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

Abstract

In this paper we prove a regularity and rigidity result for displacements in $GSBD^p$, for every $p>1$ and any dimension $n\ge 2$. We show that a displacement in $GSBD^p$ with a small jump set coincides with a $W^{1,p}$ function, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincaré-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in $GSBD^p$.

1 Introduction

The modelling and analysis of fracture in the linearised elasticity framework relies on a good understanding of the space BD 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 BD, and in the subspace SBD of special functions of bounded deformation (corresponding to the case where Eu has no Cantor part) have been better understood, and the relation between BD, SBD and the space BV of functions of bounded variation has been studied in detail (see e.g., [3, 5, 17], and [22] for the space of generalised functions of bounded deformation). For a function $u\in SBD(\Omega )$, Eu admits the decomposition

(1.1)

where e(u) is the absolutely continuous part of Eu with respect to the Lebesgue measure ${\mathcal {L}}^n$, $J_u$ the jump set of u, [u] the jump of u, $\nu _u$ the normal to $J_u$ and $[u]\odot \nu _u$ denotes the symmetric tensor product of u and $\nu _u$. 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

$$\begin{aligned} \int _{\Omega }|e(u)|^2 dx + {\mathcal {H}}^{n-1}(J_u), \end{aligned}$$

(1.2)

called the Griffith’s energy, of which the Mumford–Shah energy in SBV is the scalar counterpart. The energy (1.2) is in fact well defined in the larger space $GSBD(\Omega )$ 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, GSBD 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 $GSBD^p(\Omega )$, the space of functions $u\in GSBD(\Omega )$ for which $e(u)\in L^p(\Omega )$ and ${\mathcal {H}}^{n-1}(J_u)<+\infty$, for every dimension $n\ge 2$ and any $p>1$. More precisely, we have the following (see Theorem 4.5).

Theorem 1.1

Let $n \in {{\mathbb {N}}}$ with $n \ge 2$, $p \in (1, \infty )$, and let $\Omega \subset {{\mathbb {R}}}^n$ be a bounded, open and connected Lipschitz set. Then, there exists $c=c(n,p,\Omega )>0$ with the following property. For any $u\in GSBD^p(\Omega )$, there exist a set of finite perimeter $\omega \subset \Omega$ with ${\mathcal {H}}^{n-1}(\partial ^*\omega )\le c{\mathcal {H}}^{n-1}(J_u)$, and an infinitesimal rigid motion a (namely an affine function a, with $e(a)=0$), such that

$$\begin{aligned} \int _{\Omega {\setminus } \omega }|\nabla u - \nabla a|^p dx&\le c(n,p,\Omega ) \int _{\Omega }|e(u)|^p dx. \end{aligned}$$

(1.3)

Moreover, there exists $c=c(n,p,q,\Omega )>0$ such that

$$\begin{aligned} \Vert u - a\Vert _{L^q(\Omega {\setminus } \omega )} \le c(n,p,q,\Omega ) \Vert e(u)\Vert _{L^p(\Omega )}, \end{aligned}$$

(1.4)

with $q\le p^*$ if $p<n$, $q<\infty$ if $p=n$, and $q\le \infty$ for $p>n$.

Clearly, the volume of $\omega$ is also controlled by ${\mathcal {H}}^{n-1}(J_u)$, thanks to the isoperimetric inequality, see Remark 3.4 below.

This result is the generalisation, in dimension $n\ge 2$, 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 $\Omega$. This is in contrast with the classical Korn and Poincaré-Korn inequalities for functions $u\in W^{1,p}(\Omega ;{{\mathbb {R}}}^n)$, with $p>1$, which state that there exists an infinitesimal rigid motion a such that

$$\begin{aligned} \Vert Du-Da\Vert _{L^p(\Omega )} \le c(n,p,\Omega ) \Vert Eu\Vert _{L^p(\Omega )}, \end{aligned}$$

(1.5)

and that, thanks to the Poincaré inequality and Sobolev embeddings,

$$\begin{aligned} \Vert u-a\Vert _{L^q(\Omega )} \le c(n,p,q,\Omega ) \Vert Eu\Vert _{L^p(\Omega )}, \end{aligned}$$

(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)SBD, even for functions u with a small jump set. This is due to the possible presence of small regions of $\Omega$ 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 $u\in (G)SBD(\Omega )$, e(u) cannot control $u-a$ or its approximate gradient in the whole domain $\Omega$, 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.

Theorem 1.2

Let $n \in {{\mathbb {N}}}$ with $n \ge 2$, $p \in (1, \infty )$, and let $\Omega \subset {{\mathbb {R}}}^n$ be a bounded and open Lipschitz set. Then, there exists $c=c(n,p,\Omega )>0$ with the following property. For any $u\in GSBD^p(\Omega )$, there exists a set of finite perimeter $\omega \subset \Omega$ with ${\mathcal {H}}^{n-1}(\partial ^*\omega )\le c{\mathcal {H}}^{n-1}(J_u)$ and $v\in W^{1,p}(\Omega ;{{\mathbb {R}}}^n)$, such that $u=v$ in $\Omega {\setminus }\omega$ and $\int _\Omega |e (v)|^p dx\le c\int _\Omega |e(u)|^p dx$. If in addition u is bounded, then $\Vert v\Vert _{L^\infty (\Omega )} \le \Vert u\Vert _{L^\infty (\Omega )}$.

In Theorem 1.2, we prove ‘almost’ Sobolev regularity for functions in $GSBD^p$. More precisely we show that, given a function $u\in GSBD^p(\Omega )$, we can replace it with a function $v\in W^{1,p}(\Omega ;{{\mathbb {R}}}^n)$ outside an exceptional set $\omega \subset \Omega$, whose perimeter is controlled by ${\mathcal {H}}^{n-1}(J_u)$. Moreover, u and v have a comparable Griffith’s energy in the whole of $\Omega$. 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 $\omega =\Omega$ 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 $\Omega$ 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 $\Omega$ are ‘good’, up to a small neighbourhood of $\partial \Omega$. In this neighbourhood, the ‘bad’ cubes are cut away from the domain by connecting them to $\partial \Omega$ by means of truncated cones. In this way what remains is still a Lipschitz set (with the same Lipschitz constant as $\Omega$). 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 function ${{\tilde{u}}}\in GSBD^p(q)$, we can wipe out its jump set $J_{{{\tilde{u}}}}$ away from the boundary of q, up to a small expense in terms of the Griffith’s energy, provided ${\mathcal {H}}^{n-1}(J_{{{\tilde{u}}}})$ 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 to ${{\tilde{u}}}:=u_{|q}$ in every ‘good’ cube q, we obtain a Sobolev regularisation ${{\tilde{v}}}_q$ of $u_{|q}$ and an exceptional set ${\tilde{\omega }}_q$ with controlled perimeter, such that ${{\tilde{v}}}_q = u$ outside ${\tilde{\omega }}_q$. The function v in the statement of Theorem  1.2 is then obtained by patching together the functions ${{\tilde{v}}}_q$ on all the good cubes. The set $\omega$ where v needs not coincide with u, is then defined as the union of the exceptional sets ${\tilde{\omega }}_q$ 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 $w_0={{\tilde{u}}}$, 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 limit ${{\tilde{v}}}$ of the sequence $(w_k)$ has Sobolev regularity in a smaller ball, and satisfies ${{\tilde{u}}}={{\tilde{v}}}$ outside an exceptional set ${\tilde{\omega }}$, which is defined as the union of the coverings of each step.

1.1 Comparison with previous results

The foundations of the function spaces SBD and GSBD 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 $(G)SBD^p$, 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 $(G)SBD^p$ [16].

Our results are in between two of these avenues: we prove Sobolev regularity for functions in $GSBD^p$, for every $p>1$ and in every dimension $n\ge 2$, 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 $n\ge 2$ and every $p\ge 1$ 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 $\omega$ 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 $\omega$ 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 $GSBD^2$, up to an exceptional set with both perimeter and area bounded by (powers of) the measure of $J_u$, for $n\ge 2$. 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 $L^\infty$ 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)SBD 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 $\nabla u$ from a skew-symmetric matrix is estimated in a lower $L^q$-norm, with $q\in [1,2)$. 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 $GSBD^p$ to $W^{1,p}$, up to an exceptional set, which is now valid for every $n\ge 2$ 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 $n\ge 2$ and $p>1$, since the regularisation step is not reliant on a planar construction.

1.2 Conclusion and perspectives

The main result in this work, asserting the ‘almost’ Sobolev regularity of $GSBD^p$-functions 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 $\nabla u$ for functions in $GSBD^p$ (Corollary 5.2). Note that the existence of $\nabla u$ for functions in $GSBD^2$ had already been obtained in [28], as a consequence of the embedding $GSBD^2(\Omega ) \subset (GBV(\Omega ))^n$ (see [28, Theorem 2.9]), for $n\ge 2$.

In analogy with [16], our result has been recently used to obtain an integral representation result for functionals in $GSBD^p$ in higher dimension, see [21].

Moreover, the ‘almost’ Sobolev regularity of $GSBD^p$-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.

2 Notation

We introduce now some notation that will be used throughout the paper.

• (a) ${\mathcal {L}}^n$ denotes the Lebesgue measure on ${{\mathbb {R}}}^n$ and ${\mathcal {H}}^{n-1}$ the $(n-1)$-dimensional Hausdorff measure on ${{\mathbb {R}}}^n$.

• (b) $e_1,\dots ,e_n$ is the canonical basis of ${{\mathbb {R}}}^n$; $|\cdot |$ denotes the absolute value in ${{\mathbb {R}}}$ or the Euclidean norm in ${{\mathbb {R}}}^n$, depending on the context, and $\cdot$ denotes the Euclidean scalar product. We set ${{\mathbb {S}}}^{n-1}:=\{x\in \mathbb {R}^{n}: |x|=1\}$. We denote with ${{\mathbb {R}}}^{n\times n}_{\text {sym}}$ the set of symmetric $n\times n$ matrices.

• (c) For $x\in {\mathbb {R}}^n$ and $\rho >0$ we define the ball:

  $$\begin{aligned} B_\rho (x):=\{y\in {{\mathbb {R}}}^n: \,|y-x|<\rho \}. \end{aligned}$$

• (d) For $x\in {{\mathbb {R}}}^n$, $e \in {\mathbb {S}}^{n-1}$, and $\rho >0$, we define the cylinder:

  $$\begin{aligned}&C(x,e , h , \rho ):=\{ y\in {{\mathbb {R}}}^n:|(y-x)\cdot e |< h, |(y-x)-((y-x)\cdot e )e | < \rho \}. \end{aligned}$$

• (e) For $y\in {\mathbb {R}}^n$ and $\xi \in {{\mathbb {S}}}^{n-1}$, we set:

  $$\begin{aligned} \Pi ^{\xi }_{y}:= \{x\in {{\mathbb {R}}}^n: (x-y)\cdot \xi = 0\}, \end{aligned}$$

  and use the shorthand $\Pi ^{\xi } = \Pi ^{\xi }_{0}$.

• (f) For $a, b\in {{\mathbb {R}}}^{n}$, we denote with $a\otimes b \in {{\mathbb {R}}}^{n\times n}$ the tensor product of a and b, namely the matrix with $(a\otimes b)_{ij} = a_ib_j$ for every $i,j=1,\dots ,n$. Moreover, we denote the symmetrised tensor product as $a\odot b:= (a\otimes b+b\otimes a)/2 \in {{\mathbb {R}}}^{n\times n}_{\text {sym}}$.

• (g) ${\mathcal {R}}$ is the set of infinitesimal rigid motions in ${{\mathbb {R}}}^n$, namely $a\in {\mathcal {R}}$ if and only if $a:x\mapsto Ax+b$, with $A\in {{\mathbb {R}}}^{n\times n}$ skew-symmetric, and $b\in {{\mathbb {R}}}^n$.

• (h) For every $t\in [0,1]$ and every ${\mathcal {L}}^n$-measurable set $E\subset {{\mathbb {R}}}^n$ we denote with $E^{(t)}$ the set of all points where E has density t, namely

  $$\begin{aligned} E^{(t)}:= \left\{ x\in {{\mathbb {R}}}^n: \lim _{\rho \downarrow 0} \frac{{\mathcal {L}}^n(E\cap B_\rho (x))}{{\mathcal {L}}^n(B_\rho (x))}=t\right\} . \end{aligned}$$

• (i) An ${\mathcal {L}}^n$-measurable and bounded set $E\subset {{\mathbb {R}}}^n$ is a set of finite perimeter if its characteristic function $\chi _E$ is a function of bounded variation. The reduced boundary of E, denoted with $\partial ^*E$ is the set of points $x\in {{\,\mathrm{supp}\,}}|D\chi _E|$ where a generalised normal $\nu _E$ is defined.

• (j) For $\Omega \subset {{\mathbb {R}}}^n$ measurable, ${\mathcal {M}}_b(\Omega ;{{\mathbb {R}}}^{m})$ denotes the space of bounded Radon measures with values in ${{\mathbb {R}}}^m$, for $m\ge 1$. Moreover, for $m=1$, we denote with ${\mathcal {M}}^+_b(\Omega )$ the sub-class of positive measures.

• (k) For $k\in {\mathbb {N}}$, $\gamma _k\in {{\mathbb {R}}}$ denotes the k-dimensional Lebesgue measure of the unit ball in ${{\mathbb {R}}}^k$. With this notation, we have ${\mathcal {H}}^{n-1}({\mathbb {S}}^{n-1})=n\gamma _n$.

