Korn and Poincaré-Korn inequalities for functions with a small jump set

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Introduction
The modelling and analysis of fracture in the linearised elasticity framework relies on a good understanding of the space B D of functions of bounded deformation. These are vector-valued functions u in L 1 , whose symmetric (distributional) gradient Eu is a bounded Radon measure. Over the years, the fine properties of functions in B D, and in the subspace S B D of special functions of bounded deformation (corresponding to the case where Eu has no Cantor part) have been better understood, and the relation between B D, S B D and the space BV of functions of bounded variation has been Communicated by Y. Giga. studied in detail (see e.g., [3,5,17], and [22] for the space of generalised functions of bounded deformation). For a function u ∈ S B D( ), Eu admits the decomposition Eu = e(u)L n + [u] ν u H n−1 J u , (1.1) where e(u) is the absolutely continuous part of Eu with respect to the Lebesgue measure L n , J u the jump set of u, [u] the jump of u, ν u the normal to J u and [u] ν u denotes the symmetric tensor product of u and ν u . The decomposition (1.1) has a clear physical meaning: e(u) represents the elastic part of the strain, and J u the crack set. It is therefore natural that a model of (brittle) fracture, in the linearised setting, would involve an energy of the type |e(u)| 2 dx + H n−1 (J u ), (1.2) called the Griffith's energy, of which the Mumford-Shah energy in S BV is the scalar counterpart. The energy (1.2) is in fact well defined in the larger space G S B D( ) of generalised special functions of bounded deformation, which has been introduced by Dal Maso in [22], and is essentially designed to contain all the displacements for which the energy is finite (see Sect. 2 for the definition). Moreover, G S B D is the natural space for (1.2), where one can prove compactness and existence of minimisers under physical assumptions (see, e.g., [12,13,15]).
A key difficulty posed by the energy (1.2), compared to scalar models based on functions of bounded variation, is the lack of control on the skew-symmetric part (Du − Du T )/2 of the distributional gradient of u. The classical tool providing a relation between the full gradient and its symmetric part is the Korn inequality.
In this paper we prove Korn and Poincaré-Korn inequalities in G S B D p ( ), the space of functions u ∈ G S B D( ) for which e(u) ∈ L p ( ) and H n−1 (J u ) < +∞, for every dimension n ≥ 2 and any p > 1. More precisely, we have the following (see Theorem 4.5). Moreover, there exists c = c(n, p, q, ) > 0 such that u − a L q ( \ω) ≤ c(n, p, q, ) e(u) L p ( ) , (1.4) with q ≤ p * if p < n, q < ∞ if p = n, and q ≤ ∞ for p > n.
Clearly, the volume of ω is also controlled by H n−1 (J u ), thanks to the isoperimetric inequality, see Remark 3.4 below. This result is the generalisation, in dimension n ≥ 2, of the two-dimensional result in [16] (see also [27]). Theorem 1.1 ensures that e(u) controls u−a and its approximate gradient outside an exceptional set, and not in the whole set . This is in contrast with the classical Korn and Poincaré-Korn inequalities for functions u ∈ W 1, p ( ; R n ), with p > 1, which state that there exists an infinitesimal rigid motion a such that Du − Da L p ( ) ≤ c(n, p, ) Eu L p ( ) , (1.5) and that, thanks to the Poincaré inequality and Sobolev embeddings, u − a L q ( ) ≤ c(n, p, q, ) Eu L p ( ) , (1.6) where q depends on n and p (and q = p * for p < n).
Results like (1.5) and (1.6) are clearly out of reach in (G)S B D, even for functions u with a small jump set. This is due to the possible presence of small regions of that can be completely (or almost completely) disconnected from the domain, and where u would not necessarily be close to the infinitesimal rigid motion that achieves the smallest distance from u in the majority of the domain. Hence, in general, for a function u ∈ (G)S B D( ), e(u) cannot control u − a or its approximate gradient in the whole domain , and a result like Theorem 1.1 is the best possible.
The Korn and Poincaré-Korn inequalities in Theorem 1.1 are a corollary of the result below (see Theorem 4.1 and Remark 4.3), which is the main result of this paper. In Theorem 1.2, we prove 'almost' Sobolev regularity for functions in G S B D p . More precisely we show that, given a function u ∈ G S B D p ( ), we can replace it with a function v ∈ W 1, p ( ; R n ) outside an exceptional set ω ⊂ , whose perimeter is controlled by H n−1 (J u ). Moreover, u and v have a comparable Griffith's energy in the whole of . We observe that the conclusion of Theorem 1.2 is non-trivial only when the measure of the jump set J u is 'small' as else one could take ω = and v = 0 (see also Remark 4.2). The proof of Theorem 1.2 is done by regularising u at several scales, by means of the auxiliary results Lemma 3.1 and Theorem 3.2.
We now illustrate the idea of the proof. As a first step, we cover the domain with a family of disjoint cubes q whose size reduces towards the boundary. The cubes in the partition are then classified into 'good' and 'bad', depending on whether the amount of J u they contain is smaller or larger than a given threshold. The construction is done so that all the cubes in the covering of are 'good', up to a small neighbourhood of ∂ . In this neighbourhood, the 'bad' cubes are cut away from the domain by connecting them to ∂ by means of truncated cones. In this way what remains is still a Lipschitz set (with the same Lipschitz constant as ). Moreover, in each 'bad' cube, by definition, the perimeter of the cone is comparable to the perimeter of the cube, and hence is bounded by the measure of the jump set of u in it.
Hence it is sufficient to deal with good cubes. For each of the good cubes q we apply the auxiliary regularity result Theorem 3.2. This ensures that, given a functioñ u ∈ G S B D p (q), we can wipe out its jump set Jũ away from the boundary of q, up to a small expense in terms of the Griffith's energy, provided H n−1 (Jũ) is sufficiently smaller than the perimeter of q. This 'smallness' condition is exactly what enters in the definition of 'good' cubes. Applying Theorem 3.2 toũ := u |q in every 'good' cube q, we obtain a Sobolev regularisationṽ q of u |q and an exceptional setω q with controlled perimeter, such thatṽ q = u outsideω q . The function v in the statement of Theorem 1.2 is then obtained by patching together the functionsṽ q on all the good cubes. The set ω where v needs not coincide with u, is then defined as the union of the exceptional setsω q of the good cubes, together with the truncated cones relative to the bad cubes.
To conclude, we sketch the proof of Theorem 3.2, which is strongly inspired by its two-dimensional version [16], by Conti, Focardi and Iurlano, and involves an iterative regularisation procedure. Starting with w 0 =ũ, we construct a sequence (w k ), where w k+1 is obtained by covering a large part of J w k with a family of disjoint balls, and by replacing w k in each ball of the covering with a smoother function, provided by Lemma 3.1. The pointwise limitṽ of the sequence (w k ) has Sobolev regularity in a smaller ball, and satisfiesũ =ṽ outside an exceptional setω, which is defined as the union of the coverings of each step.

