Regularity improvement for the minimizers of the two-dimensional Griffith energy

In this paper we prove that the singular set of connected minimizers of the planar Griffith functional has Hausdorff dimension strictly less then one, together with the higher integrability of the symetrized gradient.


Introduction
In a planar elasticity setting, the Griffith energy is defined by where Ω ⊂ R 2 is a bounded open set which stands for the reference configuration of a linearized elastic body, and where λ and µ are the Lamé coefficients satisfying µ > 0 and λ+µ > 0. The energy functional is defined on pairs (u, K) composed of a displacement u : Ω \ K → R 2 and a (N − 1)-dimensional crack-set K ⊂ Ω.The notation e(u) = (∇u + ∇u T )/2 denotes the symmetric gradient of u.
The precise formulation of the Dirichlet problem is as follow.We fix a bounded open set Ω ′ containing Ω and a datum ψ ∈ W 1,∞ (Ω ′ ).We define the admissible pairs as the elements of where LD is the space of functions of bounded Lebesgue deformation, i.e., functions u ∈ W 1,2  loc (Ω ′ \ K) such that e(u) ∈ L 2 (Ω ′ \ K).We say that (u, K) ∈ A(Ω) is a minimizer for the Griffith energy if it is a solution to the problem inf ˆΩ\K Ae(v) : e(v) dx + H 1 (K) : (v, K) ∈ A(Ω), v = ψ a.e. in Ω ′ \ Ω .
A lot of attention has been given on the Griffith functional these last years (see [3,4,5,6,7,8,9,15]), and in particular it has been proved that a global minimizer (u, K) ∈ A(Ω) (with a prescribed Dirichlet boundary condition) does exist and that the crack set K is H 1 -rectifiable and locally Ahlfors-regular in Ω.The latter means that there exists C 0 ≥ 1 (depending on A) such that for all x ∈ K and all r > 0 with B(x, r) ⊂ Ω, In [3] it was proved that any isolated connected component of the singular set K of a Griffith minimizer is C 1,α a.e.It also applies to a connected minimizer K (for e.g.minimizer with connected constraints).In this paper we slightly improve the Hausdorff dimension of the singular set.We also prove some higher integrability property on the symmetrized gradient.
The main results of this paper are the following.
2. There exists C ≥ 1 and p > 1 (depending on A) such that for all x ∈ Ω and r > 0 such that B(x, r) ⊂ Ω, The proof of our main theorem follows from standard technics that was already used in the scalar context of the Mumford-Shah functional, but adapted to the vectorial Griffith functional in a non trivial manner.In particular, for (1) we follow the approach of David [10] and Rigot [17], based on uniform rectifiability of the singular set and Carleson measure estimates.The idea is to estimate to number of balls in which one can apply the ε-regularity theorem contained in [3].But the latter needs a topological separating property that one has to control in any initialized balls which is one of the main issue of the present work (Lemma 3.1).We also need to control the 2-energy by a p-energy (Corollary 5.1) which also uses a topological argument (Lemma 5.3).The proof of ( 2) is based on a strategy similar to what was first introduced by De Philippis and Figalli in [13] and also used in [16], which easily follows from the porosity of the singular set together with elliptic estimates.Since the needed elliptic estimates relatively to the Lamé system are not easy to find in the literature, we have developed an appendix containing the precise results.
Let us stress that the famous Cracktip function that arises as blow-up limits of Mumford-Shah minimizers at the tip of the crack, has a vectorial analogue.This was the purpose of the work in [2].Since the vectorial Cracktip is homogeneous of degree 1/2 (see [2,Theorem 6.4]), it is natural to conjecture that, akin to the standard Mumford-Shah functional, the integrability exponent of |e(u)| should reach every p < 4, as asked by De Giorgi for the Mumford-Shah functional.

Notation
The 1-dimensional Hausdorff measure is denoted by H 1 .If E is a measurable set, we will write |E| its Lebesgue measure.We write M 2×2 for the set of real 2 × 2 matrices, and M 2×2 sym for that of all real symmetric 2 × 2 matrices.Given two matrix A, B ∈ M 2×2 , we recall the Frobenius inner product A : B = tr( t AB) and the corresponding norm |A| = tr(A T A).
