H 1 -a.e. x 0 ∈ Γ, where Γ ⊂ K is any connected component of K ∩ Ω. The restriction to a connected component Γ is only due to ensure the separation property on K. Notice that in the following proposition we do not need the connected component to be isolated.
Proof. The initialization for the quantity β is standard (see for instance [ , we sketch below the proof for the sake of completeness. Since K is a rectifiable set, we know that there exists Z 1 ⊂ K with H 1 (Z 1 ) = 0 such that, at every point x 0 ∈ K \ Z 1 , K admits an approximate tangent line T x0 , that is for all ε ∈ (0, 1), where T x0,εr := {y ∈ R 2 : dist(y, T x0 ) ≤ εr}. Since K is also Ahlfors-regular by assumption, it is easily seen that T x0 is the usual tangent, in the sense that for all ε ∈ (0, 1) there exists r ε > 0 such that for all r ≤ r ε . Indeed, assume that there exist ε 0 ∈ (0, 1) and two sequences, r k → 0 and y k ∈ Then, by Ahlfors regularity (2.4) we have We conclude that which is against (3.10). Hence, by definition, for all x 0 ∈ K \ Z 1 and all ε ∈ (0, 1), there exists r 1 > 0 such that β(x 0 , r) ≤ ε for all r ≤ r 1 .
Now we consider ω(x 0 , r), which again can be initialized by the same argument used for the standard Mumford-Shah functional (see for instance [4,Proposition 7.9]). Let us reproduce it here. We consider the measure µ := Ae(u) : e(u)L 2 . For all t > 0, let By a standard covering argument (see [4,Theorem 2.56]) one has that But E t ⊂ K and µ(K) = 0, thus H 1 (E t ) = 0 for all t > 0. By taking a sequence t n ց 0 + and defining Z 2 := n E tn , we have that H 1 (Z 2 ) = 0 and, for all In other words, for every for all r ≤ r 2 . It remains to prove the separation property of K. To this aim, let us consider a connected component Γ of K ∩ Ω which is relatively closed in K ∩ Ω. Since Γ is a compact and connected set in R 2 with H 1 (Γ) < +∞, according to [18,Proposition 30.1] it is the range of an injective Lipschitz mapping γ : [0, 1] → Γ. This implies that Γ has an approximate tangent line L x0 for H 1 -a.e. x 0 ∈ Γ. In addition, according to [8, Proposition 2.2.(iii)], there exists an exceptional set Z 3 ⊂ Γ with H 1 (Z 3 ) = 0 and with the property that, for all x 0 ∈ Γ \ Z 3 , one can find r 3 > 0 such that, where π : R 2 → L x0 denotes the orthogonal projection onto the line L x0 . In particular, if moreover β(x 0 , r) ≤ ε ≤ 10 −3 , it follows that the balls D ± (x 0 , (1 − 10 −3 )r) are well defined and, thanks to (3.11), that Γ must separate then H 1 (Z) = 0, and we have proved that for all x 0 ∈ Γ \ Z and all r ≤ r 0 := min(r 1 , r 2 , (1 − 10 −3 )r 3 ), we obtain that 3.4. Proof of Theorem 1.1. The proof of our main result, Theorem 1.1, rests on both the following results, whose proofs are postponed to the subsequent sections. The first one is a flatness estimate in terms of the renormalized energy, which will be established in Section 4.
Proposition 3.2. There exist ε 1 > 0 and C 1 > 0 (only depending on θ 0 , the Ahlfors regularity constant of K) such that the following property holds. Let (u, K) ∈ A(Ω) be a minimizer of the Griffith functional.
For all x 0 ∈ K and r > 0 such that B(x 0 , r) ⊂ Ω, The second result is the following normalized energy decay which will be proved in Section 5.
Proposition 3.3. For all τ > 0, there exists a ∈ (0, 1) and ε 2 > 0 such that the following property holds. Let (u, K) ∈ A(Ω) be a minimizer of the Griffith functional, and let Γ be an isolated connected component of K ∩ Ω such that H 1 (Γ) > 0. Let x 0 ∈ Γ and r > 0 be such that B(x 0 , r) ⊂ Ω and With both previous results at hands, we are in position to bootstrap the preceding decay estimates in order to get a C 1,α -regularity estimate. Indeed, the conclusion of Theorem 1.1 will follow from the following result.
Proof. Let ε 1 > 0 and C 1 > 0 be the constants given by Proposition 3.2, and let a > 0 and ε 2 > 0 be the constants given by Proposition 3.3 corresponding to τ = 10 −2 . We define We can assume that H 1 (Γ) > 0, otherwise the Proposition is trivial. Using that Γ is an isolated connected component of K ∩ Ω and Proposition 3.1 (applied with ε = min(δ 1 , δ 2 )/2), we can find an exceptional set Z ⊂ Γ with H 1 (Z) = 0 such that the following property holds: for every We start by showing a first self-improving estimate which stipulates that the quantities ω(x 0 , r) and β(x 0 , r) will remain small at all smaller scales.