Comparison with previous results
The foundations of the function spaces S B D and G S B D were laid down in the papers [3,5], and [22]. Several research avenues have stemmed from them: the derivation of regularity properties for functions in (G)S B D p , and in particular of minimisers of the Griffith's energy, in the spirit of the celebrated result [23] by De Giorgi, Carriero and Leaci for the Mumford-Shah energy (see [4,10,12,13,15,17]); of Korn and Poincaré-Korn inequalities with various degrees of generality ( [9,[26][27][28]); of approximation and density results ( [7,11,[18][19][20]29]); of integral representation for functionals in (G)S B D p [16].
Our results are in between two of these avenues: we prove Sobolev regularity for functions in G S B D p , for every p > 1 and in every dimension n ≥ 2, outside an exceptional set (see Theorem 1.2) and, as a direct corollary, we obtain a Korn inequality, and a Poincaré-Korn inequality with sharp exponent (see Theorem 1.1), again outside an exceptional set.
Our work has a number of points of contact with previous results, but also a number of differences. In [9] the authors prove a Poincaré-Korn inequality like (1.4) for every n ≥ 2 and every p ≥ 1 by means of a slicing argument. Unlike our case, however, they obtain (1.4) with q = p(1 * ), rather than q = p * (which is optimal only for p = 1), and no estimate for the gradient of u is provided. Moreover, the exceptional set ω is controlled by the jump set of J u only in volume, while we also control its perimeter. A Poincaré-Korn inequality like (1.4) is proved also in [26], for n = 2 and p = 2, with an exceptional set ω whose structure is very simple, and can be related to the measure of J u . This objective is further pursued in [28], where the author proves a Poincaré-Korn inequality in G S B D 2 , up to an exceptional set with both perimeter and area bounded by (powers of) the measure of J u , for n ≥ 2. The L 2 -norm of e(u), however, only controls the distance of u from a rigid motion in the weaker norm q = 2(1 * ); additionally, one can obtain an L ∞ bound for such a distance, but the L 2 -norm of e(u) has to be weighted with a negative power of the measure of J u .
The first proofs of a Korn inequality like (1.3), in the (G)S B D context, are due to [27] and [16]. In [27] the proof is done in dimension n = 2 and for p = 2. Moreover, the distance of ∇u from a skew-symmetric matrix is estimated in a lower L q -norm, with q ∈ [1, 2). On the other hand, the exceptional set is estimated, both in perimeter and in area, with the measure of J u , and the integrability of u is improved to the sharp exponent, with consequent improvement of the Poincaré-Korn inequality. The two-dimensionality of the result is due to an approximation step, done in [26], that is only proved in the planar setting. Also the result in [16] is only proved for n = 2, and again this is due to a 'regularisation' step being done by means of a two-dimensional construction. Their approach, like ours, is based on first proving Sobolev regularity outside an exceptional set, and then deducing Korn and Poincaré-Korn inequalities as direct corollaries. Also in [16], like in our result, the exceptional set is bounded in perimeter in terms of J u , and the Poincaré-Korn inequality is proved with the sharp exponent for every p > 1.
In conclusion, our contribution is two-fold. On the one hand our result lifts the restriction to dimension n = 2 of the regularisation step from G S B D p to W 1, p , up to an exceptional set, which is now valid for every n ≥ 2 and every p > 1. In addition, the exceptional set we provide is bounded both in perimeter and in area with the measure of the jump set of the function. As a consequence, we can deduce the Korn and Poincaré-Korn inequalities up to the sharp exponent for every n ≥ 2 and p > 1, since the regularisation step is not reliant on a planar construction.

Conclusion and perspectives
The main result in this work, asserting the 'almost' Sobolev regularity of G S B D pfunctions with a small jump set, has some nontrivial consequences which are of independent interest, and which we present here. First of all, we obtain a Korn and a Poincaré-Korn inequality with sharp exponents outside an exceptional set, which is controlled in perimeter and volume by the jump set of the function (Theorem 4.5). We also prove an approximation result (Theorem 5.1) in the spirit of [11, Theorem 3.1]. Theorem 5.1 implies, in particular, the existence of the approximate gradient ∇u for functions in G S B D p (Corollary 5.2). Note that the existence of ∇u for functions in G S B D 2 had already been obtained in [28], as a consequence of the embedding G S B D 2 ( ) ⊂ (G BV ( )) n (see [28,Theorem 2.9]), for n ≥ 2.
In analogy with [16], our result has been recently used to obtain an integral representation result for functionals in G S B D p in higher dimension, see [21]. Moreover, the 'almost' Sobolev regularity of G S B D p -functions with a small jump set, Theorem 1.2, is one of the main ingredients of the extension result [6]; additionally, the Korn-Poincaré inequality (Theorem 4.5) and the approximation result (Theorem 5.1) have been used in [14] to prove compactness and lower-semicontinuity for nonhomogeneous Griffith-like energies.