Given a weakly differentiable vector field u, the symmetrized gradient of u is denoted by The p-normalized energy Let (u, K) ∈ A(Ω).Then for any x 0 ∈ Ω and r > 0 such that B(x 0 , r) ⊂ Ω, we define the normalized elastic energy of u in B(x 0 , r) by .

The flatness
Let K be a relatively closed subset of Ω.For any x 0 ∈ K and r > 0 such that B(x 0 , r) ⊂ Ω, we define the (bilateral) flatness of K in B(x 0 , r) by where L belongs to the set of lines passing through x 0 .When a minimizer (u, K) ∈ A(Ω) is given, we write simply β(x 0 , r) for β K (x 0 , r).
Remark 2.1.The flatness β K (x 0 , r) only depends on the set K ∩ B(x 0 , 2r).We have that for all 0 < t ≤ r, and for y 0 ∈ K ∩ B(x 0 , r) and t > 0 such that B(y 0 , t) ⊂ B(x 0 , r), In the sequel, we will consider the situation where x 0 ∈ K, r > 0 are such that B(x 0 , r) ⊂ Ω and β K (x 0 , r) ≤ ε, for ε ∈ (0, 1/2) 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 passing through 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 (2.1) is satisfied with β K (x 0 , r) ≤ 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.
Definition 2.1.Let K be a relatively closed set of Ω, x 0 ∈ K and r > 0 be such that B(x 0 , r) ⊂ Ω and β K (x 0 , r) ≤ 1/2.We say that K separates B(x 0 , r) if the balls D ± (x 0 , r) are contained into two different connected components of B(x 0 , r) \ K.
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).Lemma 2.1.[3, Lemma 3.1] Let τ ∈ (0, 1/16), let K ⊂ R 2 be a relatively closed subset of Ω, let x 0 ∈ K, let r > 0 be such that B(x 0 , r) ⊂ Ω and β K (x 0 , r) ≤ τ .If K separates B(x 0 , r), then for all t ∈ (16τ r, r), we have β K (x 0 , t) ≤ 1/2 and K still separates B(x 0 , t).

Local separation in many balls
The purpose of this section is the following general result on compact connected sets which are locally Ahlfors-regular.Lemma 3.1.Let K ⊂ Ω be a compact connected set which is locally Ahlforsregular in Ω, i.e., there exists C 0 ≥ 1 such that for all x ∈ K and for all r > 0 with B(x, r) ⊂ Ω, C −1 0 r ≤ H 1 (K ∩ B(x, r)) ≤ C 0 r.Then for every 0 < ε ≤ 1/2, there exists a ∈ (0, 1/2) small enough (depending on C 0 and ε) such that the following holds.For all x ∈ K and r > 0 with B(x, r) ⊂ Ω, one can find y ∈ K ∩ B(x, r/2) and t ∈ (ar, r/2) satisfying: β K (y, t) ≤ ε and K separates B(y, t) in the sense of Definition 2.1. (3.1) Proof.The letter C is a constant ≥ 1 that depends on C 0 and whose value might increase from one line to another but a finite number of times.Let x ∈ K and let r > 0 be such that B(x, r) ⊂ Ω.It will be more convenient to work under the assumption that B(x, 10r) ⊂ Ω and r ≤ diam(K)/10 and we are going to justify that we can make this assumption without loss of generality.First, we draw from the local Ahlfors-regularity that there exists a constant Let us consider the constant κ := 10C 1 , the radius r 1 = κ −1 r and some a ∈ (0, 1/2).If we solve the problem in the ball B(x, r 1 ), that is, if we find y ∈ B(x, r 1 /2) and t ∈ (ar 1 , r 1 /2) such that (3.1) holds true, then we have solved the problem in B(x, r) as well because y ∈ B(x, r/2) and t ∈ (br, r/2), where b = aκ −1 .This shows that it suffices to solve the problem in the ball B(x, r 1 ) which satisfies r 1 ≤ min(r/10, diam(K)/10).From now on, we directly assume B(x, 10r) ⊂ Ω and r ≤ diam(K)/10.