Let $\Omega \subset {{\mathbb {R}}}^n$ 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 GBD of generalised functions with bounded deformation, which is due to Dal Maso [22] and relies on slicing. Given an ${\mathcal {L}}^n$-measurable function $u:\Omega \rightarrow {{\mathbb {R}}}^n$, we say that $u\in GBD(\Omega )$ if there exists $\lambda _u \in {\mathcal {M}}_b^+(\Omega )$ such that the following is true for every $\xi \in {{\mathbb {S}}}^{n-1}$:

• For every $\tau \in C^1({{\mathbb {R}}})$ with $-\frac{1}{2}\le \tau \le \frac{1}{2}$ and $0\le \tau '\le 1$

  $$\begin{aligned} D_\xi \big (\tau (u\cdot \xi )\big ) = D\big (\tau (u\cdot \xi )\big )\cdot \xi \in {\mathcal {M}}_b(\Omega ); \end{aligned}$$

• For every Borel set $B\subset \Omega$

  $$\begin{aligned} \big |D_\xi \big (\tau (u\cdot \xi )\big )\big |(B) \le \lambda _u(B). \end{aligned}$$

We say that $u\in GSBD(\Omega )$ if in addition ${\hat{u}}_y^\xi (t) \in SBV_{\text {loc}}(\Omega _y^\xi )$ for every $\xi \in {{\mathbb {S}}}^{n-1}$ and for ${\mathcal {H}}^{n-1}$-a.e. $y\in \Pi ^\xi$, where $\Omega _y^\xi := \{t\in {{\mathbb {R}}}: y+t\xi \in \Omega \}$ and, for $t\in \Omega _y^\xi$, ${\hat{u}}_y^\xi (t):=u(y+t\xi )\cdot \xi$ denotes the slice of u in the direction $\xi$. In [22] it is shown that, given a function $u\in GSBD(\Omega )$, one can define an ‘approximate symmetrised gradient’ $e(u)\in L^1(\Omega ;{{\mathbb {R}}}^{n\times n}_{sym })$ as well as an $({\mathcal {H}}^{n-1},n-1)$-countably rectifiable jump set $J_u$, which both coincide with the standard definitions [3] if $u\in BD(\Omega )$. Finally, we recall the definition of the space $GSBD^p$, namely

$$\begin{aligned} GSBD^p(\Omega ):=\{u\in GSBD(\Omega ): e(u)\in L^p(\Omega ;{{\mathbb {R}}}^{n\times n}_{sym }), {\mathcal {H}}^{n-1}(J_u)<+\infty \}. \end{aligned}$$

3 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 $GSBD^p$-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.

Lemma 3.1

Let $n \in {{\mathbb {N}}}$ with $n \ge 2$, and let $p \in (1, \infty )$. There exist ${\bar{\delta }},c,s$ positive constants, depending only on n and p, with the following property. For every $u\in GSBD^p(B_1)$ with $\delta :={\mathcal {H}}^{n-1}(J_u)^{1/n} \le {\bar{\delta }}$, there exists ${{\tilde{u}}}\in GSBD^p(B_1)$ and $R\in (1-\sqrt{\delta },1)$ such that

• (1) ${{\tilde{u}}}\in C^\infty (B_{1-\sqrt{\delta }})$, ${{\tilde{u}}}=u$ in $B_1{\setminus } B_R$, and ${\mathcal {H}}^{n-1}(J_u\cap \partial B_R)= {\mathcal {H}}^{n-1}(J_{{{\tilde{u}}}}\cap \partial B_R)=0$;

• (2) ${\mathcal {H}}^{n-1}(J_{{{\tilde{u}}}}{\setminus } J_u)\le c\sqrt{\delta }{\mathcal {H}}^{n-1}(J_u\cap (B_1{\setminus } B_{1-\sqrt{\delta }}))$;

• (3) it holds

  $$\begin{aligned} \int _{B_1}|e({{\tilde{u}}})|^p dx \le (1+c\delta ^s)\int _{B_1}|e(u)|^p dx; \end{aligned}$$

• (4) if in addition u is bounded, then one can ensure $\Vert {{\tilde{u}}}\Vert _{L^\infty (B_1)}\le \Vert u\Vert _{L^\infty (B_1)}$.

The last point follows from Remark 6 and Lemma A.1 in [8], which can be used when building the function ${\tilde{u}}$ in the construction of [10, Theorem 3].

The following theorem is an extension in dimension $n\ge 2$ 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 $GSBD^p$-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.

Theorem 3.2

Let $n \in {{\mathbb {N}}}$ with $n \ge 2$, and let $p \in (1, \infty )$. Given $\varepsilon >0$ and $\sigma \in (0,1)$, there exist $C=C(n,p,\varepsilon )>0$ and $\tau =\tau (n,p,\varepsilon ,\sigma )>0$ with the following property. For every $\rho >0$ and $u\in GSBD^p(B_\rho )$ with ${\mathcal {H}}^{n-1}(J_u)\le \tau \rho ^{n-1}$, there exists $w\in GSBD^p(B_\rho )$ and a set of finite perimeter $\omega \subset B_\rho$, such that $w=u$ in $B_\rho {\setminus } \omega$, ${\mathcal {H}}^{n-1}(\partial ^*\omega )\le C \, {\mathcal {H}}^{n-1}(J_u)$, $w\in W^{1,p}(B_{(1-\sigma )\rho };{{\mathbb {R}}}^n)$, and

$$\begin{aligned} \int _{B_\rho }|e(w)|^p dx \le (1+\varepsilon ) \int _{B_\rho }|e(u)|^p dx, \quad {\mathcal {H}}^{n-1}(J_w)\le {\mathcal {H}}^{n-1}(J_u). \end{aligned}$$

(3.1)

Moreover if u is bounded, one can ensure $\Vert w\Vert _{L^\infty (B_\rho )}\le \Vert u\Vert _{L^\infty (B_\rho )}$.

Remark 3.3

A careful inspection of the proof shows that

$$\begin{aligned} \lim _{\sigma \rightarrow 0^+}\tau (n,p,\varepsilon ,\sigma ) = \lim _{\varepsilon \rightarrow 0^+}\tau (n,p,\varepsilon ,\sigma ) = 0, \end{aligned}$$

and

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^+}C(n,p,\varepsilon ) = +\infty . \end{aligned}$$

Remark 3.4

Note that also the volume of $\omega$ is controlled by the measure of the jump set $J_u$ of u. Indeed, the isoperimetric inequality ensures that $({\mathcal {L}}^n(\omega ))^{(n-1)/n} \le C \, {\mathcal {H}}^{n-1}(J_u)$ (possibly changing the constant C). In addition, since $\omega \subset B_\rho$, we have $( {\mathcal {L}}^n(\omega ))^{1/n} \le \gamma _n^{1/n} \rho$. Multiplying these two inequalities we obtain that ${\mathcal {L}}^n(\omega )\le C \, \rho {\mathcal {H}}^{n-1}(J_u)$.

Proof

Choose $\varepsilon >0$ and $\sigma \in (0,1)$, and let $\rho >0$ and $u\in GSBD^p(B_\rho )$. We start by assuming that

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_u)\le \tau \rho ^{n-1}, \end{aligned}$$

for a $\tau > 0$ to be determined later (see (3.27)).

The function w in the thesis of the theorem will be obtained as the pointwise limit of a sequence $(w_k)_{k\ge 0}$, 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 $(w_k)_{k\ge 0}$. We will now build a sequence of functions $(w_k)_{k\ge 0} \subset GSBD^p(B_\rho )$ by induction.

Step 1.1: Base case. Let ${\bar{\delta }}={\bar{\delta }}(n,p)$ be the constant given by Lemma 3.1. By possibly reducing its value, we assume in addition that $\gamma _{n-1} > {\bar{\delta }}^n$ (see notation (k) in Section 2). Let also $\alpha =\alpha (n,p,\varepsilon )\in (0,1)$ be a constant to be determined later (see (3.28)). We set $w_0:=u$, $\eta _0:=(\alpha {\bar{\delta }})^n$, $\rho _0:=\rho$ and

$$\begin{aligned} s_0 := \frac{1}{\rho } \bigg (\frac{{\mathcal {H}}^{n-1}(J_u)}{\eta _0}\bigg )^{\frac{1}{n-1}}. \end{aligned}$$

(3.2)

Note that by assumption $s_0\le (\tau / \eta _0)^{1/(n-1)}$. In order for the iteration to converge, we will need $s_0$ to be sufficiently small, hence the $\tau$ in the statement. We also observe that, by the definition of $s_0$, we have

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_0}\cap B_{\rho _0}) = {\mathcal {H}}^{n-1}(J_{u}\cap B_{\rho }) = \eta _0 (\rho _0 s_0)^{n-1}. \end{aligned}$$

Step 1.2: Induction step. Let $k\ge 0$, and suppose we are given $w_k\in GSBD^p(B_\rho )$, $s_k\in (0,1)$, $\rho _k\le \rho$ and $\eta _k\le {\bar{\delta }}^n$ which satisfy

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_k}\cap B_{\rho _k})\le \eta _k (s_k \rho _k )^{n-1}, \end{aligned}$$

(3.3)

as it is the case for $k=0$. We will build $w_{k+1}$, $\eta _{k+1}$, $s_{k+1}$ and $\rho _{k+1}$ (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 $B_{(1-s_k)\rho _k}$ (subsequently defined as $B_{\rho _{k+1}}$) 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 ${\mathcal {H}}^{n-1}$-a.e. $x\in J_{w_k}\cap B_{(1-s_k)\rho _k}$ there exists $r_x\in (0,s_k\rho _k]$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {H}}^{n-1}(J_{w_k}\cap B_{r_x}(x)) = \eta _kr_x^{n-1}\\ {\mathcal {H}}^{n-1}(J_{w_k}\cap B_{r}(x))\ge \eta _kr^{n-1} \quad \text {for } r\le r_x. \end{array}\right. } \end{aligned}$$

(3.4)

Indeed, if x is a point of rectifiability of $J_{w_k}\cap B_{(1-s_k)\rho _k}$ and we define

$$\begin{aligned} \phi (r):=\frac{{\mathcal {H}}^{n-1}(J_{w_k}\cap B_{r}(x))}{r^{n-1}}, \quad r\in (0, s_k\rho _k], \end{aligned}$$

then we have $\lim _{r\rightarrow 0^+}\phi (r) = \gamma _{n-1} > \eta _k$ (since $\gamma _{n-1}>{\bar{\delta }}^n)$. Moreover, since $B_{s_k\rho _k}(x) \subset B_{\rho _k}$, from (3.3) it follows that $\phi (s_k\rho _k)\le \eta _k$. Therefore, we have that $r_x:= \inf \{r \in (0, s_k\rho _k) : \phi (r)\le \eta _k \} > 0$. As $\phi$ is lower semicontinuous one has $\phi (r_x)\le \eta _k$, and as it is left-continuous, one has $\phi (r_x)\ge \eta _k$. This shows (3.4). By construction, observe also that

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_k}\cap \partial B_{r_x}(x)) =0. \end{aligned}$$

(3.5)

By the Besicovitch Covering Theorem (see, for instance, [2, Theorem 2.17]) there exists a positive integer $\xi (n)$, depending only on n, with the following property: There exist $\xi (n)$ countable families of such closed balls $({\overline{B}}_{r_{x_i^\ell }}(x_i^\ell ))$, $i\ge 1$ and $\ell =1,\dots ,\xi (n)$, with ${\overline{B}}_{r_{x_i^\ell }}(x_i^\ell )\cap {\overline{B}}_{r_{x_j^\ell }}(x_j^\ell )=\emptyset$ for $i\ne j$, and such that

$$\begin{aligned} {\mathcal {H}}^{n-1}\Bigg ((J_{w_k}\cap B_{(1-s_k)\rho _k}){\setminus } \Bigg (\bigcup _{\ell =1}^{\xi (n)}\bigcup _{i\ge 1}{\overline{B}}_{r_{x_i^\ell }}\left( x_i^\ell \right) \Bigg )\Bigg )=0. \end{aligned}$$

Let us choose ${\bar{\ell }} \in \{ 1, \ldots , \xi (n) \}$ such that

$$\begin{aligned} {\mathcal {H}}^{n-1}\Bigg ((J_{w_k}\cap B_{(1-s_k)\rho _k})\cap \Bigg (\bigcup _{i\ge 1}{\overline{B}}_{r_{x_i^{{\bar{\ell }}}}}\big (x_i^{{\bar{\ell }}}\big )\Bigg )\Bigg ) \end{aligned}$$

is maximal. Then one has

$$\begin{aligned} \sum _{i\ge 1} {\mathcal {H}}^{n-1}\left( J_{w_k} \cap {\overline{B}}_{r_{x_i^{{\bar{\ell }}}}}\big (x_i^{{\bar{\ell }}}\big ) \right)&\ge {\mathcal {H}}^{n-1}\Bigg ((J_{w_k}\cap B_{(1-s_k)\rho _k})\cap \Bigg (\bigcup _{i\ge 1}{\overline{B}}_{r_{x_i^{{\bar{\ell }}}}}\big (x_i^{{\bar{\ell }}}\big )\Bigg )\Bigg ) \nonumber \\&\ge \frac{1}{\xi (n)} {\mathcal {H}}^{n-1}(J_{w_k}\cap B_{(1-s_k)\rho _k}). \end{aligned}$$

(3.6)

In what follows we denote, to simplify, $B_i:= B_{r_{x_i^{{\bar{\ell }}}}}\big (x_i^{{\bar{\ell }}}\big )$, $x_i:=x_i^{{\bar{\ell }}}$, $r_i:=r_{x_i^{{\bar{\ell }}}}$.