Notation
We introduce now some notation that will be used throughout the paper.
(a) L n denotes the Lebesgue measure on R n and H n−1 the (n − 1)-dimensional Hausdorff measure on R n . (b) e 1 , . . . , e n is the canonical basis of R n ; | · | denotes the absolute value in R or the Euclidean norm in R n , depending on the context, and · denotes the Euclidean scalar product. We set S n−1 := {x ∈ R n : |x| = 1}. We denote with R n×n sym the set of symmetric n × n matrices. (c) For x ∈ R n and ρ > 0 we define the ball: (d) For x ∈ R n , e ∈ S n−1 , and ρ > 0, we define the cylinder: (e) For y ∈ R n and ξ ∈ S n−1 , we set: (f) For a, b ∈ R n , we denote with a ⊗ b ∈ R n×n the tensor product of a and b, namely the matrix with (a ⊗ b) i j = a i b j for every i, j = 1, . . . , n. Moreover, we denote the symmetrised tensor product as a b : the set of all points where E has density t, namely (i) An L n -measurable and bounded set E ⊂ R n is a set of finite perimeter if its characteristic function χ E is a function of bounded variation. The reduced boundary of E, denoted with ∂ * E is the set of points x ∈ supp |Dχ E | where a generalised normal ν E is defined.
(j) For ⊂ R n measurable, M b ( ; R m ) denotes the space of bounded Radon measures with values in R m , for m ≥ 1. Moreover, for m = 1, we denote with M + b ( ) the sub-class of positive measures. (k) For k ∈ N, γ k ∈ R denotes the k-dimensional Lebesgue measure of the unit ball in R k . With this notation, we have H n−1 (S n−1 ) = nγ n .
Let ⊂ R n be an open set. We now introduce the functional spaces we will work with in this paper. We first recall the definition of the space G B D of generalised functions with bounded deformation, which is due to Dal Maso [22] and relies on slicing. Given an L n -measurable function u : → R n , we say that u ∈ G B D( ) if there exists λ u ∈ M + b ( ) such that the following is true for every ξ ∈ S n−1 : We say that u ∈ G S B D( ) if in additionû ξ y (t) ∈ S BV loc ( ξ y ) for every ξ ∈ S n−1 and for H n−1 -a.e. y ∈ ξ , where ξ y := {t ∈ R : y + tξ ∈ } and, for t ∈ ξ y , u ξ y (t) := u(y + tξ) · ξ denotes the slice of u in the direction ξ . In [22] it is shown that, given a function u ∈ G S B D( ), one can define an 'approximate symmetrised gradient' e(u) ∈ L 1 ( ; R n×n sym ) as well as an (H n−1 , n − 1)-countably rectifiable jump set J u , which both coincide with the standard definitions [3] if u ∈ B D( ).

How to wipe out small jump sets
The following Lemma is a variant of [10, Theorem 3], which can be proved by adapting the arguments to the case of a ball. This result ensures that a G S B D p -function with a small jump set in the unit ball can be regularised away from the boundary, up to a small cost in Griffith's energy.
The last point follows from Remark 6 and Lemma A.1 in [8], which can be used when building the functionũ in the construction of [10, Theorem 3].
The following theorem is an extension in dimension n ≥ 2 of a planar result of Conti, Focardi and Iurlano [16]. Our proof is strongly inspired by theirs and involves an iterative regularisation procedure and a covering argument. Essentially, it shows that a G S B D p -function with a small jump set coincides, outside a small neighbourhood of the jump set, with a function that has Sobolev regularity away from the boundary. Moreover, the energy of the regularised function can be made arbitrarily close to the energy of the original function.