Step 1. We define b := a/50, and we claim that if then the following three assertions are true: Let us start with the renormalized energy. By Proposition 3.3 we get ω(x 0 , ar) ≤ 10 −2 ω(x 0 , r), which yields using (3.1) For what concerns the flatness, we can apply Proposition 3.2, so that Thus by (3.2) we get
Step 2. Iterating the decay estimate established in Step 1, we get that (3.13), (3.14), and (3.15) hold true in each ball B(x 0 , b k r), k ∈ N. We thus obtain that and subsequently, using now (3.15), If t ∈ (0, 1) we let k ≥ 0 be the integer such that Notice in particular that for some constant C > 0 only depending on C 1 and a.
Step 3. We now conclude the proof of the proposition. Indeed, according to (3.12), for every Thus, by Steps 1 and 2 applied in each ball 2 t α for all t ∈ (0, 1/2), and since this is true for all x ∈ K ∩ B(x 0 , r/2), we deduce that K ∩ B(x 0 , a 0 r) is a C 1,α curve for some a 0 ∈ (0, 1/2) thanks to Lemma 6.4 in the appendix.

Proof of the flatness estimate
In order to prove Proposition 3.2, we need to construct a competitor in a ball B(x 0 , r), where the flatness β(x 0 , r) and the renormalized energy ω(x 0 , r) are small enough. The main difficulty is to control how the crack behaves close to the boundary of the ball. A first rough competitor is constructed in Propositions 4.1 and 4.2 by introducing a wall set of length rβ(x 0 , r) on the boundary. It leads to density estimates in balls (or alternatively in rectangles) which state that, provided the crack is flat enough, the energy density scales like the diameter of the ball (or the width of the rectangle), up to a small error depending on β(x 0 , r) and ω(x 0 , r).
Unfortunately, this rough competitor is not sufficient to get a convenient flatness estimate leading to the desired regularity result. A better competitor is obtained by suitably localizing the crack in two almost opposite boxes of size η > 0, arbitrarily small (see Lemma 4.4). Then we can define a competitor inside a larger rectangle U , whose vertical sides intersect both the small boxes. The crack competitor is then defined by taking an almost horizontal segment inside the rectangle, together with a new wall set Σ ⊂ ∂U of arbitrarily small length, made of the intersection of the rectangle with the boxes. It is then possible to introduce a displacement competitor (see Lemma 4.5) by extending the value of u on ∂U \ Σ inside U . The price to pay is that the bound on the elastic energy associated to this competitor might diverge as the length of the wall set is small. It is however possible to optimize the competition between the flatness and the renormalized energy associated to this competitor by taking η = ω(x 0 , r) 1/7 , leading to the conclusion of Proposition 3.2.
4.1. Density estimates. In this section we prove some density estimates for the set K. Such estimates will be useful to select good radii, in a way that the corresponding spheres intersect the set K in two almost opposite points. One of the main tools to construct competitors will be the following extension lemma.
We next define g as the harmonic extension ofũ in B(x 0 , r which completes the proof. Lemma 4.2 (Extension lemma, first version). Let (u, K) ∈ A(Ω) be a minimizer of the Griffith functional, and let x 0 ∈ K and r > 0 be such that B(x 0 , r) ⊂ Ω and β K (x 0 , r) ≤ 1/10. Let S be the strip defined by S := {y ∈ B(x 0 , r) : dist(y, L(x 0 , r)) ≤ rβ(x 0 , r)}. Then there exist a universal constant C > 0, ρ ∈ (r/2, r), and v ± ∈ H 1 (B(x 0 , ρ); R 2 ), such that v ± = u on C ± , C ± being the connected components of ∂B(x 0 , ρ) \ S, and Proof. Let A ± be the connected components of B(x 0 , r) \ S. Since K ∩ A ± = ∅, by the Korn inequality there exist two skew-symmetric matrices R ± such that the functions x → u(x) − R ± x belong to H 1 (A ± ; R 2 ) and where the constant C > 0 is universal since the domains A ± are all uniformly Lipschitz for all possible values of β(x 0 , r) ≤ 1/10. Using the change of variables in polar coordinates, we infer that which allows us to choose a radius ρ ∈ (r/2, r) satisfying Setting C ± := ∂A ± ∩∂B(x 0 , ρ), in view of Lemma 4.1 applied to the functions u ± : x → u(x)−R ± x, which belong to H 1 (C ± ; R 2 ) since they are regular, for δ = rβ(x 0 , r) we get two functions g ± ∈ H 1 (B(x 0 , ρ); R 2 ) satisfying g ± (x) = u(x) − R ± x for H 1 -a.e. x ∈ C ± and B(x0,ρ) Finally, the functions x → v ± (x) := g ± (x) + R ± x satisfy the required properties.