Step 1. Find a smaller shifted ball with small flatness.In the sequel we want to apply the results of [12], which works with sets of infinite diameter.This explains why we are going to need to slightly modify our set K to fit in the definition of [12].Precisely, given an arbitrary line L passing through x, one can check that the set is connected and Ahlfors-regular in the exact sense of [12, Definition 1.13], that is, E is closed and there exists C ≥ 1 (that depends on C 0 as usual) such that for all y ∈ E and for all t > 0, As a consequence, it is contained in a (Ahlfors)-regular curve (see [12, (1.63)] and the discussion below) and thus is uniformly rectifiable with a constant C ≥ 1 that depends on C 0 ([12, Theorem 1.57 and Definition 1.65]).In particular, it satisfies a geometric characterisation of uniform rectifiability called Bilateral Weak Geometric Lemma ([12, Definition 2.2 and Theorem 2.4]).It means that for all ε > 0, there exists C(ε) ≥ 1 (depending on C 0 and ε) such that for all y ∈ E and all t > 0, ˆz∈E∩B(y,t) ˆρ We apply this property with y := x and ρ := r, We observe that for all z ∈ K ∩ B(x, r) and for all 0 < s < r, we have where We fix ε ∈ (0, 1/10).We let a ∈ (0, 1/2) be a small parameters that will be fixed later.Assume by contradiction that for all z ∈ K ∩B(x, r) and s ∈ (ar, r), we have β K (z, s) > ε.This means that for such pairs (z, s), we have 1 B(ε) (z, s) = 1.Moreover we have by local Ahlfors-regularity Using now (3.3), we arrive at a contradiction, provided that a is small enough (depending on C 0 and ε).Thus we have found y ∈ B(x, r/2) and t ∈ (ar, r/2) such that t ≤ diam(K)/10 and Step 2. Conclusion.We then need to find a shifted smaller ball again in which K separates.For that purpose, we will use the fact that a compact connected set with finite length is arcwise connected (see [12,Theorem 1.8]).We choose a coordinate system such that y = (0, 0) and the line L that realizes the infimum in the definition of β K (y, t) is the x axis: Since t ≤ diam(K)/10, there exists a point z ∈ K \ B(0, t) and a curve Γ ⊂ K from 0 to z.This curve touches ∂B(0, t) and we let z ′ ∈ ∂B(0, t) be the point of ∂B(0, t) which is reached by Γ for the first time.Let Γ ′ ⊂ Γ be the piece of curve from 0 to z ′ .We see that since Γ ′ starts at 0 and meets ∂B(0, t) only at its extremity.Therefore, we have The point z ′ must lie either on the arc ∂B(0, t) Let us assume that the first case occurs (for the second case we can argue similarily).The curve and runs from 0 (the center of the ball) to z ′ (on the boundary of B(0, t)) such that z ′ 1 > 0. We can therefore find a point and thus it separates B(y ′ , t/8) the two connected components of The strip in (3.5) contains the strip in (3.4) so that it is centred on y ′ .As K contains Γ ′ , then K also separates B(y ′ , t/8).Moreover, we have β K (y ′ , t/8) ≤ 16β K (y, t) ≤ 16ε and one can replace ε by ε/16 in the proof above to arrive exactly at β K (y ′ , t/8) ≤ ε.

Carleson measure estimates on ω p
We define The purpose of this section is to state the following fact.as a competitor, and the ellipticity of A. The proof in [11, Section 23] on Mumford-Shah minimizers can be followed verbatim so we prefer to omit the details and refer directly to [11].

Control of ω 2 by ω p
The main ε-regularity theorem uses an assumption on the smallness of ω 2 .Unfortunately, what we can really control in many balls (thanks to Proposition 4.1) is ω p for p < 2, which is weaker.This is why in this section we prove that ω 2 can be estimated from ω p , for a minimizer.This strategy was already used in [11] and [17] for the Mumford-Shah functional.The adaptation for the Griffith energy is not straigthforward, but can be done by following a similar approach as the one already used in [3, Section 4.1.],generalized with ω p instead of only ω 2 .Some estimates from the book [11] were also useful.