Remark 3.4
Note that also the volume of ω is controlled by the measure of the jump set J u of u. Indeed, the isoperimetric inequality ensures that (L n (ω)) (n−1)/n ≤ C H n−1 (J u ) (possibly changing the constant C). In addition, since ω ⊂ B ρ , we have (L n (ω)) 1/n ≤ γ 1/n n ρ. Multiplying these two inequalities we obtain that L n (ω) ≤ C ρH n−1 (J u ).
The function w in the thesis of the theorem will be obtained as the pointwise limit of a sequence (w k ) k≥0 , constructed iteratively starting from w 0 = u, and by progressively "wiping out" parts of the jump of u, at the expense of a controlled increase of the L p norm of the approximate symmetric gradient. We split the proof into several steps.
Step 1: Iterative construction of (w k ) k≥0 . We will now build a sequence of functions Step 1.1: Base case. Letδ =δ(n, p) be the constant given by Lemma 3.1. By possibly reducing its value, we assume in addition that γ n−1 >δ n (see notation (k) in Section 2). Let also α = α(n, p, ε) ∈ (0, 1) be a constant to be determined later (see (3.28)). We set w 0 := u, η 0 := (αδ) n , ρ 0 := ρ and Note that by assumption s 0 ≤ (τ/η 0 ) 1/(n−1) . In order for the iteration to converge, we will need s 0 to be sufficiently small, hence the τ in the statement. We also observe that, by the definition of s 0 , we have Step 1.2: Induction step. Let k ≥ 0, and suppose we are given w k ∈ G S B D p (B ρ ), s k ∈ (0, 1), ρ k ≤ ρ and η k ≤δ n which satisfy as it is the case for k = 0. We will build w k+1 , η k+1 , s k+1 and ρ k+1 (explicitly given at the end of the step) such that (3.3) is satisfied for k +1. We will divide the proof of the induction step into further substeps. Our strategy is the following. We construct a function w k+1 whose jump set is (in measure) not larger than the one of the function w k . To do so, we cover a large part of J w k in the smaller ball B (1−s k )ρ k (subsequently defined as B ρ k+1 ) with a family of disjoint balls, and we wipe out a significant part of the jump set of w k in each ball of the covering.
Step 1.2a: Construction of the covering. We claim that for As φ is lower semicontinuous one has φ(r x ) ≤ η k , and as it is left-continuous, one has φ(r x ) ≥ η k . This shows (3.4). By construction, observe also that By the Besicovitch Covering Theorem (see, for instance, [2, Theorem 2.17]) there exists a positive integer ξ(n), depending only on n, with the following property: In what follows we denote, to simplify, Step 1.2b: Definition of w k+1 . We define w k+1 in two different ways, depending on whether the amount of the jump set of w k in the annulus B ρ k \B (1−s k )ρ k is large or not. We first let In the case we let w k+1 := w k . If instead we have the reverse inequality in (3.8), and consequently we then define w k+1 as where w k,i ∈ G S B D p (B i ) denotes the function obtained by applying Lemma 3.1, after suitable translation and rescaling, to the restriction of w k in each ball B i for every i ≥ 1. Note that in this case the value of δ, by definition of the balls B i (namely by (3.4)), is given by η 1/n k , and η 1/n k ≤δ by the assumption of the induction step.
Step 1.2c: Proof of the induction step. (3.10) and, by using (3.8) and (3.3), We now assume that (3.9) holds. By Property (1) , 1) be the radius given by Lemma 3.1 and corresponding to w k,i . Setting Property (2) of Lemma 3.1 provides a control on the (possible) additional jump of w k+1 in each B i (note that this additional jump can only be in B i \B i by Property (1)): Here c depends only on n and p. We now estimate the jump of w k+1 in each B i . By property (1) of Lemma 3.1 and by (3.12) For the last term in (3.13) we have the bound where we have used properties (3.4) and (3.5) for the radii of the balls of the covering. Hence from (3.13) we have Possibly reducingδ, we may assume that (3.14) Note that (3.14) and (3.5) imply immediately that In addition, by (3.14) and (3.5) one has where the last inequality follows from (3.6). We deduce from (3.16) and (3.9) that Hence, using the value (3.7) of θ , we obtain that where in the last inequality we used (3.3). In conclusion, whether (3.8) be satisfied or not, one has that, by (3.10) and (3.15), 18) and that, by (3.11) and (3.17), We now define λ := (θ/(1 − s 0 ) n−1 ) 1/n ; recalling the definition (3.2) of s 0 , one can ensure that λ ≤ 2n √ θ < 1 by choosing τ small enough, namely, by requiring that which depends on n, p, α. Then, letting ρ k+1 : Step 2: Convergence of (w k ) k≥0 . We now start the construction of the exceptional set ω given in the statement. To this aim, for every k ≥ 0 we introduce the set ω k in the following way. If (3.8) is satisfied we let ω k := ∅, and if not, we let In both cases {w k = w k+1 } ⊂ ω k and we can estimate the perimeter of ω k , thanks to (3.3) and (3.4), as where nγ n = H n−1 (S n−1 ) (see notation (k) in Sect. 2). We now estimate the L p -norm of e(w k+1 ) in terms of the norm of e(w k ). Again, this bound is trivial if (3.8) is satisfied. If not, thanks to point (3) in Lemma 3.1, we have that in each B i of the construction for each i ≥ 1. As a consequence, by the definition of w k+1 , also Repeating the construction for all k ≥ 1 we obtain sequences Since (ρ k ) k is decreasing, there exists ρ := lim k→∞ ρ k . We claim that ρ is bounded away from zero. Indeed, using that (1 − ts 0 ) ≥ (1 − s 0 ) t for t ∈ (0, 1) and θ . Now we set, for any ≥ 0,ω := k≥ ω k . Then, thanks to (3.21) and to (3.24), where we have used the fact that ρ k ≤ ρ for k ≥ . Then, since ρ → ρ and s → 0 as → ∞, it follows that H n−1 (∂ * ω ) → 0 as → ∞. Hence by the isoperimetric inequality we also have that L n (ω ) → 0 as → ∞. Since, for k ≥ , w k = w outsideω , we conclude that, as k → ∞, w k converges L n -a.e. in B ρ to some function w. We also note that, for every k ≥ 0, by (3.23) and (3.24), Moreover, thanks to (3.18), for every k ≥ 0 we have that H n−1 (J w k ) ≤ H n−1 (J u ).
Then, thanks to [22,Theorem 11.3] (see also [12]) it follows that w ∈ G S B D p (B ρ ), Passing to the limit in (3.3) we have that H n−1 (J w ∩ B ρ ) = 0, from which it follows that w ∈ W 1, p (B ρ ; R n ), thanks to Korn's inequality. Note that by (3.24) ρ ≤ ρ, and ρ → ρ as H n−1 (J u ) → 0, thanks to (3.25) (and by the definition of s 0 ). Clearly, using (3.25) and choosing τ small enough we can ensure This holds, for instance, for which also satisfies (3.20). Here we used that Setting ω :=ω 0 , by construction we have that w = u in B ρ \ω. In addition, from Using the fact that λ ≤ 2n √ θ we finally obtain the estimates (3.28) Note that now τ = τ (n, p, ε, σ ). Correspondingly, we define as the constant in the statement of the theorem, and this concludes the proof.

Remark 3.5 From (3.22)
, it is easy to show that in fact one can refine (3.1) to In addition, one sees that C ∼ ε −n/s , where s is the exponent in Property (3) of Lemma 3.1.

Remark 3.6
It is easy to show (by modifying the proof or, in fact, using the theorem itself) a variant of Theorem 3.2 where B ρ is replaced with a cube (−ρ, ρ) n .
We can easily deduce that [16, Corollary 3.3] also holds in higher dimension. We repeat the statement here for the reader's convenience.