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 $B_{\rho _k}{\setminus } B_{(1-s_k)\rho _k}$ is large or not. We first let

$$\begin{aligned} \theta := \frac{2\xi (n)}{1+2\xi (n)} \in \left( \tfrac{2}{3},1\right) . \end{aligned}$$

(3.7)

In the case

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_{k}}\cap B_{(1-s_k)\rho _k}) \le \theta {\mathcal {H}}^{n-1}(J_{w_k}\cap B_{\rho _k}), \end{aligned}$$

(3.8)

we let $w_{k+1}:= w_k$. If instead we have the reverse inequality in (3.8), and consequently

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_{k}}\cap (B_{\rho _k}{\setminus } B_{(1-s_k)\rho _k})) \le (1-\theta ) {\mathcal {H}}^{n-1}(J_{w_k}\cap B_{\rho _k}), \end{aligned}$$

(3.9)

we then define $w_{k+1}$ as

$$\begin{aligned} w_{k+1} (x):= {\left\{ \begin{array}{ll} w_k (x) &{} \text { if } x \in B_{\rho }{\setminus } \left( \bigcup _{i\ge 1} {\overline{B}}_i\right) , \\ {{\widetilde{w}}}_{k, i} (x) &{} \text { if } x \in B_i \text { for some } i \in {{\mathbb {N}}}, \end{array}\right. } \end{aligned}$$

where ${{\widetilde{w}}}_{k, i} \in GSBD^p (B_i)$ 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 $i\ge 1$. Note that in this case the value of $\delta$, by definition of the balls $B_i$ (namely by (3.4)), is given by $\eta _k^{1/n}$, and $\eta _k^{1/n}\le {\bar{\delta }}$ by the assumption of the induction step.

Step 1.2c: Proof of the induction step. In case (3.8) is satisfied, we have $w_{k+1}=w_k\in GSBD^p(B_\rho )$, so that

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_{k+1}})= {\mathcal {H}}^{n-1}(J_{w_k}), \end{aligned}$$

(3.10)

and, by using (3.8) and (3.3),

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_{k+1}}\cap B_{(1-s_k)\rho _k}) \le \theta \eta _k (s_k\rho _k)^{n-1}. \end{aligned}$$

(3.11)

We now assume that (3.9) holds. By Property (1) of Lemma 3.1 we have $w_{k+1}\in GSBD^p(B_\rho )$. Moreover, let $R_{k, i} \in (1-\eta _k^{1/(2n)},1)$ be the radius given by Lemma 3.1 and corresponding to ${{\widetilde{w}}}_{k, i}$. Setting $B'_i:= B_{(1-\eta _k^{1/(2n)})r_i}(x_i)$ and $B''_i:= B_{R_{k, i} r_i}(x_i)$, we have in particular that $w_{k+1}\in C^\infty (B_i')$, $w_{k+1}=w_k$ in $B_i{\setminus } B_i''$, and ${\mathcal {H}}^{n-1}(J_{w_k}\cap \partial B_i'') = {\mathcal {H}}^{n-1}(J_{w_{k+1}}\cap \partial B_i'') =0$.

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 $B_i''{\setminus } B_i'$ by Property (1)):

$$\begin{aligned} {\mathcal {H}}^{n-1}((J_{w_{k+1}}{\setminus } J_{w_k})\cap B_i)&= {\mathcal {H}}^{n-1}((J_{w_{k+1}}{\setminus } J_{w_k})\cap (B''_i{\setminus } B_i')) \nonumber \\&\le c \eta _k^{\frac{1}{2n}} {\mathcal {H}}^{n-1}(J_{w_k}\cap (B_i{\setminus } B'_i)). \end{aligned}$$

(3.12)

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)

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_{k+1}}\cap B_i)&= {\mathcal {H}}^{n-1}(J_{w_{k+1}}\cap (B_i{\setminus } B_i')) \nonumber \\&\le {\mathcal {H}}^{n-1}((J_{w_{k+1}}{\setminus } J_{w_k})\cap (B''_i{\setminus } B'_i))+ {\mathcal {H}}^{n-1}(J_{w_{k}}\cap (B_i{\setminus } B_i')) \nonumber \\&\le \big (1+c \eta _k^{\frac{1}{2n}}\big ){\mathcal {H}}^{n-1}(J_{w_k}\cap (B_i{\setminus } B'_i)). \end{aligned}$$

(3.13)

For the last term in (3.13) we have the bound

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_k}\cap (B_i{\setminus } B'_i))&= {\mathcal {H}}^{n-1}(J_{w_k}\cap B_i)-{\mathcal {H}}^{n-1}(J_{w_k}\cap B'_i) \\&\le \eta _k r_i^{n-1} - \eta _k \big ( (1-\eta _k^{\frac{1}{2n}} )r_i\big )^{n-1} \\&= \big (1-\big (1-\eta _k^{\frac{1}{2n}}\big )^{n-1}\big ) {\mathcal {H}}^{n-1}(J_{w_k}\cap B_i), \end{aligned}$$

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

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_{k+1}}\cap B_i)\le \big (1+c \eta _k^{\frac{1}{2n}}\big ) \big (1-\big (1-\eta _k^{\frac{1}{2n}})^{n-1}\big ) {\mathcal {H}}^{n-1}(J_{w_k}\cap B_i\big ). \end{aligned}$$

Possibly reducing ${\bar{\delta }}$, we may assume that

$$\begin{aligned}\big (1+c \eta _k^{\frac{1}{2n}}\big ) \left( 1-\big (1-\eta _k^{\frac{1}{2n}}\big )^{n-1}\right) \le \Big (1+c \sqrt{{\bar{\delta }}}\Big ) \left( 1-\Big (1-\sqrt{{\bar{\delta }}}\Big )^{n-1}\right) \le \frac{1}{2}, \end{aligned}$$

so that

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_{k+1}}\cap B_i)\le \frac{1}{2} {\mathcal {H}}^{n-1}(J_{w_k}\cap B_i). \end{aligned}$$

(3.14)

Note that (3.14) and (3.5) imply immediately that

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_{k+1}})\le {\mathcal {H}}^{n-1}(J_{w_k}). \end{aligned}$$

(3.15)

In addition, by (3.14) and (3.5) one has

$$\begin{aligned}&{\mathcal {H}}^{n-1}(J_{w_{k+1}}\cap B_{\rho _k}) - {\mathcal {H}}^{n-1}(J_{w_{k}}\cap B_{\rho _k}) \nonumber \\&\qquad \le \sum _{i\ge 1} \left( {\mathcal {H}}^{n-1} (J_{w_{k+1}}\cap B_i) - {\mathcal {H}}^{n-1} (J_{w_{k}}\cap B_ i) \right) \nonumber \\&\qquad \le -\frac{1}{2} \sum _{i\ge 1} {\mathcal {H}}^{n-1} (J_{w_{k}}\cap B_i) \le - \frac{1}{2\xi (n)} {\mathcal {H}}^{n-1}(J_{w_k}\cap B_{(1-s_k)\rho _k}), \end{aligned}$$

(3.16)

where the last inequality follows from (3.6). We deduce from (3.16) and (3.9) that

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_{k+1}}\cap B_{\rho _k})&\le \left( 1-\frac{1}{2\xi (n)} \right) {\mathcal {H}}^{n-1}(J_{w_k}\cap B_{\rho _k}) \\&\qquad + \frac{1}{2\xi (n)} {\mathcal {H}}^{n-1}(J_{w_{k}}\cap (B_{\rho _k}{\setminus } B_{(1-s_k)\rho _k})) \\&\le \left( 1-\frac{\theta }{2\xi (n)} \right) {\mathcal {H}}^{n-1}(J_{w_k}\cap B_{\rho _k}). \end{aligned}$$

Hence, using the value (3.7) of $\theta$, we obtain that

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_{k+1}}\cap B_{(1-s_k)\rho _k}) \le \theta {\mathcal {H}}^{n-1}(J_{w_k}\cap B_{\rho _k}) \le \theta \eta _k (s_k\rho _k)^{n-1}, \end{aligned}$$

(3.17)

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),

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_{k+1}})\le {\mathcal {H}}^{n-1}(J_{w_k}), \end{aligned}$$

(3.18)

and that, by (3.11) and (3.17),

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_{k+1}}\cap B_{(1-s_k)\rho _k}) \le \theta {\mathcal {H}}^{n-1}(J_{w_k}\cap B_{\rho _k}) \le \theta \eta _k (s_k\rho _k)^{n-1}. \end{aligned}$$

(3.19)

We now define $\lambda := (\theta /(1-s_0)^{n-1})^{1/n}$; recalling the definition (3.2) of $s_0$, one can ensure that $\lambda \le \root 2n \of {\theta }<1$ by choosing $\tau$ small enough, namely, by requiring that

$$\begin{aligned} \tau \le \eta _0\big (1-\theta ^{\frac{1}{2(n-1)}}\big )^{n-1} = (\alpha {\bar{\delta }})^n\big (1-\theta ^{\frac{1}{2(n-1)}}\big )^{n-1}, \end{aligned}$$

(3.20)

which depends on $n,p,\alpha$. Then, letting $\rho _{k+1}:=(1-s_k)\rho _k$, $\eta _{k+1}:=\lambda \eta _k$, $s_{k+1}:=\lambda s_k$, we deduce from (3.19) that

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_{w_{k+1}}\cap B_{\rho _{k+1}})\le \eta _{k+1}(s_{k+1}\rho _{k+1})^{n-1}, \end{aligned}$$

which is (3.3) at step $k+1$.

Step 2: Convergence of $(w_k)_{k\ge 0}$. We now start the construction of the exceptional set $\omega$ given in the statement. To this aim, for every $k\ge 0$ we introduce the set $\omega _k$ in the following way. If (3.8) is satisfied we let $\omega _k:=\emptyset$, and if not, we let $\omega _k:=\bigcup _{i\ge 1}{\overline{B}}_i$ (note that $\omega _k \subseteq B_{\rho _k}$). In both cases $\{w_k\ne w_{k+1}\}\subset \omega _k$ and we can estimate the perimeter of $\omega _k$, thanks to (3.3) and (3.4), as

$$\begin{aligned} {\mathcal {H}}^{n-1}(\partial ^* \omega _k)&\le n\gamma _n\sum _{i\ge 1} r_i^{n-1} = \frac{n \gamma _n}{\eta _k} \sum _{i\ge 1} {\mathcal {H}}^{n-1}(J_{w_k}\cap B_i) \nonumber \\&\le \frac{n\gamma _n}{\eta _k}{\mathcal {H}}^{n-1}(J_{w_k}\cap B_{\rho _k}) \le n\gamma _n (s_k\rho _k)^{n-1}, \end{aligned}$$

(3.21)

where $n\gamma _n = {\mathcal {H}}^{n-1}({{\mathbb {S}}}^{n-1})$ (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

$$\begin{aligned} \int _{B_i} |e(w_{k+1})|^pdx \le \left( 1+c\eta _k^{\frac{s}{n}}\right) \int _{B_i}|e(w_k)|^p dx \end{aligned}$$

(3.22)

for each $i\ge 1$. As a consequence, by the definition of $w_{k+1}$, also

$$\begin{aligned} \int _{B_\rho } |e(w_{k+1})|^p dx \le \left( 1+c\eta _k^{\frac{s}{n}}\right) \int _{B_\rho }|e(w_k)|^p dx. \end{aligned}$$

(3.23)

Repeating the construction for all $k\ge 1$ we obtain sequences $(w_k)_{k\ge 0}$, $(s_k)_{k\ge 0}$, $(\eta _k)_{k\ge 0}$, $(\rho _k)_{k\ge 0}$ and $(\omega _k)_{k\ge 0}$ with:

$$\begin{aligned} \eta _k = \lambda ^k(\alpha {\bar{\delta }})^n,\quad s_k =\lambda ^k s_0, \quad \rho _k = \rho \prod _{\ell =0}^{k-1}(1-\lambda ^\ell s_0). \end{aligned}$$

(3.24)

Since $(\rho _k)_k$ is decreasing, there exists $\rho ' :=\lim _{k\rightarrow \infty }\rho _k$. We claim that $\rho '$ is bounded away from zero. Indeed, using that $(1-t s_0) \ge (1-s_0)^t$ for $t \in (0,1)$ and $s_0<1$

$$\begin{aligned} \rho ' = \rho \prod _{\ell =0}^{\infty }(1-\lambda ^\ell s_0) \ge \rho \prod _{\ell =0}^{\infty }(1- s_0)^{\lambda ^\ell } = \rho (1- s_0)^{\frac{1}{1 - \lambda }} \ge \rho (1- s_0)^{\frac{1}{1 - \root 2n \of {\theta }}}, \nonumber \\ \end{aligned}$$

(3.25)

since $\lambda \le \root 2n \of {\theta }$.