We now use the extension to prove two density estimates, first in smaller balls, then in smaller rectangles.
Proposition 4.1 (Density estimate in a ball). Let (u, K) ∈ A(Ω) be a minimizer of the Griffith functional, and let x 0 ∈ K and r > 0 be such that B(x 0 , r) ⊂ Ω and β K (x 0 , r) ≤ 1/10. Then there exist a universal constant C > 0 and a radius ρ ∈ (r/2, r) such that Proof. We keep using the same notation than that used in the proof of Lemma 4.2. Let ρ ∈ (r/2, r) and v ± ∈ H 1 (B(x 0 , ρ); R 2 ) be given by the conclusion of Lemma 4.2. We now construct a competitor in B(x 0 , ρ) as follows. First, we consider a "wall" set Z ⊂ ∂B(x 0 , ρ) defined by and that We are now ready to define the competitor (v, K ′ ) by setting and, denoting by V ± the connected components of Since H 1 (K ′ ∩ B(x 0 , ρ)) ≤ 2ρ + 4rβ(x 0 , r), we deduce that and the proposition follows.
The following proposition is similar to Proposition 4.1, but here balls are replaced by rectangles. The assumption that K separates D ± (x 0 , r) in B(x 0 , r) is not crucial here and could be removed. We will keep it to simplify the proof of the proposition, since nothing changes for the purpose of proving Theorem 1.1.
Proposition 4.2 (Density estimates in a rectangle). Let (u, K) ∈ A(Ω) be a minimizer of the Griffith functional, and let x 0 ∈ K and r > 0 be such that B(x 0 , r) ⊂ Ω, β K (x 0 , r) ≤ 1/10, and K separates D ± (x 0 , r) in B(x 0 , r). Let {e 1 , e 2 } be an orthogonal system such that L(x 0 , r) is directed by e 1 . Then there exists a universal constant C * > 0, such that Proof. We first apply Lemma 4.2 to get a radius ρ ∈ (r/2, r) and functions v ± ∈ H 1 (B(x 0 , ρ); R 2 ) which satisfy the conclusion of that result. In order to construct a competitor for K in B(x 0 , ρ), we would like to replace the set K inside the rectangle by the segment L(x 0 , r) ∩ R which has length exactly equal to r/5. Such a competitor may not separate the balls D ± (x 0 , ρ) in B(x 0 , ρ). If D ± (x 0 , ρ) belonged to the same connected component, we could only take v + (or v − ) as a competitor of u, introducing a big jump on the boundary of B(x 0 , ρ) and removing completely the jump on K. To overcome this problem, we consider a "wall set" (inside the vertical boundaries of R) as well as a second wall set on ∂B(x 0 , r) as before, defined by Z := {y ∈ ∂B(x 0 , ρ) : dist(y, L(x 0 , r)) ≤ rβ(x 0 , r)}.
We define Note that K ′ is now separating the ball B(x 0 , ρ) (thanks to the wall set Z ′ ) and Now we define the competitor for the function u in B(x 0 , ρ). To this aim, using that K ′ separates the ball B(x 0 , ρ), we can find two connected components V ± of B(x 0 , ρ) \ K ′ whose closure intersect C ± and define v : Let us recall that u = v ± on ∂B(x 0 , ρ) \ Z. Note that the presence of Z in the singular set K ′ is due to the fact that v ± does not match u on Z. The pair (v, K ′ ) is then a competitor for (u, K) in B(x 0 , ρ), and thus, from which we deduce that which completes the proof of the result.
An interesting consequence of the previous density estimates is a selection result of good radii, in a way that the corresponding spheres intersect the set K at only two almost opposite points.  Finding a good radius). There exists a universal constant ε 0 > 0 such that the following property holds: let (u, K) ∈ A(Ω) be a minimizer of the Griffith functional and let x 0 ∈ K and r > 0 be such that B(x 0 , r) ⊂ Ω and ω u (x 0 , r) + β K (x 0 , r) ≤ ε 0 .

4.2.
The main extension result. The first rough density estimate given by Proposition 4.1 is based on the property that the crack is always contained in a small strip of thickness rβ(x 0 , r). This enables one to construct a competitor outside a wall set with height of order rβ(x 0 , r). However, in order to bootstrap the estimates on our main quantities, β and ω, we need to slightly improve such a density estimate obtaining a remainder of order rη, η well chosen (of order ω(x 0 , r) 1/7 ), instead of rβ(x 0 , r).