Corollary 3.7
Under the same assumptions and notation of Theorem 3.2, there exists an infinitesimal rigid motion a ∈ R such that

Regularity and rigidity in a general domain
The main result of this section is the following regularity result.   Note that property (i) follows from the fact that is a Lipschitz domain, while the existence of N satisfying property (ii) is shown in the proof of Theorem 4.1. Moreover, N satisfies the estimate For the proof of Theorem 4.1 we will use the following lemma.  ∈ (0, 1). There exist c > 0 depending only on D, α and p, such that for any w ∈ W 1, p (D; R n ) and any Lebesgue measurable where a E := arg min a∈R E |w − a| p dx. (4.1) Proof Such a lemma is standard and easily proved by contradiction. Suppose that for every k ∈ N there exist a function w k ∈ W 1, p (D; R n ) and a Lebesgue measurable set where a E k is as in (4.1). Setting we have that, by the definition of a E k , u k satisfies Moreover, by (4.2), so that by the classical Korn inequality for some constant C = C(n, p, D) > 0. Hence, there exists u ∈ W 1, p (D; R n ) such that, up to subsequences, u k u weakly in W 1, p (D; R n ) as k → +∞ (and strongly in L p (D; R n )). Note that, up to subsequences, the characteristic functions χ E k of E k converge weakly * in L ∞ (D) to some function φ ∈ L ∞ (D) with 0 ≤ φ ≤ 1 and such that Therefore, passing to the limit in (4.3) and (4.4), we have that e(u) = 0 in D and Since e(u) = 0 in D, by the classical Poincaré-Korn inequality (see, e.g., (1.6)) we deduce that there exists a ∈ R such that u = a in D. Choosing a = a in (4.6) we then have D |a| p φ dx = 0.
Since a ∈ R and taking into account (4.5) it follows that a = 0 and hence u = 0. This is however incompatible with u L p (D;R n ) = 1. (4.7) In the following, we will use the shorthand C i := C(x i , e(x i ), 4Lr, 2r ) for every i = 1, . . . , N . Note that B r (x i ) ⊂ B 2r (x i ) ⊂ C i and that dist(∂C i , B r (x i )) = r for every i = 1, . . . , N . Moreover, from (4.7) it follows that Step 2: We show that, at any given point of , the maximal number of overlapping sets in the covering {C 0 , C 1 , . . . , C N } only depends on L and n. To this aim, it will be enough to prove the statement for the sets C 1 , . . . , C N . Let z ∈ , and let Our goal is to show that the cardinality of A(z) is bounded by a number that only depends on n and L. Note that, if z is 'far' from ∂ , it can be A(z) = ∅, but in this case there is nothing to prove. Since the diameter of each cylinder is given by 4r 1 + 4L 2 , we have |z − x| < 4r 1 + 4L 2 for every x ∈ A(z).