Lemma 5.1 (Harmonic extension in a ball from an arc of circle).Let p ∈ (1, 2], 0 < δ ≤ 1/2, x 0 ∈ R 2 , r > 0 and let C δ ⊂ ∂B(x 0 , r) be the arc of circle defined by Then, there exists a constant C ≥ 1 (depending on p) such that every function Proof.Let Φ : C δ → C 0 be a bijective and bilipschitz mapping from C δ to C 0 .As δ ≤ 1/2, we can can request that the biLipschitz constant of Φ is universal.Since u • Φ −1 ∈ W 1,p (C 0 ; R 2 ), we can define the extension by reflection ũ ∈ W 1,p (∂B(x 0 , r); R 2 ) on the whole circle ∂B(x 0 , r), that satisfies where C ≥ 1 is a universal constant.We next define g as the harmonic extension of ũ in B(x 0 , r).Using [11,Lemma 22.16], we obtain , which completes the proof.
Lemma 5.2.Let p ∈ (1/2], let (u, K) ∈ A(Ω) be a minimizer of the Griffith functional.Let x 0 ∈ K and r > 0 be such that B(x 0 , r) ⊂ Ω and β(x 0 , r) ≤ 1/2.Let S be the strip defined by where L is the line passing through x 0 which achieves the infimum in β K (x 0 , r).
Then there exists a constant C ≥ 1 (depending on p), a radius ρ ∈ (r/2, r) and , where C ± are the connected components of ∂B(x 0 , ρ) \ S Proof.Let A ± be the connected components of B(x 0 , r) \ S. Since K ∩ A ± = ∅, by Korn inequality there exists two skew-symmetric matrices R ± such that the functions x → u(x) − R ± x belong to W 1,p (A ± ; R 2 ) and where the constant C ≥ 1 is universal since the domains A ± are all uniformly Lipschitz for all possible values of β(x 0 , r) ≤ 1/2.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 5.1 applied to the functions u ± : x → u(x) − R ± x, which belong to W 1,p (C ± ; R 2 ) since they are regular, for δ = β(x 0 , r) we get two functions .
Using the competitor above, we can obtain the following.
Proof.We keep using the same notation than that used in the proof of Lemma 5.2.Let ρ ∈ (r/2, r) and v ± ∈ W 1,2 (B(x 0 , ρ); R 2 ) be given by Lemma 5.2.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 B(x 0 , ρ) \ L(x 0 , r) which intersect A ± , and the proposition follows.
The next Lemma is of purely topological nature.
Lemma 5.3.Let K ⊂ R 2 be a compact connected set with H 1 (K) < +∞.For all x ∈ K and r ∈ (0, diam(K)/2), we have Proof.The inequality is trivial if β K (x, r) ≥ 1/4 so we can assume β K (x, r) ≤ 1/4 without loss of generality.The difficulty here is that even though K is connected, it may be that K ∩ B(x, r) is not.Let ε be such that β K (x, r) < ε ≤ 1/2.To simplify the notations, we assume that x = (0, 0) and that horizontal axis L = R × { 0 } achieves the infimum in the definition of β K (x, r).Therefore, K ∩ B(x, r) is contained in the strip S defined by Let π 1 : R 2 → L be the orthogonal projection onto L. As π 1 is 1-Lipschitz, we know that Let us define E := π 1 (K ∩ B(x, r)) and introduce the constant We are going to use the fact that K is connected to show that L ∩ [−rc ε , rc ε ] \ E is an interval (we identify L with the real axis).For each a ∈ E, there exists |t| ≤ εr such that z 0 := (a, t) ∈ K, and since r < diam(K)/2 there exists a curve Γ that connects z 0 to some point z 1 ∈ K \ B(x, r).But then E has to contain π 1 (Γ), and since Indeed, the curve Γ is contained in the strip S and has to "escape the ball" B(x, r) either from the right or from the left.The projection with minimal length would be when Γ escapes exactly at the corner of S ∩ B(x, r) which gives the definition of c ε (see the picture below).This holds true for all a ∈ E, which necessarily imply that [−c ε r, c ε r] \ E is an interval, that we denote by I.As (I × [−εr, εr]) ∩ K = ∅, we must have |I| ≤ 2εr otherwise the center of I is a distance ≥ εr from K whence β K (x, r) ≥ ε, which would contredict the definition of ε.All in all we have proved that Since ε can be chosen arbitrary close to β K (x, r), the result follows.We now come to the interesting "reverse Hölder" type estimate that will be needed later.