Now we set, for any $\ell \ge 0$, ${\tilde{\omega }}_\ell := \bigcup _{k\ge \ell }\omega _k$. Then, thanks to (3.21) and to (3.24),

$$\begin{aligned} {\mathcal {H}}^{n-1}(\partial ^*{\tilde{\omega }}_\ell )&\le \sum _{k\ge \ell } {\mathcal {H}}^{n-1}(\partial ^*\omega _k) \le n\gamma _n\sum _{k\ge \ell }(s_k\rho _k)^{n-1} \nonumber \\&\le n\gamma _n (s_\ell \rho _\ell )^{n-1} \sum _{k\ge \ell }(\lambda ^{k-\ell })^{n-1} \le n\gamma _n (s_\ell \rho _\ell )^{n-1} \frac{1}{1-\lambda ^{n-1}}, \end{aligned}$$

(3.26)

where we have used the fact that $\rho _k\le \rho _\ell$ for $k\ge \ell$. Then, since $\rho _\ell \rightarrow \rho '$ and $s_\ell \rightarrow 0$ as $\ell \rightarrow \infty$, it follows that ${\mathcal {H}}^{n-1}(\partial ^*{\tilde{\omega }}_\ell ) \rightarrow 0$ as $\ell \rightarrow \infty$. Hence by the isoperimetric inequality we also have that ${\mathcal {L}}^n({\tilde{\omega }}_\ell )\rightarrow 0$ as $\ell \rightarrow \infty$. Since, for $k\ge \ell$, $w_k=w_\ell$ outside ${\tilde{\omega }}_\ell$, we conclude that, as $k\rightarrow \infty$, $w_k$ converges ${\mathcal {L}}^n$-a.e. in $B_\rho$ to some function w. We also note that, for every $k\ge 0$, by (3.23) and (3.24),

$$\begin{aligned} \int _{B_\rho } |e(w_{k})|^pdx&\le \prod _{i=0}^{k-1}\left( 1+c\eta _i^{\frac{s}{n}}\right) \int _{B_\rho }|e(u)|^p dx \\&\le \prod _{i=0}^\infty \big (1+c(\alpha {\bar{\delta }})^s \lambda ^{\frac{is}{n}}\big )\int _{B_\rho }|e(u)|^p dx. \end{aligned}$$

Using $1+t \le e^t$, we estimate

$$\begin{aligned} \prod _{i=0}^\infty \big (1+c(\alpha {\bar{\delta }})^s \lambda ^{\frac{is}{n}}\big ) \le \exp \Big ( c(\alpha {\bar{\delta }})^s \sum _{i\ge 0} \big (\lambda ^{\frac{s}{n}}\big )^i \Big ) = \exp \Big (c(\alpha {\bar{\delta }})^s \frac{1}{1-\lambda ^{\frac{s}{n}}}\Big ), \end{aligned}$$

so that

$$\begin{aligned} \int _{B_\rho } |e(w_{k})|^pdx \le \exp \Big (c(\alpha {\bar{\delta }})^s \frac{1}{1-\lambda ^{\frac{s}{n}}}\Big )\int _{B_\rho }|e(u)|^p dx. \end{aligned}$$

Moreover, thanks to (3.18), for every $k\ge 0$ we have that ${\mathcal {H}}^{n-1}(J_{w_k})\le {\mathcal {H}}^{n-1}(J_{u})$. Then, thanks to [22, Theorem 11.3] (see also [12]) it follows that $w\in GSBD^p(B_\rho )$,

$$\begin{aligned}{\mathcal {H}}^{n-1}(J_w)\le \liminf _{k\rightarrow \infty }{\mathcal {H}}^{n-1}(J_{w_k})\le {\mathcal {H}}^{n-1}(J_u), \end{aligned}$$

and

$$\begin{aligned} \int _{B_\rho } |e(w)|^p dx \le \liminf _{k\rightarrow \infty } \int _{B_\rho }|e(w_k)|^p dx \le \exp \Big (c(\alpha {\bar{\delta }})^s \frac{1}{1-\lambda ^{\frac{s}{n}}}\Big ) \int _{B_\rho }|e(u)|^p dx. \end{aligned}$$

Passing to the limit in (3.3) we have that ${\mathcal {H}}^{n-1}(J_{w}\cap B_{\rho '})=0$, from which it follows that $w\in W^{1,p}(B_{\rho '};{{\mathbb {R}}}^n)$, thanks to Korn’s inequality. Note that by (3.24) $\rho '\le \rho$, and $\rho '\rightarrow \rho$ as ${\mathcal {H}}^{n-1}(J_u)\rightarrow 0$, thanks to (3.25) (and by the definition of $s_0$). Clearly, using (3.25) and choosing $\tau$ small enough we can ensure

$$\begin{aligned} \rho \ge \rho (1- s_0)^{ \frac{1}{1 - \root 2n \of {\theta }} } \ge \rho (1-\sigma ). \end{aligned}$$

This holds, for instance, for

$$\begin{aligned} \tau :=\eta _0 (1-\root 2n \of {\theta } )^{n-1} \sigma ^{n-1}, \end{aligned}$$

(3.27)

which also satisfies (3.20). Here we used that $( 1-\root 2n \of {\theta } ) \sigma \le 1 - (1- \sigma )^{1-\root 2n \of {\theta }}$. Note that $\tau =\tau (n,p,\alpha ,\sigma )$. With this choice we obtain that $w\in W^{1,p}(B_{\rho (1 - \sigma )};{{\mathbb {R}}}^n)$.

Setting $\omega := {\tilde{\omega }}_0$, by construction we have that $w=u$ in $B_{\rho } {\setminus } \omega$. In addition, from (3.26) we have that ${\mathcal {H}}^{n-1}(\partial ^*\omega )\le n\gamma _n (s_0\rho )^{n-1}/(1-\lambda ^{n-1})$. By our choice (3.2) of $s_0$, this implies

$$\begin{aligned} {\mathcal {H}}^{n-1}(\partial ^* \omega )\le \frac{n\gamma _n}{(\alpha {\bar{\delta }})^n(1-\lambda ^{n-1})} {\mathcal {H}}^{n-1}(J_u). \end{aligned}$$

Using the fact that $\lambda \le \root 2n \of {\theta }$ we finally obtain the estimates

$$\begin{aligned} \int _{B_\rho } |e(w)|^p dx&\le \exp \bigg ( \frac{c(\alpha {\bar{\delta }})^s}{1-\theta ^{\frac{s}{2n^2}}}\bigg ) \int _{B_\rho }|e(u)|^p dx,\\ {\mathcal {H}}^{n-1}(\partial ^* \omega )&\le \frac{n\gamma _n}{(\alpha {\bar{\delta }})^n(1-\theta ^{\frac{n-1}{2n}})} {\mathcal {H}}^{n-1}(J_u). \end{aligned}$$

Now we choose $\alpha =\alpha (n,p,\varepsilon )\in (0,1)$ such that

$$\begin{aligned} \exp \bigg (\frac{c(\alpha {\bar{\delta }})^s}{1-\theta ^{\frac{s}{2n^2}}}\bigg )\le (1+\varepsilon ). \end{aligned}$$

(3.28)

Note that now $\tau =\tau (n,p,\varepsilon ,\sigma )$. Correspondingly, we define

$$\begin{aligned} C = C(n,p,\varepsilon ):= \frac{n\gamma _n}{(\alpha {\bar{\delta }})^n(1-\theta ^{\frac{n-1}{2n}})} \end{aligned}$$

as the constant in the statement of the theorem, and this concludes the proof. $\square$

Remark 3.5

From (3.22), it is easy to show that in fact one can refine (3.1) to

$$\begin{aligned} \int _{\omega } |e(w)|^pdx \le (1+\varepsilon )\int _{\omega } |e(u)|^p dx. \end{aligned}$$

In addition, one sees that $C\sim \varepsilon ^{-n/s}$, 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 $B_\rho$ is replaced with a cube $(-\rho ,\rho )^n$.

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 $a \in {\mathcal {R}}$ such that

$$\begin{aligned} \int _{B_{(1-\sigma )\rho }{\setminus } \omega } |\nabla u-\nabla a|^p dx \le c(n,p) \int _{B_{\rho }}|e(u)|^p dx, \end{aligned}$$

and

$$\begin{aligned} \int _{B_{(1-\sigma )\rho }{\setminus } \omega } |u-a|^p dx \le c(n,p)\rho ^p \int _{B_{\rho }}|e(u)|^p dx. \end{aligned}$$

4 Regularity and rigidity in a general domain

The main result of this section is the following regularity result.

Theorem 4.1

Let $n \in {{\mathbb {N}}}$ with $n \ge 2$, $p \in (1, \infty )$, and let $\Omega \subset {{\mathbb {R}}}^n$ be a bounded and open Lipschitz set. There exists $c=c(n,p,\Omega )>0$ such that, for any $u\in GSBD^p(\Omega )$, there is a set of finite perimeter $\omega \subset \Omega$ with ${\mathcal {H}}^{n-1}(\partial ^*\omega )\le c{\mathcal {H}}^{n-1}(J_u)$ and $v\in W^{1,p}(\Omega ;{{\mathbb {R}}}^n)$ such that $u=v$ in $\Omega {\setminus }\omega$ and $\int _\Omega |e (v)|^p dx\le c\int _\Omega |e(u)|^p dx$. If in addition u is bounded, then $\Vert v\Vert _{L^\infty (\Omega )} \le \Vert u\Vert _{L^\infty (\Omega )}$. The constant c is invariant under uniform scalings of the domain.

Remark 4.2

Note that the conclusion of Theorem 4.1 is non-trivial only when the measure of the jump set $J_u$ is small, since otherwise one can simply take $\omega :=\Omega$ and $v:=0$ (see the proof of Theorem 4.1 for more details).

Remark 4.3

A careful inspection of the proof of Theorem 4.1 shows that the constant $c = c (n, p, \Omega )$ depends on $\Omega$ via the triple (N, r, L) defined as follows:

• (i) for every $x\in \partial \Omega$ there exists $e (x) \in {\mathbb {S}}^{n-1}$ such that $\partial \Omega \cap C(x,e (x),4Lr,2r)$ is the graph of an L-Lipschitz function defined on the $(n-1)$-dimensional ball $\{ y\in \Pi _{x}^{e (x)}: |(y-x)-((y-x)\cdot e (x))e (x)| < 2r \}$;

• (ii) there exist N points $x_1,\dots ,x_N\in \partial \Omega$ such that $\partial \Omega \subset \bigcup _{i = 1}^N B_{r} (x_{i})$ and $B_{r/5}(x_1),\dots , B_{r/5}(x_N)$ are mutually disjoint.

Note that property (i) follows from the fact that $\Omega$ 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

$$\begin{aligned} N \gamma _{n-1} \left( r/5\right) ^{n-1} \le {\mathcal {H}}^{n-1}(\partial \Omega ) \le N \gamma _{n-1} r^{n-1}\sqrt{1+L^2}. \end{aligned}$$

For the proof of Theorem 4.1 we will use the following lemma.

Lemma 4.4

Let $n \in {{\mathbb {N}}}$ with $n \ge 2$, $p \in (1, \infty )$, and let $D \subset {{\mathbb {R}}}^n$ be a bounded, open and connected Lipschitz set. Let $\alpha \in (0,1)$. There exist $c>0$ depending only on D, $\alpha$ and p, such that for any $w\in W^{1,p}(D;{{\mathbb {R}}}^n)$ and any Lebesgue measurable set $E\subset D$ with ${\mathcal {L}}^n(E)\ge \alpha {\mathcal {L}}^n(D)$, one has

$$\begin{aligned} \int _D |w-a_E|^p dx \le c \int _D |e(w)|^p dx, \end{aligned}$$

where

$$\begin{aligned} a_E:= {{\,\mathrm{\mathrm{arg\,min}}\,}}_{a\in {\mathcal {R}}} \int _E |w-a|^p dx. \end{aligned}$$

(4.1)

Proof

Such a lemma is standard and easily proved by contradiction. Suppose that for every $k\in {{\mathbb {N}}}$ there exist a function $w_k\in W^{1,p}(D;{{\mathbb {R}}}^n)$ and a Lebesgue measurable set $E_k\subset D$ with ${\mathcal {L}}^n(E_k)\ge \alpha {\mathcal {L}}^n(D)$ such that

$$\begin{aligned} \int _D |w_k-a_{E_k}|^p dx > k \int _D |e(w_k)|^p dx, \end{aligned}$$

(4.2)

where $a_{E_k}$ is as in (4.1). Setting

$$\begin{aligned} u_k:= \frac{w_k-a_{E_k}}{\Vert w_k-a_{E_k}\Vert _{L^p(D;{{\mathbb {R}}}^n)}}, \end{aligned}$$

we have that, by the definition of $a_{E_k}$, $u_k$ satisfies

$$\begin{aligned} \int _{E_k} |u_k|^p dx \le \int _{E_k} |u_k-a|^p dx \quad \forall \, a\in {\mathcal {R}}. \end{aligned}$$

(4.3)

Moreover, by (4.2),

$$\begin{aligned} \Vert u_k\Vert _{L^p(D;{{\mathbb {R}}}^n)}=1, \quad \Vert e(u_k)\Vert ^p_{L^p(D;{{\mathbb {R}}}_{\text {sym}}^{n\times n})}<\frac{1}{k}, \end{aligned}$$