Therefore,
Recalling that the family {B r /5 (x i )} i=1,...,N is composed of mutually disjoint balls, for every i ∈ {1, . . . , N } with i = j we have |x i − x j | > 2r /5. Then, the cardinality of A(z) is bounded by the maximum number of disjoint balls of radius r /5 which can intersect a ball of radius 4r 1 + 4L 2 , that we denote with κ. By scaling, one can check that κ does not depend on r , but only onL (i.e. on L) and on the dimension n.
Step 3: We show that it is enough to prove the theorem in the set C i ∩ , for every i ∈ {0, 1 . . . , N }. We introduce a partition of unity of subordinate to the open covering Our construction will be simpler in the case i = 0 and, when necessary, we will explicitly point this out in the proof. If i = 0, without loss of generality we can assume that C i = C(0, e n , 4Lr, 2r ), with e n being the n-th coordinate unit vector, and that ∩ C i = {x = (x , x n ) ∈ C i : x n < g(x )} for a givenL-Lipschitz function g defined on the ((n − 1)-dimensional) ball centred at 0 and of radius r in e n 0 , with g(0) = 0.
We now build v i and ω i for the set C i ∩ . Let δ > 0, and let C i denote the union of all the n-dimensional cubes q ∈ {z + (0, δ] n : z ∈ δZ n } with q ⊂ C i . Since dist(supp φ i , ∂C i ) > r /24, we can assume that δ is small enough so that supp φ i ⊂ C i . Note that the choice of δ/r depends only on n.
Then we build recursively the set Q of dyadic cubes of edge size δ2 −k , k ≥ 0, which refine towards the boundary ∂ , as follows. As a first step, we denote with Q 0 the set of cubes q ∈ {z + (0, δ] n : z ∈ δZ n }, q ⊂ C i ∩ , such that dist(q, ∂ ) > δ. Then, for k ≥ 1, having built Q for < k, we define Q k as the set of all the smaller cubes q ∈ {z + (0, δ2 −k ] n : z ∈ δ2 −k Z n }, q ⊂ C i ∩ , such that dist(q, ∂ ) > δ2 −k , and q does not intersect cubes of <k q∈Q q. Note that, if i = 0, we can assume that all the cubes in C 0 belong to the family Q 0 (by e.g. choosing δ < r /8).
Finally, we let Q := ∞ k=0 Q k ; note that q∈Q q = C i ∩ ⊂ C i ∩ covers entirely supp φ i ∩ . Now, for each q ∈ Q, let q and q denote cubes concentric with q, and with edge size 10% and 20% longer, respectively. Then the cubes q (as well as q ), for q ∈ Q, form a sort of Whitney covering of C i ∩ , at least covering supp φ i ∩ . Moreover, since for every k ≥ 0 any q ∈ Q k satisfies dist(q, ∂ ) > δ2 −k , clearly also q , q ⊂ . Note that, for fixed k ≥ 0, an enlarged cube q of some cube q ∈ Q k can only intersect cubes belonging to Q k , Q k+1 and, if k ≥ 1, Q k−1 .
Next, we choose the constantc =c( ) introduced at the start of the proof to bē where τ is given by Theorem 3.2 (or, more precisely, by the version of Theorem 3.2 for a cube, following Remark 3.6), corresponding to σ = 1/12 and ε = 1. Hence, by the initial assumption H n−1 (J u ) ≤c we have Then, by applying Theorem 3.2 (with ε = 1) to u ∈ G S B D p (q ), for each q ∈ Q 0 , we find a function w q ∈ G S B D p (q ) and a set of finite perimeter ω q ⊂ q such that For smaller cubes q ∈ Q k , k ≥ 1, we proceed as follows: if H n−1 (J u ∩ q ) ≤ τ (δ/2 k+1 ) n−1 , we say that q is "good", we apply Theorem 3.2 to the restriction of u to q , and find w q and ω q as in the case k = 0 (note that all the cubes in Q 0 are "good" and that, in particular, C 0 is made of "good" cubes). In conclusion, for q "good", we find a function w q ∈ G S B D p (q ) and a set of finite perimeter ω q ⊂ q such that w q = u in q \ω q and (4.12) where C = C(n, p). If instead H n−1 (J u ∩ q ) > τ(δ/2 k+1 ) n−1 , we say that q is "bad" and we definẽ namely we connect q with ∂ via a sort of truncated cone with an opening controlled by the Lipschitz constant L of . Scaling arguments (and the fact that where the constant c = c(n, L) depends only on L and the dimension. It follows that in this case, namely for q "bad", We letω := q∈Q bω q , G := ( q∈Q q)\ω, andω := q∈Q g (ω q ∩ q ), where we denoted with Q b , Q g ⊂ Q the "bad" and "good" cubes in Q, respectively. By construction, there exists a (2L)-Lipschitz function f such that G is the subgraph of f , with g − 2δ ≤ f ≤ g. Moreover, for some constant c (depending on L, n and τ ), and using that the cubes q have finite overlap, one has, by (4.12) and (4.13), (4.14) where we set ω i :=ω ∪ω.
We then let, for x ∈ ∪ q∈Q g q,ṽ i (x) := q∈Q g w q (x)ϕ q (x). First of all, we extend v i|G from G to C i ∩ . This can be done, for instance, by following the procedure in [31,Lemma 4], since G is a special Lipschitz set (according to [31, property (49)]) andṽ i|G ∈ W 1, p (G; R n ), as each w q belongs to W 1, p (q ; R n ) for q ∈ Q g , by (4.10). We denote this extension by v i . Then v i ∈ W 1, p (C i ∩ ; R n ), v i =ṽ i in G, and by [31, property (50)] we have that (4.15) where the constant c depends only on the dimension n, on p, and on the Lipschitz constant of G (namely of f ), which is 2L, hence c = c(n, p, L).
To conclude the proof of this step, it remains to show that (4.16) for some c = c(n, p, L). By (4.15), it is sufficient to show that ∪ q∈Qg q |e(ṽ i )| p dx ≤ c C i ∩ |e(u)| p dx. By the definition ofṽ i , one has We need therefore to estimate the L p norm of q∈Q g w q ∇ϕ q in terms of the L p norm of e(u), since the other term in the sum satisfies the bound by (4.11). Notice that as q ϕ q ≡ 1 in ∪ q∈Q g q, we have that q ∇ϕ q = 0 in ∪ q∈Q g q (where here and in what follows the sums run on cubes in Q g ). Then, if we fix q ∈ Q g and Note that the last equality in (4.18) follows since the only terms in the sum that have a non-zero contribution are the ones corresponding to cubesq such thatq intersects q , whose number is bounded by 2 n . Now we observe that, if q ∩q = ∅, then there are two cases: either q andq are of the same size, or, alternatively, the edge length of one is twice the edge length of the other one. In either case where β 1 = β 1 (n) > 0 is an explicit constant depending only on the dimension. Now, to fix the ideas, assume that q ∈ Q k andq ∈ Q k+1 ; then, by Remark 3.4 and (4.12) (where we recall that C = C(n, p)), and since q,q ∈ Q g , where c(n, p) denotes possibly different constants. Therefore, for every q,q ∈ Q g with q ∩q = ∅, Hence, up to possibly reducing τ , we have that for some β 2 > 0 depending on n and p.
We now apply Lemma 4.4 to w q in q and to wq inq , with E = (q ∩q )\(ω q ∪ωq ) and α = β 1 β 2 . Note that the constant c in the lemma scales with the size of the domain; more precisely, for a dyadic cube q with side length , c = c(n, α, p)( ) p , with c(n, α, p) being the constant for the unit cube in R n . Then On the other hand, since w q = wq = u in E, we have that a q = aq = a, and hence, thanks to (4.19)-(4.20) and (4.11), In conclusion, for a given q ∈ Q g , by (4.17), (4.18), (4.11) and the previous estimate, with c = c(n, p), where we have used the fact that ∇ϕq L ∞ (q ) ≤ c/ˆ . Using that the cubes q have finite overlap, we have and, by (4.15) we obtain (4.16). We have then proved the estimates where c i = c i (n, p, L) is the maximum of the two constants in (4.14) and (4.16).
Step 5: Step 3), and that the number of C i 's intersecting at every point of is at most κ + 1 (see Step 2), the statement holds true by setting Note that, since H n−1 (∂ * ) can be estimated in terms of the parameters N , r and L introduced in Step 1, we have that c (n, p, ) = c(n, p, r , N , L).
An immediate consequence of Theorem 4.1 is the Korn's inequality below, whose proof is a direct adaptation of [16,Corollary 3.3]. where q ≤ p * if p < n, q < ∞ if p = n, and q ≤ ∞ for p > n.