To this aim, we need a refined version of the extension Lemma 4.2, in which the boundary value of the competitor displacement is prescribed outside a wall set of height rη, instead of rβ(x 0 , r). To construct such a suitable small wall set, we first find a nice region in the annulus B(x 0 , 2r 5 ) \ B(x 0 , r 5 ) where to cut, i.e. we find some little boxes in which the set K is totally trapped. This is the purpose of the following lemma. Notice that in all this subsection and in the next one we never use any connectedness assumption on K, but we rather use a separating assumption only. where θ 0 > 0 is the Ahlfors regularity constant of K, and C * > 0 is the universal constant given in Proposition 4.2. We also assume that K separates D ± (x 0 , r) in B(x 0 , r). Let {e 1 , e 2 } be an orthogonal system such that L(x 0 , r) is directed by e 1 . Then for every η ∈ (0, 10 −2 ) there exist two points y 0 and z 0 ∈ R 2 such that Proof. It is enough to prove the existence of a point y 0 since the argument leading to the existence of z 0 is similar. For simplicity, we will denote by β := β(x 0 , r), ω := ω(x 0 , r).
We start by finding a good vertical strip in which K has small length. Let us define the vertical strip S := y ∈ B(x 0 , r) : Let η < 1/10 and let N ∈ N, N ≥ 2, be such that 1 We split S into the pairwise disjoint union of N smaller sets S 1 , . . . , S N defined, for all k ∈ {1, . . . , N }, by S k =: y ∈ S : Since β ≤ 1/10 we can apply Proposition 4.2 which implies with E := 5C * (β + ω), where we recall that θ 0 is the Ahlfors regularity constant of K, and C * > 0 is the universal constant given in Proposition 4.2. As it will be used later, we notice that under our assumptions we have in particular that (4.8) E ≤ min(1, 10 −2 θ 0 ).
From (4.7) we deduce the existence of k 0 ∈ {1, . . . , N } such that (see Figure 1) By the separation property of K, one can find inside K ∩ S k0 , an injective Lipschitz curve Γ connecting both vertical sides of ∂S k0 (see Lemma 3.2). In particular, we have and thus (4.9) leads to (4.11) Thanks to the length estimate (4.10), if we denote by z, z ′ ∈ ∂S k0 both points of Γ on the boundary of S k0 , we have in particular, for every point y ∈ Γ, because E ≤ 1. In other words, We now finally give a bound on the distance from the points of K to the curve Γ in a strip slightly thinner than S k0 , by use of the Ahlfors-regularity of K. For that purpose we let S ′ ⊂ S k0 be defined by with δ := 2ηE θ0 . Since E ≤ 10 −2 θ 0 , we deduce that δ ≤ 10 −1 5N , so that S ′ is not empty. We claim that Indeed, if y ∈ K ∩ S ′ is such that d := dist(y, Γ) > δr = 2ηE θ0 r, then B(y, δr) ⊂ S k0 \ Γ and, by Ahlfors regularity, H 1 (K ∩ B(y, δr)) ≥ θ 0 δr = 2ηEr, which is a contradiction with (4.11) and proves (4.13).
To conclude, gathering (4.13) and (4.12), we have obtained (4.14) sup since 2E/θ 0 ≤ 1. Therefore, if we define y 0 as being the middle point of the segment [z, z + r 5N e 1 ] (in particular in the middle of S ′ ), the conclusion (4.2) and (4.3) of the lemma are satisfied.
Next, we notice that by (4.6), the width of S ′ is exactly provided that E ≤ θ0 16 , which is valid thanks to (4.8) (see Figure 2). Consequently, using (4.14) and that (y 0 ) 2 = z 2 , we deduce that with this choice of y 0 it holds Figure 2. The set K is trapped into a rectangle of size ≃ ηr.
The proof of the lemma follows by relabeling η/8 as η.
We are now in the position to establish an improved version of the extension lemma. Its proof is similar to that of Proposition 4.1, the difference being the definition of the wall set that has now size ηr instead of rβ(x 0 , r).

Lemma 4.5 (Extension Lemma)
. Let (u, K) ∈ A(Ω) be a minimizer of the Griffith functional, and let x 0 ∈ K and r > 0 be such that B(x 0 , r) ⊂ Ω and where θ 0 is the Ahlfors regularity constant of K and C * > 0 is the universal constant given in Proposition 4.2. We also assume that K separates D ± (x 0 , r) in B(x 0 , r). Then for all 0 < η < 10 −4 there exist: • an open rectangle U such that B(x 0 , r/5) ⊂ U ⊂ B(x 0 , r), • a wall set (i.e. union of two vertical segments) Σ ⊂ ∂U such that K ∩∂U ⊂ Σ, u ∈ H 1 (∂U \Σ; R 2 ) and H 1 (Σ) ≤ 120ηr. In addition, if K ′ ⊂ Ω is a closed set such that K ′ \ U = K \ U and D ± (x 0 , r/5) are contained in two different connected components of U \ K ′ , then there exists a function v ∈ H 1 (Ω \ K ′ ; where C > 0 is universal. Proof. We denote by {e 1 , e 2 } an orthogonal system such that L(x 0 , r) is directed by e 1 and we apply Lemma 4.4, which gives the existence of two points y 0 and z 0 ∈ B(x 0 , 2r/5) \ B(x 0 , r/5) satisfying (4.2)-(4.5). In order to construct the rectangle U and the wall set Σ, we need to introduce a domain A which is a "rectangular annulus" of thickness of order ηr.