(4.4)

so that by the classical Korn inequality

$$\begin{aligned} \Vert u_k\Vert _{W^{1,p}(D;{{\mathbb {R}}}^n)}\le C\left( \Vert u_k\Vert _{L^p(D;{{\mathbb {R}}}^n)} + \Vert e(u_k)\Vert _{L^p(D;{{\mathbb {R}}}_{\text {sym}}^{n\times n})}\right) \le 2C, \end{aligned}$$

for some constant $C=C(n,p,D)>0$. Hence, there exists $u\in W^{1,p}(D;{{\mathbb {R}}}^n)$ such that, up to subsequences, $u_k\rightharpoonup u$ weakly in $W^{1,p}(D;{{\mathbb {R}}}^n)$ as $k\rightarrow +\infty$ (and strongly in $L^p(D;{{\mathbb {R}}}^n)$). Note that, up to subsequences, the characteristic functions $\chi _{E_k}$ of $E_k$ converge weakly$^*$ in $L^{\infty }(D)$ to some function $\phi \in L^{\infty }(D)$ with $0 \le \phi \le 1$ and such that

$$\begin{aligned} \alpha {\mathcal {L}}^n(D) \le \int _D \phi \, dx. \end{aligned}$$

(4.5)

Therefore, passing to the limit in (4.3) and (4.4), we have that $e(u)=0$ in D and

$$\begin{aligned} \int _{D} |u|^p \phi \, dx \le \int _{D} |u-a|^p \phi \, dx \quad \forall \, a\in {\mathcal {R}} , \quad \Vert u\Vert _{L^p(D;{{\mathbb {R}}}^n)}=1. \end{aligned}$$

(4.6)

Since $e(u)=0$ in D, by the classical Poincaré-Korn inequality (see, e.g., (1.6)) we deduce that there exists ${\overline{a}} \in {\mathcal {R}}$ such that $u={\overline{a}}$ in D. Choosing $a = {\overline{a}}$ in (4.6) we then have

$$\begin{aligned} \int _D | {\overline{a}} |^p \phi \, dx = 0. \end{aligned}$$

Since ${\overline{a}} \in {\mathcal {R}}$ and taking into account (4.5) it follows that ${\overline{a}} =0$ and hence $u=0$. This is however incompatible with $\Vert u\Vert _{L^p(D;{{\mathbb {R}}}^n)}=~1$. $\square$

Proof

[Proof of Theorem 4.1] It is enough to prove the result in the case ${\mathcal {H}}^{n-1}(J_u)\le {\bar{c}}$, for some constant ${\bar{c}}>0$ to be determined later on (see (4.9)). Indeed, if ${\mathcal {H}}^{n-1}(J_u)>{\bar{c}}$, we can simply set $\omega :=\Omega$ and $v:=0$, which clearly satisfy the required bounds, since ${\mathcal {H}}^{n-1}(\partial ^*\omega )\le ({\mathcal {H}}^{n-1}(\partial ^*\Omega )/{\bar{c}}){\mathcal {H}}^{n-1}(J_u)$.

We divide the proof into several steps.

Step 1: We introduce a finite open cover $\{ C_0, C_1, \ldots , C_N \}$ of $\overline{\Omega }$. Since $\Omega$ is Lipschitz and bounded (see [1, Sect. 4.9]), there exist $r > 0$ such that for every $x \in \partial \Omega$, there exists $e (x) \in {\mathbb {S}}^{n-1}$ such that $\partial \Omega \cap C(x,e (x), 4 L r , 2r)$ (see (d) in the Notation Section) is the graph of an L-Lipschitz function defined on the $(n-1)$-dimensional ball $\{ y\in \Pi _{x}^{e (x)}: |(y-x)-((y-x)\cdot e (x))e (x)| < 2r \}$. Setting ${\tilde{L}}:= \max \{ 1, L \}$, by possibly reducing r we still have that $\partial \Omega \cap C(x,e (x), 4 {\tilde{L}} r , 2r)$ is the graph of an ${\tilde{L}}$-Lipschitz function for every $x \in \partial \Omega$. Consider now the family of open balls $\{ B_{r/5} (x) \}_{x \in \partial \Omega }$. By Vitali’s Covering Theorem [24, Section 1.5.1], there exist $N \in {\mathbb {N}}$ and $\{ x_1, \ldots , x_N \} \subset \partial \Omega$ such that the family $\{ B_{r/5} (x_{i}) \}_{i = 1, \ldots , N}$ is composed of mutually disjoint balls and

$$\begin{aligned} \partial \Omega \subset \bigcup _{x\in \partial \Omega } B_{r/5} (x) \subset \bigcup _{i = 1}^N B_{r} (x_{i}). \end{aligned}$$

(4.7)

In the following, we will use the shorthand $C_i:= C (x_{i},e (x_{i}), 4 {\tilde{L}} r , 2r)$ for every $i = 1, \ldots , N$. Note that $B_r(x_i)\subset B_{2r}(x_i)\subset C_i$ and that $\mathrm{dist}(\partial C_i,B_r(x_i))=r$ for every $i = 1, \ldots , N$. Moreover, from (4.7) it follows that

$$\begin{aligned} \left\{ x\in \Omega : \mathrm{dist}(x,\partial \Omega )\le \frac{r}{6} \right\} \subset \left\{ x\in \Omega : \mathrm{dist}(x,\partial \Omega ) < \frac{r}{5} \right\} \subset \bigcup _{i=1}^N B_{r}(x_i). \end{aligned}$$

(4.8)

Setting

$$\begin{aligned} {{\widetilde{C}}}_0:= \left\{ x\in \Omega : \mathrm{dist}(x,\partial \Omega )> \frac{r}{6} \right\} , \quad C_0:= \left\{ x\in \Omega : \mathrm{dist}(x,\partial \Omega ) > \frac{r}{8} \right\} , \end{aligned}$$

thanks to (4.8) it follows that $\overline{\Omega } \subset {{\widetilde{C}}}_0\cup \Big ( \bigcup _{i=1}^N B_r(x_i)\Big )\subset \bigcup _{i=0}^N C_i$.

Step 2: We show that, at any given point of $\Omega$, the maximal number of overlapping sets in the covering $\{ C_0, C_1, \ldots , C_N\}$ only depends on L and n. To this aim, it will be enough to prove the statement for the sets $C_1, \ldots , C_N$. Let $z \in \Omega$, and let

$$\begin{aligned} A (z) := \left\{ x \in \{ x_{1}, \ldots , x_{N} \} : z \in C(x,e (x), 4 {\tilde{L}} r , 2r) \right\} . \end{aligned}$$

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 $\partial \Omega$, it can be $A (z) = \emptyset$, but in this case there is nothing to prove. Since the diameter of each cylinder is given by $4r\sqrt{1 + 4 {\tilde{L}}^2}$, we have

$$\begin{aligned} | z - x | < 4r \sqrt{1 + 4 {\tilde{L}}^2} \quad \text { for every } x \in A (z). \end{aligned}$$

Therefore,

$$\begin{aligned} A (z) \subset B_{ 4r \sqrt{1 + 4 {\tilde{L}}^2}} (z). \end{aligned}$$

Recalling that the family $\{ B_{r/5} (x_{i}) \}_{i = 1, \ldots , N}$ is composed of mutually disjoint balls, for every $i \in \{1, \ldots , N \}$ with $i \ne j$ we have $| x_i - x_j | > 2 r/5$. 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 $4r \sqrt{1 + 4 {\tilde{L}}^2}$, that we denote with $\kappa$. By scaling, one can check that $\kappa$ does not depend on r, but only on ${\tilde{L}}$ (i.e. on L) and on the dimension n.

Step 3: We show that it is enough to prove the theorem in the set $C_i \cap \Omega$, for every $i \in \{ 0, 1 \ldots , N \}$. We introduce a partition of unity of $\Omega$ subordinate to the open covering $\{{{\widetilde{C}}}_0, B_r(x_1), \dots , B_r(x_N)\}$, namely maps $\phi _0\in C_c^\infty ({{\widetilde{C}}}_0,[0,1])$, and $\phi _i\in C_c^\infty (B_r(x_i),[0,1])$ for $i=1,\dots ,N$, with $\sum _{i=0}^N \phi _i=1$ in $\Omega$ (see, e.g. [30, Definition A.13]). Note that this is also a partition of unity of $\Omega$ subordinate to the open covering $\{C_i\}$, for $i=0,\dots ,N$. Moreover, by construction $\mathrm{dist}({{\,\mathrm{supp}\,}}\phi _0,\partial C_0) > r/24$, while $\mathrm{dist}({{\,\mathrm{supp}\,}}\phi _i, \partial C_i) \ge r$ for $i =1,\dots ,N$. Assuming that for every i we can find a function $v_i$ and a set $\omega _i$ satisfying the thesis of the theorem in $C_i\cap \Omega$, then $v=\sum _{i=0}^N \big ({\phi _i}_{|{C_i\cap \Omega }}\big ) v_i$ and $\omega =\bigcup _{i=0}^N\omega _i$ satisfy the claim in $\Omega$.

Step 4: We fix $i \in \{ 0, 1, \ldots , N \}$ and prove the theorem in $C_i \cap \Omega$. Let $i \in \{ 0, 1, \ldots , N \}$. Our construction will be simpler in the case $i = 0$ and, when necessary, we will explicitly point this out in the proof. If $i \ne 0$, without loss of generality we can assume that $C_i=C(0,e_n, 4 {\tilde{L}} r,2r)$, with $e_n$ being the n-th coordinate unit vector, and that $\Omega \cap C_i = \{ x=(x',x_n)\in C_i: x_n<g(x')\}$ for a given ${\tilde{L}}$-Lipschitz function g defined on the ($(n-1)$-dimensional) ball centred at 0 and of radius r in $\Pi ^{e_n}_0$, with $g(0)=0$.

We now build $v_i$ and $\omega _i$ for the set $C_i\cap \Omega$. Let $\delta >0$, and let ${\mathbf {C}}_i$ denote the union of all the n-dimensional cubes $q\in \{ z+(0,\delta ]^{n}: z\in \delta {{\mathbb {Z}}}^{n}\}$ with $q\subset {C}_i$. Since $\mathrm{dist}({{\,\mathrm{supp}\,}}\phi _i, \partial C_i) > r/24$, we can assume that $\delta$ is small enough so that ${{\,\mathrm{supp}\,}}\phi _i \subset {\mathbf {C}}_i$. Note that the choice of $\delta /r$ depends only on n.

Then we build recursively the set ${\mathcal {Q}}$ of dyadic cubes of edge size $\delta 2^{-k}$, $k\ge 0$, which refine towards the boundary $\partial \Omega$, as follows. As a first step, we denote with ${\mathcal {Q}}_0$ the set of cubes $q \in \{ z+(0,\delta ]^{n}: z\in \delta {{\mathbb {Z}}}^{n}\}$, $q\subset {\mathbf {C}}_i\cap \Omega$, such that $\mathrm{dist}(q,\partial \Omega )> \delta$. Then, for $k\ge 1$, having built ${\mathcal {Q}}_\ell$ for $\ell <k$, we define ${\mathcal {Q}}_k$ as the set of all the smaller cubes $q\in \{z+(0,\delta 2^{-k}]^n: z\in \delta 2^{-k}{{\mathbb {Z}}}^n\}$, $q\subset {\mathbf {C}}_i\cap \Omega$, such that $\mathrm{dist}(q,\partial \Omega )> \delta 2^{-k}$, and q does not intersect cubes of $\bigcup _{\ell <k}\bigcup _{{{\hat{q}}}\in {\mathcal {Q}}_\ell }{{\hat{q}}}$. Note that, if $i = 0$, we can assume that all the cubes in ${\mathbf {C}}_0$ belong to the family ${\mathcal {Q}}_0$ (by e.g. choosing $\delta < r/8$).

Finally, we let ${\mathcal {Q}}:=\bigcup _{k=0}^\infty {\mathcal {Q}}_k$; note that $\bigcup _{q\in {\mathcal {Q}}} q = {\mathbf {C}}_i \cap \Omega \subset C_i\cap \Omega$ covers entirely ${{\,\mathrm{supp}\,}}\phi _i \cap \Omega$. Now, for each $q\in {\mathcal {Q}}$, 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 $q\in {\mathcal {Q}}$, form a sort of Whitney covering of $C_i\cap \Omega$, at least covering ${{\,\mathrm{supp}\,}}\phi _i\cap \Omega$. Moreover, since for every $k\ge 0$ any $q \in {\mathcal {Q}}_k$ satisfies $\mathrm{dist}(q,\partial \Omega )> \delta 2^{-k}$, clearly also $q',q''\subset \Omega$. Note that, for fixed $k\ge 0$, an enlarged cube $q''$ of some cube $q \in {\mathcal {Q}}_k$ can only intersect cubes belonging to ${\mathcal {Q}}_k$, ${\mathcal {Q}}_{k+1}$ and, if $k\ge 1$, ${\mathcal {Q}}_{k-1}$.

Next, we choose the constant ${\bar{c}}= {\bar{c}} (\Omega )$ introduced at the start of the proof to be

$$\begin{aligned} {\bar{c}}:= \tau (\delta /2)^{n-1}, \end{aligned}$$

(4.9)

where $\tau$ 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 $\sigma = 1/12$ and $\varepsilon =1$. Hence, by the initial assumption ${\mathcal {H}}^{n-1}(J_u)\le {\bar{c}}$ we have

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_u)\le \tau (\delta /2)^{n-1}. \end{aligned}$$

Then, by applying Theorem 3.2 (with $\varepsilon =1$) to $u\in GSBD^p(q'')$, for each $q\in {\mathcal {Q}}_0$, we find a function $w_q\in GSBD^p(q'')$ and a set of finite perimeter $\omega _q\subset q''$ such that $w_q=u$ in $q''{\setminus } \omega _q$, $\int _{q''}|e(w_q)|^pdx \le C\int _{q''}|e(u)|^p dx$, ${\mathcal {H}}^{n-1}(\partial ^*\omega _q)\le C{\mathcal {H}}^{n-1}(J_u\cap q'')$ and $w_q\in W^{1,p}(q';{{\mathbb {R}}}^n)$, where $C = C (n, p)$.