Proof
Let v ∈ W 1, p ( ; R n ) be given by We now define a(x) := Ax + b; then a ∈ R. Since ∇v = ∇u L n -a.e. on {v = u}, we have that where the last inequality follows by Theorem 4.1. This proves (4.21). Moreover, we can improve the norm on the left-hand side of (4.23) to the exponent q of the Sobolev embedding of W 1, p into L q . Then, since v = u in \ω, we have that which proves the estimate (4.22). Note that, if p < n, we can take q = p * , and that if p > n we can estimate v − a in the Hölder seminorm C 0,α , with α = 1 − n p .

An approximation result
In this last section, as an application, we show an approximation result in the spirit of where w ± and u ± denote the traces of w and u on the two sides of .

Corollary 5.2
Under the same assumptions and notation of Theorem 5.1, for u ∈ G S B D p ( ) the approximate gradient ∇u exists L n -a.e. in .
Proof of Corollary 5.2 Let k ∈ N, and letω k ,ω k and w k be as in Theorem 5.1, for we have in particular that ∇w k exists L n -a.e. in \ω k . Moreover, as u = w k in \(ω k ∪ω k ), it follows that ∇u exists L n -a.e. in \(ω k ∪ω k ) (note thatω k is a finite union of cubes, and hence its boundary is L n -negligible). By repeating this argument for every k ∈ N we have that ∇u exists L n -a.e. in \ω, where Since by (5.1) and Remark 3.4 we have that L n (ω k ∪ω k ) ≤ C( 1 k ) n/(n−1) for every k ∈ N, where C = C(n), it follows that L n (ω) = 0. Hence we can conclude that ∇u exists L n -a.e. in .
Note that in the case p = 2 the result in Corollary 5.2 has been obtained in [28], as a consequence of the embedding G S B D 2 ( ) ⊂ (G BV ( )) n (see [28,Theorem 2.9]), for n ≥ 2.
Theorem 5.1 will follow as a special case of the following technical proposition.
We recall that, for u ∈ G S B D(R n ), the set J u is countably (H n−1 , n −1) rectifiable [22,Section 6] (see [25,Section 3.2.14] for the definition), so that the assumption J u ⊂ J is not restrictive.
Theorem 5.1 is deduced from Proposition 5.3 in the following way. Let ⊂ R n and u ∈ G S B D p ( ) as in the assumptions of Theorem 5.1, and letũ denote the extension of u to R n obtained by settingũ := 0 outside . Thenũ ∈ G S B D p (R n ), and by applying Proposition 5.3 toũ and J = Jũ we obtain the claim.