Step 1: Construction of a rectangular annulus A. The vertical parts of A are defined as being the following open rectangles Notice that (y 0 − x 0 ) 1 ≤ 2 5 r and ηr ≤ 10 −2 r, so that which means that the right corners of V 1 have a distance to x 0 bounded by 41 2 100 2 + 1 9 r < r and therefore V 1 ⊂ B(x 0 , r).
Now the horizontal parts of A are given by the following open rectangles Note that the four rectangles V 1 , V 2 , H 1 , and H 2 are all contained in the ball B(x 0 , r). Finally, we define the "rectangular annulus" A by Figure 3). Next, we consider the two closed boxes Let us finally consider the subset of A outside the cutting boxes, and let A ± be both connected components of A ′ . The open sets A ± are Lipschitz domains, and they are actually unions of vertical and horizontal rectangles of thickness of order η and lengths of order r (notice that 30η ≤ 10 −2 ). In addition, since by construction we have K ∩ A ± = ∅, it follows that u ∈ H 1 (A ± ; R 2 ) Figure 3. The rectangular annulus A. and that the Korn inequality (see Lemma 6.2) applies in each rectangle composing A ± . Therefore, there exist two skew-symmetric matrices R ± such that (4.16) for some constant C > 0 universal, where η −5 appears estimating the distance between the skewsymmetric matrices in the intersection of two overlapping vertical and horizontal rectangles.
Step 2: Construction of the rectangle U . Let us denote by R : For any t ∈ [−ηr, ηr] we denote the vertical line passing through y(t) := y 0 + te 1 by L t := y(t) + Re 2 . According to Fubini's Theorem, we have We can thus find t 0 ∈ [−ηr, ηr] such that u ∈ H 1 (L t0 ∩ A ′ ; R 2 ) and 2ηr We perform the same argument at the point z 0 , finding some t 1 ∈ [−ηr, ηr] such that, denoting by L t1 the line z 0 + t 1 e 1 + Re 2 , one has u ∈ H 1 (L t1 ∩ A ′ ; R 2 ) and Arguing similarly for the top horizontal part of A + , we get a horizontal line L H + such that u ∈ H 1 (L H + ∩ A + ; R 2 ) and The vertical line L t0 intersects L H + at a single point a + 0 , and L t1 intersects L H + at another single point a + 1 .
We perform a similar construction on the lower part A − of A ′ which leads to another horizontal line The vertical line L t0 intersects L H − at a single point a − 0 , and L t1 intersects L H − at another single point a − 1 .
Finally, we define U as the rectangle with vertices (a − 0 , a − 1 , a + 0 , a + 1 ) (See Figure 4) and we define Σ as Σ : Figure 4. The rectangular domain U and the wall set Σ.
Step 3: Construction of the competitor v. Since U is a rectangle with "uniform shape", there exists a bilipschitz mapping Φ : R 2 → R 2 such that Φ(U ) = B := B(0, 1), Φ(∂U ) = ∂B and Φ(∂U + ) = C δ for some δ < 1/2, where ∂U + := ∂U ∩ A + and C δ is as in the statement of Lemma 4.1. Note that the Lipschitz constants of Φ and Φ −1 are bounded by Cr −1 and Cr where C is universal. Let R + be the skew symmetric matrix appearing in (4.16). Since u ∈ H 1 (∂U + ; R 2 ), we infer that the function x → u•Φ −1 (x)−R + Φ −1 (x) belongs to H 1 (C δ ; R 2 ). Applying Lemma 4.1, we obtain a function h + ∈ H 1 (B; where C > 0 is a universal constant. Then, defining v + := h + • Φ ∈ H 1 (U ; R 2 ) and noticing that if τ is a tangent vector to ∂B, then ∇Φ −1 τ is a tangent vector to ∂U + H 1 -a.e. in ∂U + , we infer that v + (x) = u(x) − R + x for H 1 -a.e. x ∈ ∂U + and Arguing similarly for where R − is the skew-symmetric matrix appearing in (4.16). Let K ′ ⊂ Ω be as in the statement. We construct a function v ∈ H 1 (Ω \ K ′ ; if x belongs to the connected component of U \ K ′ containing D ± (x 0 , r/5), and v := u otherwise. Note that by construction v = u on ∂U \ Σ, and Notice that K ∩ ∂U = K ′ ∩ ∂U by assumption, because K \ U = K ′ \ U and U is open. Thus, from (4.17) it follows that as required.