Proof.If β(x 0 , r) ≥ 1/2, the inequality is straightforward.Indeed we know by the usual upper bound (4.1) that we have ω 2 (x 0 , ρ) ≤ C for some constant C ≥ 1. Therefore in this case, it suffices to choose C p ≥ 2C in Corollary 5.1.We now assume that β(x 0 , r) ≤ 1/2.By Proposition 5.1, we already know that there exists a constant C ≥ 1 and a radius ρ ∈ (r/2, r) such that Then, we apply Lemma 5.3 in B(x 0 , ρ) which yields Finally, by ellipticity of A we get which finishes the proof.

Porosity of the bad set
Given 0 < α < 1, x 0 ∈ K and r > 0 such that B(x, r) ⊂ Ω, we say that the crack-set K is C 1,α -regular in the ball B(x 0 , r) if it is the graph of a C 1,α function f such that, in a convenient coordinate system it holds f (0) = 0, f ′ (0) = 0 and r α f ′ C α ≤ 1/16.We recall the following ε-regularity theorem coming from [3].
Proof.Unfortunately, the above statement is not explicitely stated in [3], but it directly follows from the proof of [3,Proposition 3.4].Indeed, in the latter proof, some explicit thresholds δ 1 > 0 and δ 2 > 0 and an exponent α ∈ (0, 1) are given so that, provided that and K separates B(x 0 , r), then β(y, t) ≤ C t r α for all y ∈ B(x 0 , r/2) and t ∈ (0, r/2).It implies that there exists a ∈ (0, 1) (which depends on C and α) such that B(x 0 , ar) is a C 1,α and 10 −2 -Lipschitz graph above a linear line (thanks to [3, Lemma 6.4]).The line can be chosen to be the tangent line to K at x 0 so that the graph satisfies f ′ (0) = 0.In addition the estimate (6.8) in [3] says moreover r α f ′ C 0,α ≤ C, from which we easily get (ar) α f ′ C 0,α ≤ 1/16 up to take a smaller constant a.The fact that α and a depend only on A, follows from a careful inspection of the proof in [3].
We are now in position to prove the following, which says that the singular set is porous in K. Proposition 6.1.Let (u, K) ∈ A(Ω) be a minimizer of the Griffith functional with K connected.There exists constants a ∈ (0, 1/2) (depending on A) such that the following property holds true.For all x 0 ∈ K and r > 0 such that B(x 0 , r) ⊂ Ω, there exists y ∈ K ∩ B(x 0 , r/2) such that K ∩ B(y, ar) is C 1,α -regular (where α is the constant of Theorem 6.1).
Proof.In view Theorem 6.1, it is enough to prove the following fact: there exists a ∈ (0, 1/2) such that for all x ∈ K and r > 0 with B(x, r) ⊂ Ω, there exists y ∈ K ∩ B(x, r/2) and ar < s < r/2 such that ω 2 (y, s) + β(y, s) ≤ ε 2 and K separates B(y, s), where ε 2 is the constant of Theorem 6.1.We already know from Lemma 3.1 how to control β and the separation.We therefore need to add a control on ω 2 , and this will be done by applying successively Proposition 4.1 and Corollary 5.1, but we need to fix carefully the constants so that it compiles well.
Let us pick any p ∈ (1, 2) and let C p be a constant which is bigger than the constant of Proposition 4.1 and the constant of Corollary 5.1.Let also C 0 be the constant used for Ahlfors-regularity, see (1.1).Then we define and we let ε 0 ∈ (0, 1/2) be a small constant which will be fixed during the proof and will depend only on A. We fix x ∈ K and r > 0 such that B(x, r) ⊂ Ω.As noticed at the beginning of the proof of Lemma 3.1, we can assume without loss of generality that B(x, 10 r) ⊂ Ω and r ≤ diam(K)/10.We apply Lemma 3.1 with the previous definition of ε 0 and we get that for some a ∈ (0, 1/2) (depending on A), there exists y ∈ B(x, r/2) and t ∈ (ar, r/2) satisfying: β(y, t) ≤ ε 0 and K separates B(y, t) as in Definition 2.1.
Note that the assumption of Lemma and here we can assume ε 0 small enough such that β(z, s) ≤ ε 2 /(4C 0 ).Next, we apply Corollary 5.1 which says that there exists s ′ ∈ (s/2, s) such that The ball B(z, s ′ ) satisfies all the required properties because β(z, s as required. It remains to see that K still separates the ball B(z, s ′ ).But since we know that β(y, t) ≤ ε 0 , that K separates B(y, t) and that z ∈ B(y, t/2), s ∈ (bt, t/2), it follows from Lemma 2.1 that ε 0 can be chosen small enough so that K separates B(z, s ′ ).