For smaller cubes $q\in {\mathcal {Q}}_k$, $k\ge 1$, we proceed as follows: if ${\mathcal {H}}^{n-1}(J_u\cap q'')\le \tau (\delta /2^{k+1})^{n-1}$, we say that q is “good", we apply Theorem 3.2 to the restriction of u to $q''$, and find $w_q$ and $\omega _q$ as in the case $k=0$ (note that all the cubes in ${\mathcal {Q}}_0$ are “good” and that, in particular, ${\mathbf {C}}_0$ is made of “good” cubes). In conclusion, for q “good”, we find a function $w_q\in GSBD^p(q'')$ and a set of finite perimeter $\omega _q\subset q''$ such that $w_q=u$ in $q''{\setminus } \omega _q$ and

$$\begin{aligned}&w_q\in W^{1,p}(q';{{\mathbb {R}}}^n), \end{aligned}$$

(4.10)

$$\begin{aligned}&\int _{q''}|e(w_q)|^pdx \le C\int _{q''}|e(u)|^p dx, \end{aligned}$$

(4.11)

$$\begin{aligned}&{\mathcal {H}}^{n-1}(\partial ^*\omega _q)\le C{\mathcal {H}}^{n-1}(J_u\cap q''), \end{aligned}$$

(4.12)

where $C = C (n, p)$. If instead ${\mathcal {H}}^{n-1}(J_u\cap q'')> \tau (\delta /2^{k+1})^{n-1}$, we say that q is “bad” and we define

$$\begin{aligned} {\tilde{\omega }}_q := \Omega \cap (q+\{ x=(x',x_n): x_n>2 {\tilde{L}} |x'| \}), \end{aligned}$$

namely we connect q with $\partial \Omega$ via a sort of truncated cone with an opening controlled by the Lipschitz constant L of $\Omega$. Scaling arguments (and the fact that $\mathrm{dist}(q,\partial \Omega )\le \delta 2^{-k+1}$) show that ${\mathcal {H}}^{n-1}(\partial ^*{\tilde{\omega }}_q)\le c(\delta 2^{-k})^{n-1}$ where the constant $c=c(n,L)$ depends only on L and the dimension. It follows that in this case, namely for q “bad",

$$\begin{aligned} {\mathcal {H}}^{n-1}(\partial ^*{\tilde{\omega }}_q) \le c(n,L) \frac{2^{n-1}}{\tau }{\mathcal {H}}^{n-1}(J_u\cap q''). \end{aligned}$$

(4.13)

We let ${\tilde{\omega }}:=\bigcup _{q \in {\mathcal {Q}}_b} {\tilde{\omega }}_q$, $G:= (\bigcup _{q\in {\mathcal {Q}}} q){\setminus } {\tilde{\omega }}$, and ${\hat{\omega }} := \bigcup _{q \in {\mathcal {Q}}_g} (\omega _q\cap q')$, where we denoted with ${\mathcal {Q}}_b, {\mathcal {Q}}_g \subset {\mathcal {Q}}$ the “bad” and “good” cubes in ${\mathcal {Q}}$, respectively. By construction, there exists a $(2 {\tilde{L}})$-Lipschitz function f such that G is the subgraph of f, with $g-2\delta \le f\le g$. Moreover, for some constant c (depending on L, n and $\tau$), and using that the cubes $q''$ have finite overlap, one has, by (4.12) and (4.13),

$$\begin{aligned} {\mathcal {H}}^{n-1}(\partial ^*\omega _i) \le {\mathcal {H}}^{n-1}(\partial ^*{\tilde{\omega }}) + {\mathcal {H}}^{n-1}(\partial ^* {\hat{\omega }})\le c{\mathcal {H}}^{n-1}(J_u\cap (C_i\cap \Omega )), \end{aligned}$$

(4.14)

where we set $\omega _i:= {\tilde{\omega }} \cup {\hat{\omega }}$.

We now construct a regularised function $v_i$ as a convex combination of the functions $w_q$ relative to “good" cubes $q\in {\mathcal {Q}}_g$ only. More precisely, let $\psi \in C_c^\infty ((0,1.1)^n;[0,1])$ be a smooth cut-off function with $\psi =1$ on $[0,1]^n$. For any $k\ge 0$ and any $q\in {\mathcal {Q}}_g\cap {\mathcal {Q}}_k$ with centre $c_q$, we define the translated and rescaled version of $\psi _q$, $\psi _q(x):= \psi ((x-c_q)/(\delta 2^{-k}))\in C_c^\infty (q';[0,1])$, so that $\psi _q=1$ on q. Finally, we define the ‘normalised’ cut-off function $\varphi _q(x) := \psi _q(x)/(\sum _{{{\hat{q}}} \in {\mathcal {Q}}_g}\psi _{{{\hat{q}}}}(x))$ for $x\in \cup _{q\in {\mathcal {Q}}_g}q$.

We then let, for $x\in \cup _{q \in {\mathcal {Q}}_g} q$, ${{\tilde{v}}}_i(x):=\sum _{{q\in {\mathcal {Q}}_g}}w_q(x)\varphi _{q}(x)$. First of all, we extend ${{{\tilde{v}}}_{i|G}}$ from G to ${\mathbf {C}}_i \cap \Omega$. 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)]) and ${{{\tilde{v}}}_{i|G}}\in W^{1,p}(G;{{\mathbb {R}}}^n)$, as each $w_q$ belongs to $W^{1,p}(q';{{\mathbb {R}}}^n)$ for $q \in {\mathcal {Q}}_g$, by (4.10). We denote this extension by $v_i$. Then $v_i \in W^{1,p}({\textbf {C}}_i \cap \Omega ; {{\mathbb {R}}}^n)$, $v_i = {{\tilde{v}}}_i$ in G, and by [31, property (50)] we have that

$$\begin{aligned} \int _{{\mathbf {C}}_i \cap \Omega } |e(v_i)|^p dx \le c \int _{G} |e({{\tilde{v}}}_i)|^p dx \le c \int _{\cup _{q \in {\mathcal {Q}}_g}q} |e({{\tilde{v}}}_i)|^p dx, \end{aligned}$$

(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 $2 {\tilde{L}}$, hence $c=c(n,p, L)$.

Moreover, $v_i = u$ in $({\mathbf {C}}_i \cap \Omega ){\setminus } \omega _i$. Indeed, ${{{\tilde{v}}}_{i|G}} =u$ in $G {\setminus } {\hat{\omega }}$ by construction, $v_i = {{\tilde{v}}}_i$ in G, and $G = ({\mathbf {C}}_i\cap \Omega ) {\setminus } {\tilde{\omega }}$.

To conclude the proof of this step, it remains to show that

$$\begin{aligned} \int _{{\mathbf {C}}_i \cap \Omega }|e(v_i)|^p dx \le c\int _{C_i\cap \Omega }|e(u)|^p dx, \end{aligned}$$

(4.16)

for some $c = c(n, p, L)$. By (4.15), it is sufficient to show that $\int _{\cup _{q\in {\mathcal {Q}}_g}q} |e({{\tilde{v}}}_i)|^p dx \le c\int _{C_i\cap \Omega }|e(u)|^p dx$. By the definition of ${{\tilde{v}}}_i$, one has

$$\begin{aligned} e({{\tilde{v}}}_i) = \sum _{q \in {\mathcal {Q}}_g} \big (e(w_q)\varphi _q + w_q\odot \nabla \varphi _q\big ). \end{aligned}$$

(4.17)

We need therefore to estimate the $L^p$ norm of $\sum _{q \in {\mathcal {Q}}_g} w_q\odot \nabla \varphi _q$ 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 $\sum _q\varphi _q\equiv 1$ in $\cup _{q \in {\mathcal {Q}}_g} q$, we have that $\sum _{q}\nabla \varphi _{q}=0$ in $\cup _{q \in {\mathcal {Q}}_g} q$ (where here and in what follows the sums run on cubes in ${\mathcal {Q}}_g$). Then, if we fix $q\in {\mathcal {Q}}_g$ and $x\in q' \cap (\cup _{{{\hat{q}}}\in {\mathcal {Q}}_g}{{\hat{q}}})$, we have

$$\begin{aligned} \sum _{{{\hat{q}}}}w_{{{\hat{q}}}}(x)\odot \nabla \varphi _{{{\hat{q}}}}(x)&= \sum _{{{\hat{q}}}}w_{{{\hat{q}}}}(x)\odot \nabla \varphi _{{{\hat{q}}}}(x) - w_{q}(x)\odot \sum _{{{\hat{q}}}}\nabla \varphi _{{{\hat{q}}}}(x)\nonumber \\&= \sum _{{{\hat{q}}}} (w_{{{\hat{q}}}}(x) - w_q(x))\odot \nabla \varphi _{{{\hat{q}}}}(x)\nonumber \\&= \sum _{{{\hat{q}}}: q'\cap {{\hat{q}}}'\ne \emptyset }(w_{{{\hat{q}}}}(x)-w_q(x))\odot \nabla \varphi _{{{\hat{q}}}}(x). \end{aligned}$$

(4.18)

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 cubes ${{\hat{q}}}$ such that ${{\hat{q}}}'$ intersects $q'$, whose number is bounded by $2^n$.

Now we observe that, if $q'\cap {{\hat{q}}}' \ne \emptyset$, then there are two cases: either q and ${{\hat{q}}}$ are of the same size, or, alternatively, the edge length of one is twice the edge length of the other one. In either case

$$\begin{aligned} {\mathcal {L}}^n(q'\cap {{\hat{q}}}') \ge \beta _1 \max \{{\mathcal {L}}^n(q'),{\mathcal {L}}^n({{\hat{q}}}')\}, \end{aligned}$$

where $\beta _1=\beta _1(n) >0$ is an explicit constant depending only on the dimension. Now, to fix the ideas, assume that $q\in {\mathcal {Q}}_k$ and ${{\hat{q}}}\in {\mathcal {Q}}_{k+1}$; then, by Remark 3.4 and (4.12) (where we recall that $C=C(n,p)$), and since $q,{{\hat{q}}}\in {\mathcal {Q}}_g$,

$$\begin{aligned} {\mathcal {L}}^n(\omega _q\cup \omega _{{{\hat{q}}}})&\le C \delta {2^{-k}}\left( {\mathcal {H}}^{n-1}(J_u\cap q'') + {\mathcal {H}}^{n-1}(J_u \cap {{{\hat{q}}}}'')\right) \\&\le c(n,p) \,\tau \left( \frac{\delta }{2^{k+1}}\right) ^{n} \le c(n,p) \,\tau {\mathcal {L}}^n(q'\cap {{\hat{q}}}'), \end{aligned}$$

where c(n, p) denotes possibly different constants. Therefore, for every $q,{{\hat{q}}} \in {\mathcal {Q}}_g$ with $q'\cap {{\hat{q}}}'\ne \emptyset$,

$$\begin{aligned} {\mathcal {L}}^n(\omega _q\cup \omega _{{{\hat{q}}}})\le c(n,p) \tau {\mathcal {L}}^n(q'\cap {{\hat{q}}}'). \end{aligned}$$

Hence, up to possibly reducing $\tau$, we have that

$$\begin{aligned} {\mathcal {L}}^n((q'\cap {{\hat{q}}}'){\setminus } (\omega _q\cup \omega _{{{\hat{q}}}})) \ge \beta _2 {\mathcal {L}}^n(q'\cap {{\hat{q}}}') \ge \beta _1 \beta _2 \max \{{\mathcal {L}}^n(q'), {\mathcal {L}}^n({{\hat{q}}}')\}, \end{aligned}$$

for some $\beta _2>0$ depending on n and p.