Proof of Proposition 3.2.
In Lemma 4.5 we have constructed the key displacement competitor associated to a separating crack competitor, which will be employed to show the flatness estimate. The construction of the crack competitor will be similar to that of Proposition 4.1, i.e. it will be obtained by replacing K by a segment in some ball. The difference here will be in the error appearing in the density estimate, which will depend only on ω(x 0 , r), and not anymore on β(x 0 , r).
By construction K ′ \ U = K \ U and D ± (x 0 , r/5) are contained in different connected components of U \ K ′ . Then, Lemma 4.5 provides a function v ∈ H 1 (Ω \ K ′ ; R 2 ) which coincides with u on (Ω \ U ) \ Σ and satisfies (4.15). The pair (v, K ′′ ), with is thus a competitor for (u, K), and by minimality of (u, K) we have that B(x 0 , r)).
We are now ready to prove the main flatness estimate.
Proof of Proposition 3.2. Let us define where ε ′ 0 > 0 is the threshold of Proposition 4.3 and C ′ > 0 is the universal constant in (4.18). By Proposition 4.3, we know that there exists s ∈ (r/40, r/5) such that K ∩ ∂B(x 0 , s) = {z, z ′ }, for some z = z ′ , and (4.20) (4.19). Let L be the line passing through x 0 which is parallel to the segment [z, z ′ ].

Proof of the normalized energy decay
In this section we prove a decay estimate for the normalized energy of a Griffith minimizer. The strategy is based upon a compactness argument and a Γ-convergence type analysis where one shows the stability of the Neumann problem in planar elasticity along a sequence of sets K n which converge in the Hausdorff sense to a diameter within a ball. It gives an alternative approach even for the scalar case (albeit only 2-dimensional and under topological conditions) to the corresponding decay estimates of the normalized energy in the standard proofs of regularity for the Mumford-Shah minimizers ( [4], [29,Theorem 1.10]).
We will start establishing some auxiliary results on the Airy function.
5.1. The Airy function. We state here a general result concerning the existence of the Airy function associated to a minimizer of the Griffith energy. It follows a construction similar to that in [6, Proposition 4.3], itself inspired by that introduced in [9]. The Airy function will be useful in order to get compactness results along a sequence of minimizers. The main difference with the situation in [6] is that now K is not assumed to be connected. The proofs are very similar to those of [6], and for that reason we do not write all the arguments but only point out the main changes with respect to the original proof.
. From the previous Lemma, one can construct the "harmonic conjugate" v associated to a minimizer (u, K) of the Griffith functional. The proof follows the lines of that in [6,Proposition 4.2]. The main difference with [6] is that, here, the singular set K is not assumed to be connected. This implies that it is not in general possible to conclude that v vanishes on the full crack K. However, the following proof makes it possible to ensure that, in some suitable weak sense, v is constant in each connected component of K, but the constants might depend on the associated connected component. This is the reason why we renormalize the harmonic conjugate v to vanish only on an arbitrary connected component of the crack of positive length.
It remains to show that if A ⊂ R 2 is an open set with A ⊂ Ω, then w ∈ H 2 0,L (A). We first note that w ∈ H 1 0,L (Ω) ∩ H 2 (Ω) with ∇w ∈ H 1 0,L (Ω; R 2 ). In particular, since w ∈ H 2 (Ω), it has a (Hölder) continuous representative, still denoted w, so that it makes sense to consider its pointwise values.
Since A \ L is not smooth, in order to show that w ∈ H 2 0,L (A), we will use a capacity argument similar to that used in [6,Proposition 4.3] and in [9,Theorem 1].
As a consequence of [1, Theorem 9.1.3], we conclude that z ∈ H 2 0 (Ω \ L), or in other words, that there exists a sequence (z n ) ⊂ C ∞ c (Ω \ L) such that z n → z = ηw in H 2 (Ω \ L). Note in particular that z n ∈ C ∞ (A) and that z n vanishes in a neighborhood of L ∩ A. Therefore, since z = w and ∇z = ∇w in A, we deduce that w ∈ H 2 0,L (A). We argue by contradiction by assuming that the statement of the proposition is false. Then, there exists τ 0 > 0 such that for every n ∈ N, one can find a minimizer (û n ,K n ) ∈ A(Ω) of the Griffith functional (with the same Dirichlet boundary data ψ), an isolated connected componentΓ n ofK n ∩ Ω, points x n ∈Γ n , radii r n > 0 with B(x n , r n ) ⊂ Ω such that K n ∩ B(x n , r n ) =Γ n ∩ B(x n , r n ), βK n (x n , r n ) → 0, and ωû n (x n , ar n ) > τ 0 ωû n (x n , r n ), for some a ∈ (0, 1) (to be fixed later).