We are now ready to state one of our main results about the the Hausdorff dimension of the singular set.Corollary 6.1.Let (u, K) ∈ A(Ω) be a minimizer of the Griffith functional with K connected.Then there exists a closed set Σ ⊂ K such that dim H (Σ) < 1 and Proof.The proof is standard now that Proposition 6.1 is established.Indeed, we can argue exactly as Rigot in [17,Remark 3.29] which we refer to for more detail.
We rely on a higher integrability lemma ([16, Lemma 4.2]) which is inspired by the technique of [13].We recall that given 0 < α < 1, a closed set K, x 0 ∈ K and r > 0, we say that K is C 1,α -regular in the ball B(x 0 , r) if it is the graph of a C 1,α function f such that, in a convenient coordinate system it holds f (0) = x 0 , f ′ (0) = 0 and r α f ′ C α ≤ 1/16.We take the convention that the C 1,α norm is small enough because we don't want it to interfere with the boundary gradient estimates for the Lamé's equations.It is also required by the covering lemma [16,Lemma 4.3] on which [16, Lemma 4.2] is based.Lemma 7.1.We fix a radius R > 0. Let K be a closed subset of B(0, R) ⊂ R 2 and v : B(0, R) → R + be a non-negative Borel function.We assume that there exists C 0 ≥ 1 and 0 < α ≤ 1 such that the following holds true.
(ii) For each ball B(x, r) ⊂ B(0, R) with x ∈ K, there exists a smaller ball B(y, C −1 0 r) ⊂ B(x, r) with y ∈ K in which K is C 1,α -regular.(iii) For each ball B(x, r) ⊂ B(0, R) such that either K is disjoint from B(x, r) or such that x ∈ K and K is C 1,α -regular in B(x, r), we have .
Proof.It suffices to prove that for all solution of A. From now on, we deal with (A.2).The letter C plays the role of a constant ≥ 1 that depends on λ, µ and α, A. We refer to the proof [14,Theorem 3.18] which itself refers to the proof of [1,Theorem 7.53].We straighten the boundary Γ R via the C 1,α diffeomorphism φ : x → x ′ + (x n − f (x ′ ))e n .We observe that φ(V R ) contains a half-ball ball B + = B(0, C −1 0 R) + , where C 0 ≥ 1 is a constant that depends on λ, µ, α.The Neumann problem satisfied by u in V R is transformed into a Neumann problem satisfied by a function v in B(0, C −1 0 R) + .Then we symmetrize the elliptic system to the whole ball B = B(0, C −1 0 R) as in [14,Theorem 3.18].Following the proof of [1,Theorem 7.53] (in the special case where the right-hand side h is zero), we arrive to the fact there exists q > n = 2 (depending on λ, µ, α) such that for all x 0 ∈ 1 2 B and 0 < ρ ≤ r ≤ C , where σ = q−2 2 , and this implies sup This property is inherited by u via the diffeomorphism φ, sup We can finally bound the supremum of |∇u| on 1 2 V R by a covering argument.
For a and b ∈ R 2 , we write a•b = 2 i=1 a i b i 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)).

Figure 1 :
Figure 1: estimating the length of K ∩ B(x, r).
Then there exists p > 1 and C ≥ 1 (depending on C 0 ) such thatB(0,R/2) v p ≤ C.Proof of Theorem 7.1.We apply the Lemma 7.1.More precisely, for all x ∈ Ω and all R > 0 such that B(x, R) ⊂ Ω, one can applies Lemma[16, Lemma 4.2] in the ball B(x, R) to the function v := R|e(u)| 2 .The assumption (i) follows from the local Ahlfors-regularity of K.The assumption (ii) follows from the porosity (Proposition 6.1).The assumption (iii) follows from interior/boundary gradient estimates for the Lame's equations and from the local Ahlfors-regularity.In particular, the boundary estimate is detailed in Lemma A.1 in Appendix A.Lemma A.1.Let us assume n = 2.There exists C ≥ 1 (depending on α, A, λ, µ)