Proof of Proposition 5.3
Let u, J and ε be as in the statement, and let ρ > 0 and α > 0 be constants to be determined later. We split the proof into several steps.
Step 1: Covering the jump set Since J is countably (H n−1 , n − 1) rectifiable and H n−1 (J ) < +∞, by [25,Theorem 3.2.29] there exists a countable family (M k ) k∈N of C 1 hypersurfaces such that With no loss of generality we can assume that for each k ∈ N the manifold M k is a Lipschitz graph with Lipschitz constant less than 1/4 [2, Theorem 2.76]. Then, for every k ∈ N, H n−1 -a.e. point in J ∩ M k is a point of H n−1 -density 1 both for J and J ∩ M k , namely for every k ∈ N and H n−1 -a.e. x ∈ J ∩ M k . From this, it follows that for every k ∈ N and for H n−1 -a.e. x ∈ J ∩ M k there exists η(α, x) ∈ (0, ρ) such that and for every r ≤ η(α, x). In other words, up to sufficiently restricting the radius of the ball, we can assume that the main content of J in a ball centred at a point x ∈ J ∩ M k comes from M k , and not from the other components M j , for j = k. Moreover, the subfamily above is countable, since it is composed of disjoint sets with nonempty interior. Hence, there exists a sequence {x i } i∈N ⊂ ∪ k M k such that where B i := B r i (x i ) for every i ∈ N, and where we set r i := r (α, x i ). Finally, note that from the identity above it follows that there exists N = N (α) ∈ N such that Properties a), b), c) follow immediately. We now prove property d). First, note that Hence, by (5.2) and by property c) which shows d). To see e) note that, since the closed balls are disjoint, where we have also used b). Finally, letting := N i=1 i , one has that is a finite union of disjoint C 1 manifolds with C 1 boundary. Moreover, thanks to c) and d), Step 2: Cleaning the jump set in the balls B i . We split this step into further substeps. Step where we have used that From this it follows thatN + i can be chosen to be depending only on n. In conclusion, (N + i ,r + i ,L + i ) can be chosen uniformly in i. Since, by Remark 4.3, the constant c + i depends on B + i only via (N + i ,r + i ,L + i ), we finally conclude that the constant c := max{c ± i : i = 1, . . . , N } can be bounded uniformly in N , and hence, as N = N (α), uniformly in α. In particular, in (5.4)-(5.6), we can replace c ± i with the uniform constant c.
Step 2.3: Conclusion. Thanks to b) and c) in Step 1, we have that and hence from (5.6) and Step 2.
. (Note that, in the case where ω ± i = B ± i , we can simply let v ± i = 0; however by choosing α > 0 small enough we can assume with no loss of generality that this does not happen.) It follows that on ∂ B i the trace of each v ± i coincides with the trace of u, except on a set of total measure at most 2c α(1 + α)γ n−1 r n−1 Then v ∈ G S B D p (R n ), and we have the following properties: Property 1) follows from the definition of v. For property 2), note that by c) and (5.6) Let us show property 3). By (5.2), 1) and 2) we have that , one has that v = u L n -a.e. in R n \ω B and, by (5.6), H n−1 (∂ * ω B ) ≤ 2c αH n−1 (J ).
Step 3: Cleaning the jump set in the rest of the domain. We now pick δ > 0 with and consider the covering of R n \ N i=1 B i made of: • the family Q 1 of cubes δz + [0, δ] n , z ∈ Z n , which intersect R n \ N i=1 B i ; • the family Q 2 of cubes δz + [0, δ] n , z ∈ Z n , which are not in Q 1 , but intersect some cubes in Q 1 .
We set Q = Q 1 ∪ Q 2 . For each q ∈ Q, we denote with q ⊂ q ⊂ q the concentric cubes q and q with edges (9/8)δ and (10/8)δ, respectively; we also denote with < < the lengths of the edges of q, q and q , respectively, so that in particular, we observe that, since i are equi-Lipschitz with constant less than 1 2 , one has for some constant c = c(n), where in the last inequality we used property b). Hence, since by the definition of δ we have that q ∩ B i = ∅ for each q ∈ Q and for every i, we have that Then, recalling 1), and using (5.2), (5.7) and 2), we have for a constant c depending only on the dimension. We now invoke Theorem 3.2 (in its version for cubes, as noted in Remark 3.6) for parameters ε = 1 (which thus needs not be the ε of the statement), and σ = 0.1, and find constants C = C(n, p) and τ = τ (n, p) satisfying the thesis of the theorem.
Let Q g ⊂ Q denote the set of cubes q such that H n−1 (J v ∩ q ) ≤ τ δ n−1 , let Q b := Q\Q g , andω := q∈Q b q. Since for q ∈ Q b one has H n−1 (J v ∩q ) > τδ n−1 , there can be only a finite number of such cubes. Moreover, thanks to (5.8) we have that τ δ α(1 + (1 + 3c)H n−1 (J )).
Now, let q ∈ Q g . By Theorem 3.2 there exist w q ∈ G S B D p (q ) ∩ W 1, p (q ; R n ) and ω q ⊂ q , with w q = v in q \ω q , and where (5.10) follows by Remark 3.5.
Possibly reducing τ , we may assume that if q ∩ i = ∅ for some i = 1, . . . , N , then H n−1 ( i ∩ q ) ≥ τ δ n−1 (see point a) in Step 1), so that q / ∈ Q g . It then follows that for any q ∈ Q g , when q ⊂ B i for some i (or more precisely q ⊂ B ± i , since q ∈ Q g is such that q does not intersect i ), then w q = v in q .
We now 'glue' the functions w q in order to find a global W 1, p loc function as in the claim of the theorem. To do so, we introduce a cut-off function ψ ∈ C ∞ c ((−9/16, 9/16) n ; [0, 1]) with η = 1 on [−1/2, 1/2] n . Then for each q ∈ Q g , with center c q , we define ψ q (x) , so that ψ q = 1 on q. We then let, for x ∈ G := q∈Q g q, ϕ q (x) := ψ q (x)/( q∈Q g ψq (x)) ∈ [0, 1], and if x ∈ R n \(G ∪ω). (5.12) By construction we have that w ∈ W 1, p (B R (0)\( ∪ω); R n ) for any R > 0. Indeed, we observe that R n \(G ∪ω) ⊂ i B i , and hence (by the definition of v), in this set the function w is Sobolev outside . Moreover, w does not jump on the intersection between the boundaries of G and R n \(G ∪ω). Indeed, if q ∈ Q g is any cube touching the set R n \(G ∪ω), then it has to be that q ∈ Q 2 and q ⊂ B i for some i (and therefore, as observed before, w q = v in q ).
Step 3.1: Traces of w on . We now compare the traces of w and of u on the two sides of . We have already observed that w = u in R n \(ω ∪ω), whereω = ω B ∪ ω G .
Step 3.3: L p -estimate of e(w). We start by estimating ω G |e(w)| p dx. From (5.12) we have, for x ∈ G, e(w)(x) = q∈Q g e(w q )(x)ϕ q (x) + w q (x) ∇ϕ q (x) . (5.19) Note that, since the cubes q have finite overlap, the sum in the right-hand side of (5.19) is done, at each point, over a uniformly bounded number of terms, depending on the dimension.
We estimate the L p norm of the two terms of the sum in (5.19) where in the last step we used (5.10). Since the cubes q have finite overlap, from (5.20) we conclude that Therefore, (5.21) where in the last inequality we have used the definition of v, and in particular the fact that v = u outside ∪ i B i , and the estimate of e(v ± i ) in terms of e(u) (see Step 2). We now estimate the second term of the sum in (5.19). For x ∈ G we define Q x g := {q ∈ Q g : ϕ q (x) > 0}, and denote N x Q := #Q x g (which, as already observed, is uniformly bounded by a quantity depending only on the dimension, namely 2 n ).
Using that q∈Q x g ∇ϕ q (x) = 0, one has Since q,q ∈ Q x g ⇒ x ∈ q ∩q , to bound the L p norm of the above expression, it is enough to estimate q ∩q |w q − wq | p |∇ϕ q | p dx (5.22) for any pair of neighbouring cubes q,q ∈ Q x g . Note that w q − wq ∈ W 1, p (q ∩q ; R n ) and w q −wq = 0 in (q ∩q )\(ω q ∪ωq ), since both functions coincide with v. Moreover, since L n (q ∩q ) ≥ δ n /8 n and L n (ω q ∪ ωq ) ≤ Cτ n/(n−1) δ n for some dimensional constant C, provided τ is chosen small enough one can ensure that L n {x ∈ q ∩q : w q − wq = 0} ≥ 1 2 L n (q ∩q ).
One can then easily deduce from Lemma 4.4 that, for some constant c (depending on p and on the dimension): q ∩q |w q −wq | p dx ≤ cδ p q ∩q |e(w q −wq )| p dx ≤ cδ p ω q ∪ωq |e(w q − wq )| p dx.
Since |∇ϕ q | ≤ C/δ in each cube, we can estimate ( since w = u in R n \(ω ∪ω). Using that w = v outsideω ∪ ω G we have