Rescaling and compactness. In order to prove compactness properties on the sequences of sets and displacements, we need to rescale them into a unit configuration. For simplicity, from now on, we denote by B := B(0, 1). Let us first rescale the setsK n andΓ n by setting, for all n ∈ N, K n :=K n − x n r n , Γ n :=Γ n − x n r n .
LetL n := L(x n , r n ) be an affine line such that d H (L n ∩ B(x n , r n ),K n ∩ B(x n , r n )) ≤ r n βK n (x n , r n ), and L n :=L n −xn rn its rescaling. Up to a subsequence, and up to a change of orthonormal basis, we can assume that L n ∩ B → T ∩ B in the sense of Hausdorff, where T := Re 1 . Then, since we deduce that Γ n ∩ B = K n ∩ B → T ∩ B in the sense of Hausdorff. We next rescale the displacementsû n by setting, for all n ∈ N and a.e. y ∈ B, u n (y) :=û n (x n + r n y) ωû n (x n , r n )r n .
Then, we have . We next show that e is the symmetrized gradient of some displacement. To this aim, we consider, for any 0 < δ < 1/10, the Lipschitz domain δ are both connected components of A δ . Note that for such δ, D ± := B (0, ± 3 4 ), 1 4 ) ⊂ A ± δ and K n ∩ U δ = ∅ for n large enough (depending on δ). Denoting by the rigid body motion associated to u n in D ± , by virtue of the Poincaré-Korn inequality [2, Theorem 5.2 and Example 5.3], we get that , for some constant c δ > 0 depending on δ. Thanks to a diagonalisation argument, for a further subsequence (not relabeled), we obtain a function v ∈ LD(B \ T ) such that u n − r ± n ⇀ v weakly in H 1 (A ± δ ; R 2 ), for any 0 < δ < 1/10. Necessarily we must have that e = e(v) and thus, e(u n )1 B\Kn ⇀ e(v) weakly in L 2 (B; M 2×2 sym ).
Minimality. We next show that v satisfies the minimality property for all ϕ ∈ LD(B \ T ) such that ϕ = 0 on ∂B \ T . According to [9,Theorem 1], it is enough to consider competitors ϕ ∈ H 1 (B \ T ; R 2 ) such that ϕ = 0 on ∂B \ T . For a given arbitrary competitor ϕ ∈ H 1 (B \ T ; R 2 ) such that ϕ = 0 on ∂B \ T , we construct a sequence of competitors for the minimisation problems (5.6) using a jump transfert type argument (see [22] and [6]). To this aim, we denote by C ± n the connected component of B \ K n which contains the point (0, ±1/2), and we define ϕ n as follows Then, one can check that ϕ n ∈ H 1 (B \ K n ; R 2 ) and ϕ n = 0 on ∂B \ K n . Moreover, ϕ n → ϕ strongly in L 2 (B; R 2 ) and e(ϕ n )1 B\Kn → e(ϕ) strongly in L 2 (B; M 2×2 sym ). Therefore, thanks to the minimality property satisfied by u n , we infer that B\Kn Ae(u n ) : e(u n ) dx ≤ B\Kn Ae(u n + ϕ n ) : e(u n + ϕ n ) dx, which implies, by expanding the squares, that 0 ≤ 2 B\Kn Ae(u n ) : e(ϕ n ) dx + B\Kn Ae(ϕ n ) : e(ϕ n ) dx.
Using that e(ϕ n )1 B\Kn → e(ϕ) strongly in L 2 (B; M 2×2 sym ) and e(u n )1 B\Kn ⇀ e(v) weakly in L 2 (B; M 2×2 sym ), we can pass to the limit as n → +∞ to get that As a consequence, v is a smooth function in B\T . Moreover, due to the Korn inequality in both connected components of B \ T (which are Lipschitz domains), we get that v ∈ H 1 (B \ T ; R 2 ).
Convergence of the elastic energy. In order to pass to the limit in inequality (5.5), we need to show the convergence of the elastic energy, or in other words, the strong convergence of the sequence of elastic strains (e(u n )) n∈N . This will be achieved by using Proposition 5.2 which provides an Airy functionŵ n associated to the displacementû n , satisfyingŵ n ∈ H 2 (Ω) ∩ H 1 0,Γn (Ω) ∩ H 2 0,Γn (A) for all open set A ⊂ R 2 with A ⊂ Ω, (see also Remark 5.1) such that ∆ 2ŵ n = 0 in Ω \K n , and in Ω.