We now apply Lemma 4.4 to $w_q$ in $q'$ and to $w_{{{\hat{q}}}}$ in ${{\hat{q}}}'$, with $E=(q'\cap {{\hat{q}}}'){\setminus } (\omega _q\cup \omega _{{{\hat{q}}}})$ and $\alpha =\beta _1\beta _2$. 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 $\ell '$, $c = c(n,\alpha ,p) (\ell ')^p$, with $c(n,\alpha ,p)$ being the constant for the unit cube in ${{\mathbb {R}}}^n$. Then

$$\begin{aligned} \int _{q'} |w_q-a_q|^p dx&\le c (\ell ')^p\int _{q'}|e(w_q)|^p dx, \end{aligned}$$

(4.19)

$$\begin{aligned} \int _{{{\hat{q}}}'} |w_{{{\hat{q}}}}-a_{{{\hat{q}}}}|^p dx&\le c ({\hat{\ell }}')^p \int _{{{\hat{q}}}'}|e(w_{{{\hat{q}}}})|^p dx, \end{aligned}$$

(4.20)

and

$$\begin{aligned} a_{q}:= {{\,\mathrm{\mathrm{arg\,min}}\,}}_{a\in {\mathcal {R}}} \int _{E} |w_q-a|^p dx, \quad a_{{{\hat{q}}}}:= {{\,\mathrm{\mathrm{arg\,min}}\,}}_{a\in {\mathcal {R}}} \int _{E} |w_{{{\hat{q}}}}-a|^p dx. \end{aligned}$$

On the other hand, since $w_q = w_{{{\hat{q}}}} = u$ in E, we have that $a_{q} = a_{{{\hat{q}}}} = a$, and hence, thanks to (4.19)–(4.20) and (4.11),

$$\begin{aligned} \int _{q'\cap {{\hat{q}}}'} |w_q-w_{{{\hat{q}}}}|^p dx&\le c(p)\int _{q'\cap {{\hat{q}}}'} |w_q-a|^p dx + c(p)\int _{q'\cap {{\hat{q}}}'} |w_{{{\hat{q}}}} - a|^p dx\\&\le c(n, p) \left( (\ell ')^p\int _{q'}|e(w_q)|^p dx + ({\hat{\ell }}')^p\int _{{{\hat{q}}}'}|e(w_{{{\hat{q}}}})|^p dx\right) \\&\le c(n, p) \, \ell ^p \left( \int _{q''}|e(u)|^p dx+ \int _{{{\hat{q}}}''}|e(u)|^p dx\right) . \end{aligned}$$

In conclusion, for a given $q\in {\mathcal {Q}}_g$, by (4.17), (4.18), (4.11) and the previous estimate,

$$\begin{aligned} \int _{q}|e({{\tilde{v}}}_i)|^p dx&\le c \sum _{{{\hat{q}}}: q'\cap {{\hat{q}}}'\ne \emptyset }\int _{q\cap {{\hat{q}}}'} |e(w_{{{\hat{q}}}})|^p dx \\&\qquad + c \sum _{{{\hat{q}}}: q'\cap {{\hat{q}}}'\ne \emptyset } \Vert \nabla \varphi _{{{\hat{q}}}}\Vert ^p_{L^\infty ({{\hat{q}}}')} \int _{q\cap {{\hat{q}}}'} |w_q-w_{{{\hat{q}}}}|^p dx \\&\le c \sum _{{{\hat{q}}}: q'\cap {{\hat{q}}}'\ne \emptyset } \int _{{{\hat{q}}}''}|e(u)|^p dx, \end{aligned}$$

with $c = c (n, p)$, where we have used the fact that $\Vert \nabla \varphi _{{{\hat{q}}}}\Vert _{L^\infty ({{\hat{q}}}')} \le c /{\hat{\ell }}$. Using that the cubes $q''$ have finite overlap, we have

$$\begin{aligned} \int _{\cup _{q \in {\mathcal {Q}}_g}q} |e({{\tilde{v}}}_i)|^p dx \le c \int _{C_i \cap \Omega } |e(u)|^p dx, \end{aligned}$$

and, by (4.15) we obtain (4.16). We have then proved the estimates

$$\begin{aligned} \int _{{\mathbf {C}}_i \cap \Omega }|e(v_i)|^p dx \le c_i \int _{C_i\cap \Omega }|e(u)|^p dx, \qquad {\mathcal {H}}^{n-1}(\partial ^*\omega _i) \le c_i {\mathcal {H}}^{n-1}(J_u\cap (C_i\cap \Omega )), \end{aligned}$$

where $c_i = c_i (n, p, L)$ is the maximum of the two constants in (4.14) and (4.16).

Step 5: Conclusion. Recalling that $v=\sum _{i=0}^N \big ({\phi _i}_{|{C_i\cap \Omega }}\big ) v_i =\sum _{i=0}^N \big ({\phi _i}_{|{{\mathbf {C}}_i \cap \Omega }}\big ) v_i$ and $\omega =\bigcup _{i=0}^N\omega _i$ (see Step 3), and that the number of $C_i$’s intersecting at every point of $\Omega$ is at most $\kappa + 1$ (see Step 2), the statement holds true by setting

$$\begin{aligned} c (n, p, \Omega ):= (\kappa + 1) \max \left\{ c_0 , c_1, \dots , c_N, {\tilde{c}} \right\} , \end{aligned}$$

where ${\tilde{c}} = {\mathcal {H}}^{n-1} (\partial ^*\Omega ) / {\bar{c}}$.

Note that, since ${\mathcal {H}}^{n-1} (\partial ^*\Omega )$ can be estimated in terms of the parameters N, r and L introduced in Step 1, we have that $c (n, p, \Omega ) = c(n,p, r, N, L)$. $\square$

An immediate consequence of Theorem 4.1 is the Korn’s inequality below, whose proof is a direct adaptation of [16, Corollary 3.3].

Theorem 4.5

Under the same assumptions and notation of Theorem 4.1, and under the additional requirement that $\Omega$ is connected, there exists an affine function $a\in {\mathcal {R}}$ such that

$$\begin{aligned} \int _{\Omega {\setminus } \omega }|\nabla u - \nabla a|^p dx&\le c(n,p,\Omega ) \int _{\Omega }|e(u)|^p dx. \end{aligned}$$

(4.21)

Moreover,

$$\begin{aligned} \left( \int _{\Omega {\setminus } \omega }|u - a|^q dx\right) ^{\frac{1}{q}} \le c(n,p,q,\Omega ) \left( \int _{\Omega }|e(u)|^p dx\right) ^{\frac{1}{p}}, \end{aligned}$$

(4.22)

where $q\le p^*$ if $p<n$, $q<\infty$ if $p=n$, and $q\le \infty$ for $p>n$.

Proof

Let $v\in W^{1,p}(\Omega ;{{\mathbb {R}}}^n)$ be given by Theorem 4.1. By Korn’s inequality applied to v, there exists $A\in {{\mathbb {R}}}^{n\times n}_{\text {skw}}$ such that

$$\begin{aligned} \int _{\Omega }|\nabla v - A|^p dx \le c(n,p,\Omega ) \int _{\Omega }|e(v)|^p dx; \end{aligned}$$

moreover, by applying Poincaré’s inequality to the function $x\mapsto v(x)-Ax$, there exists $b\in {{\mathbb {R}}}^n$ such that

$$\begin{aligned} \int _{\Omega }|v(x) - Ax-b|^p dx \le c(n,p,\Omega ) \int _{\Omega }|\nabla v - A|^p dx. \end{aligned}$$

(4.23)

We now define $a(x):= Ax+b$; then $a\in {\mathcal {R}}$. Since $\nabla v = \nabla u$ ${\mathcal {L}}^n$-a.e. on $\{v=u\}$, we have that

$$\begin{aligned} \int _{\Omega {\setminus } \omega }|\nabla u - \nabla a|^p dx&= \int _{\Omega {\setminus } \omega }|\nabla v - \nabla a|^p dx\\&\le c(n,p,\Omega ) \int _{\Omega }|e(v)|^p dx \le {{\tilde{c}}}(n,p,\Omega ) \int _{\Omega }|e(u)|^p dx, \end{aligned}$$

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 $\Omega {\setminus } \omega$, we have that

$$\begin{aligned} \int _{\Omega {\setminus } \omega }|u - a|^q dx&= \int _{\Omega {\setminus } \omega }|v - a|^q dx \\&\le c(n,p,q,\Omega ) \int _{\Omega }|e(v)|^p dx \le {{\tilde{c}}}(n,p,q,\Omega ) \int _{\Omega }|e(u)|^p dx, \end{aligned}$$

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 $C^{0,\alpha }$, with $\alpha =1-\frac{n}{p}$. $\square$

5 An approximation result

In this last section, as an application, we show an approximation result in the spirit of [11, Theorem 3.1].

Theorem 5.1

Let $n \in {{\mathbb {N}}}$ with $n \ge 2$, $p \in (1, \infty )$, and let $\Omega \subset {{\mathbb {R}}}^n$ be a bounded open set of finite perimeter. Let $\varepsilon >0$. Then, for any $u\in GSBD^p(\Omega )$, there exist

• a closed set $\Gamma$, finite union of disjoint $(n-1)$-dimensional $C^1$ manifolds with $C^1$ boundary;

• a set ${\tilde{\omega }}$, finite union of cubes;

• a set of finite perimeter ${\hat{\omega }}$;

such that

$$\begin{aligned} {\mathcal {H}}^{n-1}(J_u\triangle \Gamma ) + {\mathcal {H}}^{n-1}(\partial ^*{\tilde{\omega }})+ {\mathcal {H}}^{n-1}(\partial ^*{\hat{\omega }}) < \varepsilon . \end{aligned}$$

(5.1)

Moreover, there exists a function $w\in GSBD^p(\Omega )\cap W^{1,p}(\Omega {\setminus } (\Gamma \cup \overline{{\tilde{\omega }}}); {{\mathbb {R}}}^n)$ such that $\{w\ne u\}\subset {\tilde{\omega }}\cup {\hat{\omega }}$,

$$\begin{aligned} \int _{\Omega {\setminus }{\tilde{\omega }}}|e(w)|^p\,dx\le (1+\varepsilon ) \int _{\Omega }|e(u)|^p dx, \end{aligned}$$

and

$$\begin{aligned} {\mathcal {H}}^{n-1}(\Gamma \cap \{w^\pm \ne u^\pm \}) < \varepsilon , \end{aligned}$$

where $w^\pm$ and $u^\pm$ denote the traces of w and u on the two sides of $\Gamma$.

Corollary 5.2

Under the same assumptions and notation of Theorem 5.1, for $u\in GSBD^p(\Omega )$ the approximate gradient $\nabla u$ exists ${\mathcal {L}}^n$-a.e. in $\Omega$.

Proof of Corollary 5.2

Let $k\in {{\mathbb {N}}}$, and let ${\tilde{\omega }}_k$, ${\hat{\omega _k}}$ and $w_k$ be as in Theorem 5.1, for $\varepsilon =\frac{1}{k}$. Since $w_k\in GSBD^p(\Omega )\cap W^{1,p}(\Omega {\setminus } (\Gamma \cup \overline{{\tilde{\omega }}_k}); {{\mathbb {R}}}^n)$, we have in particular that $\nabla w_k$ exists ${\mathcal {L}}^n$-a.e. in $\Omega {\setminus }{\tilde{\omega }}_k$. Moreover, as $u=w_k$ in $\Omega {\setminus } ({\tilde{\omega }}_k\cup {\hat{\omega _k}})$, it follows that $\nabla u$ exists ${\mathcal {L}}^n$-a.e. in $\Omega {\setminus }({\tilde{\omega }}_k\cup {\hat{\omega _k}})$ (note that ${\tilde{\omega }}_k$ is a finite union of cubes, and hence its boundary is ${\mathcal {L}}^n$-negligible). By repeating this argument for every $k\in {{\mathbb {N}}}$ we have that $\nabla u$ exists ${\mathcal {L}}^n$-a.e. in $\Omega {\setminus } \omega$, where

$$\begin{aligned} \omega := \bigcap _{k\in {{\mathbb {N}}}} ({\tilde{\omega }}_k\cup {\hat{\omega _k}}). \end{aligned}$$

Since by (5.1) and Remark 3.4 we have that ${\mathcal {L}}^n ({\tilde{\omega }}_k\cup {\hat{\omega _k}}) \le C(\frac{1}{k})^{n/(n-1)}$ for every $k\in {{\mathbb {N}}}$, where $C=C(n)$, it follows that ${\mathcal {L}}^n(\omega )=0$. Hence we can conclude that $\nabla u$ exists ${\mathcal {L}}^n$-a.e. in $\Omega$. $\square$

Note that in the case $p=2$ the result in Corollary 5.2 has been obtained in [28], as a consequence of the embedding $GSBD^2(\Omega ) \subset (GBV(\Omega ))^n$ (see [28, Theorem 2.9]), for $n\ge 2$.

Theorem 5.1 will follow as a special case of the following technical proposition.

Proposition 5.3

Let $n \in {{\mathbb {N}}}$ with $n \ge 2$, and $p \in (1, \infty )$. Let $u\in GSBD^p({{\mathbb {R}}}^n)$ and let J be a countably $({\mathcal {H}}^{n-1},n-1)$ rectifiable set with $J_u\subset J$ and ${\mathcal {H}}^{n-1}(J)<+\infty$. Let $\varepsilon >0$. Then there exist

• a closed set $\Gamma$, finite union of disjoint $(n-1)$-dimensional $C^1$ manifolds with $C^1$ boundary;

• a set ${\tilde{\omega }}$, finite union of cubes;

• a set of finite perimeter ${\hat{\omega }}$;

• a function $w \in GSBD^p({{\mathbb {R}}}^n)$, with $w\in W^{1,p}(B_R(0){\setminus } (\Gamma \cup \overline{{\tilde{\omega }}}); {{\mathbb {R}}}^n)$ for any $R>0$;

such that $w = u$ ${\mathcal {L}}^n$-a.e. in ${{\mathbb {R}}}^n{\setminus } ({\tilde{\omega }}\cup {\hat{\omega }})$, and

$$\begin{aligned}&{\mathcal {H}}^{n-1}(J\triangle \Gamma )\le \varepsilon \,,\\&\int _{{{\mathbb {R}}}^n{\setminus }{\tilde{\omega }}} |e(w)|^pdx \le (1+\varepsilon )\int _{{{\mathbb {R}}}^n} |e(u)|^p dx\,,\\&{\mathcal {H}}^{n-1}(\partial ^*{\tilde{\omega }})+{\mathcal {H}}^{n-1}(\partial ^*{\hat{\omega }}) \le \varepsilon \,. \end{aligned}$$

Moreover, $u^\pm (x) = w^\pm (x)$ for ${\mathcal {H}}^{n-1}$-a.e. $x\in \Gamma {\setminus } (\overline{{\tilde{\omega }}}\cup {\hat{\omega }}^{(1)}\cup \partial ^*{\hat{\omega }})$, and ${\mathcal {H}}^{n-1}(\Gamma \cap \{w^\pm \ne u^\pm \}) < \varepsilon$, where we used (h) in Sect. 2.