SinceK n ∩ B(x n , r n ) =Γ n ∩ B(x n , r n ) and B(x n , r n ) ⊂ Ω, we infer thatŵ n ∈ H 2 0,Kn (B(x n , r n )). We rescale the Airy functionŵ n by setting, for all n ∈ N and a.e. y ∈ B, w n (y) :=ŵ n (x n + r n y) ωû n (x n , r n )r n in such a way thatw n ∈ H 2 0,Kn (B), ∆ 2 w n = 0 in B \ K n , and Ae(u n ) = ∂ 22 w n −∂ 12 w n −∂ 12 w n ∂ 11 w n .
In addition, since then Poincaré's inequality ensures that the sequence (w n ) n∈N is bounded in H 2 (B), and thus, up to a subsequence w n ⇀ w weakly in H 2 (B), for some w ∈ H 2 (B). A similar capacity argument than that used in the proof of [6, Proposition 6.1] shows that w ∈ H 2 0,T (B(0, r)) for all r < 1, and In addition, using that the biharmonicity of w n is equivalent to the minimality for all z ∈ w n + H 2 0,Kn (B), we can again reproduce the proof of [6, Proposition 6.1] to get that w n → w strongly in H 2 (B(0, r)) for all r < 1. In particular, it implies that e(u n )1 B\Kn → e(v) strongly in L 2 (B(0, r); M 2×2 sym ), and thus passing to the limit in inequalities (5.4) and (5.5) yields (5.8) ω v (0, 1) ≤ 1 and ω v (0, a) ≥ τ 0 .
According to inequality (5.8), we infer that Without loss of generality, we assume that Decay of the elastic energy. We finally want to show a decay estimate on the elastic energy which will give a contradiction to (5.9). To this aim, denoting by B ± = B ∩ {±x 2 > 0}, we will work on the Airy function w to construct an extension of v| B + on B which still solves the elasticity system in B.
According to (formula (3.28) in) [34] (see also [21]), since w ∈ C ∞ (B + ) is a solution of we can consider the biharmonic reflexionw ∈ C ∞ (B) of w| B + in B defined bỹ which satisfies ∆ 2w = 0 in B. Thanks to this biharmonic extension, we are going to extend the displacement v| B + on the whole ball B into a functionṽ which minimizes the elastic energy. To this aim, let us define the stress byσ ensures the existence of someṽ ∈ C ∞ (B; R 2 ) such thatẽ = e(ṽ) in B. In particular, according to (5.7), we have which shows that e(ṽ) = e(v) in B + , and thus that v andṽ only differ from a rigid body motion in B + . We have thus constructed an extensionṽ of v| B + which satisfies −div(Ae(ṽ)) = 0 in B, or equivalently, According to (5.9), we have that where the last inequality comes from (5.8), possibly changing c γ . Taking γ = 1/2 and r = a yields 1 a B(0,a) Ae(ṽ) : e(ṽ) dx ≤ c 1/2 a 1/2 , which is against (5.10) provided we choose a < ( τ0 2c 1/2 ) 2 . ✷

Appendix
The following lemma is an easy consequence of the coarea formula.
Lemma 6.1. Let K ⊂ R 2 be a H 1 -rectifiable set. Then for all 0 < s < r and x 0 ∈ R 2 we have where, H 1 -a.e. in E, Jd E f denotes the 1-dimensional coarea factor associated to the tangential differential df E . Since E admits an approximate tangent line oriented by a unit vector τ at H 1 -a.e. points, we deduce that which leads to (6.1).
We next recall a version of the Korn inequality in a rectangle.
Gathering both the above inequalities, we obtain d S (τ (x, r k ), τ (x, r k+1 )) ≤ d S (τ (x, r k ), v) + d S (v, τ (x, r k+1 )) ≤ 9Cr α k = 9Cr α 0 2 −kα , as claimed. It follows that for all k, l ≥ k 0 , Since the latter can be made arbitrarily small provided k 0 is large enough, we deduce that τ (x, r k ) is a Cauchy sequence in S 1 /{±1}, and therefore, it converges to some vector denoted by τ (x). In particular, letting l → +∞, we get the following estimate for all k ≥ 0 Moreover, it can be easily seen through the distance estimate (6.3), that T x := x + Rτ (x) is a tangent line for the set K at the point x.
Step 3. K ∩ B(0, a) is a Lipschitz graph. We first show that for a > 0 small enough (to be fixed later), the set K ∩ B(0, a) is a graph above the line Rτ (0), that we assume for simplicity to be oriented by e 1 := τ (0). Notice that for all x ∈ K ∩ B(0, a), (6.10) d S (τ (x), e 1 ) ≤ C ′′ a α ,