We recall that, for $u\in GSBD({{\mathbb {R}}}^n)$, the set $J_u$ is countably $({\mathcal {H}}^{n-1},n-1)$ rectifiable [22, Section 6] (see [25, Section 3.2.14] for the definition), so that the assumption $J_u\subset J$ is not restrictive.

Theorem 5.1 is deduced from Proposition 5.3 in the following way. Let $\Omega \subset {{\mathbb {R}}}^n$ and $u\in GSBD^p(\Omega )$ as in the assumptions of Theorem 5.1, and let ${{\tilde{u}}}$ denote the extension of u to ${{\mathbb {R}}}^n$ obtained by setting ${{\tilde{u}}}:= 0$ outside $\Omega$. Then ${{\tilde{u}}}\in GSBD^p({{\mathbb {R}}}^n)$, and by applying Proposition 5.3 to ${{\tilde{u}}}$ and $J=J_{{{\tilde{u}}}}$ we obtain the claim.

Proof of Proposition 5.3

Let u, J and $\varepsilon$ be as in the statement, and let $\rho >0$ and $\alpha >0$ be constants to be determined later. We split the proof into several steps.

Step 1: Covering the jump set Since J is countably $({\mathcal {H}}^{n-1},n-1)$ rectifiable and ${\mathcal {H}}^{n-1}(J)<+\infty$, by [25, Theorem 3.2.29] there exists a countable family $(M_k)_{k \in {\mathbb {N}}}$ of $C^1$ hypersurfaces such that

$$\begin{aligned} {\mathcal {H}}^{n-1} \left( J {\setminus } \bigcup _{k=1}^{\infty } M_k \right) = 0. \end{aligned}$$

With no loss of generality we can assume that for each $k\in {{\mathbb {N}}}$ the manifold $M_k$ is a Lipschitz graph with Lipschitz constant less than 1/4 [2, Theorem 2.76]. Then, for every $k\in {{\mathbb {N}}}$, ${\mathcal {H}}^{n-1}$-a.e. point in $J\cap M_k$ is a point of ${\mathcal {H}}^{n-1}$-density 1 both for J and $J\cap M_k$, namely

$$\begin{aligned} \lim _{r \rightarrow 0^+} \frac{{\mathcal {H}}^{n-1} (J \cap {\overline{B}}_r (x))}{\gamma _{n-1} r^{n-1}} = \lim _{r \rightarrow 0^+} \frac{{\mathcal {H}}^{n-1} ((J\cap M_k) \cap {\overline{B}}_r (x))}{\gamma _{n-1} r^{n-1}} = 1, \end{aligned}$$

for every $k\in {{\mathbb {N}}}$ and ${\mathcal {H}}^{n-1}$-a.e. $x\in J\cap M_k$. From this, it follows that for every $k\in {\mathbb {N}}$ and for ${\mathcal {H}}^{n-1}$-a.e. $x \in J\cap M_k$ there exists $\eta (\alpha , x) \in (0, \rho )$ such that

$$\begin{aligned}&|{\mathcal {H}}^{n-1}({\overline{B}}_r(x)\cap J)- \gamma _{n-1} r^{n-1}|\le \alpha \gamma _{n-1} r^{n-1},\\&|{\mathcal {H}}^{n-1}({\overline{B}}_r(x)\cap (J\cap M_k))- \gamma _{n-1} r^{n-1}|\le \alpha \gamma _{n-1} r^{n-1}, \end{aligned}$$

and

$$\begin{aligned} {\mathcal {H}}^{n-1}({\overline{B}}_r(x)\cap (J\Delta M_k))\le \alpha {\mathcal {H}}^{n-1}({\overline{B}}_r(x)\cap J), \end{aligned}$$

for every $r\le \eta (\alpha , x)$. 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 $x\in J\cap M_k$ comes from $M_k$, and not from the other components $M_j$, for $j\ne k$.

Let $M:=J\cap \cup _k M_k$. Note that the family $\left\{ \overline{B}_r(x): x \in M, r\le \eta (\alpha , x) \right\}$ is a fine cover of M (see [2, Section 2.4]). Then, applying the Vitali-Besicovitch’s Covering Theorem [2, Theorem 2.19] to $A = M$ and , there exists a disjoint subfamily $\left\{ \overline{B}_{r(\alpha ,x)}(x): x \in M' \right\}$, for some $M'\subset M$ and $r(\alpha ,x)\le \eta (\alpha , x)$, such that

$$\begin{aligned} {\mathcal {H}}^{n-1} \bigg ( J {\setminus } \bigcup _{x \in M'} \overline{B}_{r (\alpha , x)} (x) \bigg ) =0. \end{aligned}$$

Moreover, the subfamily above is countable, since it is composed of disjoint sets with nonempty interior. Hence, there exists a sequence $\{ x_i \}_{i \in {\mathbb {N}}} \subset \cup _k M_k$ such that

$$\begin{aligned} {\mathcal {H}}^{n-1} \bigg ( J {\setminus } \bigcup _{i \in {\mathbb {N}}} \overline{B}_i \bigg ) =0, \end{aligned}$$

where $B_i:= {B}_{r_i} (x_i)$ for every $i \in {\mathbb {N}}$, and where we set $r_i := r (\alpha , x_i)$. Finally, note that from the identity above it follows that there exists $N = N (\alpha ) \in {\mathbb {N}}$ such that

$$\begin{aligned} {\mathcal {H}}^{n-1} \bigg ( J {\setminus } \bigcup _{i =1}^{N} \overline{B}_i \bigg ) < \alpha . \end{aligned}$$

(5.2)

Given $i\in \{1,\dots , N\}$, let $k(i)\in {{\mathbb {N}}}$ be such that $x_i\in M_{k(i)}$ and define $\Gamma _i:= M_{k(i)}\cap \overline{B}_i$. Then $B_i{\setminus } \Gamma _i$ has two (Lipschitz) connected components, and the following properties are satisfied:

• a) $\Gamma _i$ is a Lipschitz graph with constant less than 1/4;

• b) $|{\mathcal {H}}^{n-1}({\overline{B}}_r(x_i)\cap J)-\gamma _{n-1} r^{n-1}|\le \alpha \gamma _{n-1} r^{n-1}$ for all $r\le r_i$;

• c) ${\mathcal {H}}^{n-1}({\overline{B}}_i\cap (J\triangle \Gamma _i)) \le \alpha {\mathcal {H}}^{n-1}({\overline{B}}_i\cap J)$;

• d) ${\mathcal {H}}^{n-1}\left( J{\setminus }\bigcup _{i=1}^{N} \Gamma _i\right) \le \alpha (1+ {\mathcal {H}}^{n-1}(J))$;

• e) ${\mathcal {L}}^n\left( \bigcup _{i=1}^{ N} {\overline{B}}_i\right) \le \frac{\gamma _n}{\gamma _{n-1}}\frac{\rho }{1-\alpha } {\mathcal {H}}^{n-1}(J)$.

Properties a), b), c) follow immediately. We now prove property d). First, note that

$$\begin{aligned} J{\setminus }\bigcup _{i=1}^{N} \Gamma _i = \left( J{\setminus }\bigcup _{i=1}^{N} {\overline{B}}_i\right) \cup \left( \bigcup _{i=1}^{N} J\cap ({\overline{B}}_i{\setminus } \Gamma _i)\right) . \end{aligned}$$

Hence, by (5.2) and by property c)

$$\begin{aligned} {\mathcal {H}}^{n-1}\left( J{\setminus }\bigcup _{i=1}^{N} \Gamma _i\right)&<\alpha + {\mathcal {H}}^{n-1} \left( \bigcup _{i=1}^{N} J\cap ({\overline{B}}_i{\setminus } \Gamma _i)\right) \\&\le \alpha + \sum _{i=1}^N {\mathcal {H}}^{n-1}\left( J\cap ({\overline{B}}_i{\setminus } \Gamma _i)\right) \le \alpha (1+ {\mathcal {H}}^{n-1}(J)), \end{aligned}$$

which shows d). To see e) note that, since the closed balls are disjoint,

$$\begin{aligned} {\mathcal {L}}^n\bigg (\bigcup _{i=1}^{N} B_i\bigg )&= \sum _{i = 1}^{N} \gamma _n r_i^n \le \frac{\gamma _n \rho }{\gamma _{n-1}} \sum _{i = 1}^{N} \gamma _{n-1} r_i^{n-1} \\&\le \frac{\gamma _n}{\gamma _{n-1}} \frac{\rho }{1-\alpha } \sum _{i = 1}^{N} {\mathcal {H}}^{n-1}({\overline{B}}_i\cap J) \le \frac{\gamma _n}{\gamma _{n-1}} \frac{\rho }{1-\alpha } {\mathcal {H}}^{n-1} (J), \end{aligned}$$

where we have also used b).

Finally, letting $\Gamma := \bigcup _{i=1}^{N} \Gamma _i$, one has that $\Gamma$ is a finite union of disjoint $C^1$ manifolds with $C^1$ boundary. Moreover, thanks to c) and d),

$$\begin{aligned} {\mathcal {H}}^{n-1}(J\triangle \Gamma ) \le \alpha (1+2{\mathcal {H}}^{n-1}(J)). \end{aligned}$$

(5.3)

Step 2: Cleaning the jump set in the balls $B_i$. We split this step into further substeps.

Step 2.1: Application of Theorem 4.1in the balls $B_i$. Let us denote $B_i^+$, $B_i^-$ the connected components of $B_i{\setminus } \Gamma _i$. Thanks to Theorem 4.1, in each $B_i^\pm$, $i=1,\dots , {N}$, there exists a set of finite perimeter $\omega _i^\pm$ and a function $v_i^\pm \in W^{1,p}(B_i^\pm ; {{\mathbb {R}}}^n)$ such that

$$\begin{aligned} v_i^\pm&= u \quad \text { in } B_i^\pm {\setminus } \omega _i^\pm , \end{aligned}$$

(5.4)

$$\begin{aligned} \int _{B_i^\pm }|e(v_i^\pm )|^p dx&\le c_i^\pm \int _{B_i^\pm }|e(u)|^p dx \end{aligned}$$

(5.5)

$$\begin{aligned} {\mathcal {H}}^{n-1}(\partial ^*\omega _i^\pm )&\le c_i^\pm {\mathcal {H}}^{n-1}(J\cap B_i^\pm ), \end{aligned}$$

(5.6)

where $c_i^\pm =c_i^\pm (n,p)$.

Step 2.2: The constant $c:=\max \{c_i^\pm : i=1,\dots ,N\}$ is bounded uniformly in N (and hence in $\alpha$). First we note that, due to the invariance of $c_i^\pm$ under uniform rescalings of the domain, it is not restrictive to assume that $B_i$ has unit radius for every $i=1,\dots , N$.

Now, let $i=1,\dots ,N$ be fixed, and consider for instance the set $B_i^+$. Since $B_i^+$ is a Lipschitz set, there exist ${{\hat{r}}}_i^+$ and ${{\hat{L}}}_i^+$ such that for every point of $\partial B_i^+$ we can construct a cylinder with radius $2{{\hat{r}}}_i^+$ and half-height $4{{\hat{L}}}_i^+{{\hat{r}}}_i^+$ where $\partial B_i^+$ is the graph of an ${{\hat{L}}}_i^+$-Lipschitz function. Since $\Gamma _i$ is a Lipschitz graph with constant less than 1/4, a careful construction shows that one can find ${{\hat{r}}}_i^+={{\hat{r}}}$ and ${{\hat{L}}}_i^+={{\hat{L}}}$ independent of the particular manifold $\Gamma _i$. Finally, let ${{\hat{N}}}_i^+\in {\mathbb {N}}$ be given by point (ii) of Remark 4.3, for the domain $B_i^+$. By the same remark it follows that ${{\hat{N}}}_i^+$ satisfies the estimates

$$\begin{aligned} {\left\{ \begin{array}{ll} {{\hat{N}}}_i^+{{\hat{r}}}^{n-1} \le (5^{n-1}/\gamma _{n-1}) {\mathcal {H}}^{n-1}(\partial B_i^+) \le (5^{n-1}/\gamma _{n-1}) C_2(n),\\ C_1(n)\le {\mathcal {H}}^{n-1}(\partial B_i^+)\le {{\hat{N}}}_i^+ \gamma _{n-1} {{\hat{r}}}^{n-1}\sqrt{1+{{\hat{L}}}^2}, \end{array}\right. } \end{aligned}$$

where we have used that

$$\begin{aligned} C_1(n)&:= {\mathcal {H}}^{n-1} ({{\mathbb {S}}}^{n-1}\cap \{x_n \ge 1/4\}) \le {\mathcal {H}}^{n-1} (\partial B_i^+)\\&\quad \le {\mathcal {H}}^{n-1} ({{\mathbb {S}}}^{n-1}) + {\mathcal {H}}^{n-1} (\Gamma _i)\le n\gamma _n + \gamma _{n-1}\sqrt{1+(1/4)^2}=:C_2(n). \end{aligned}$$

From this it follows that ${{\hat{N}}}^+_i$ can be chosen to be depending only on n.

In conclusion, $({{\hat{N}}}_i^+,{{\hat{r}}}_i^+,{{\hat{L}}}_i^+)$ can be chosen uniformly in i. Since, by Remark 4.3, the constant $c_i^+$ depends on $B_i^+$ only via $({{\hat{N}}}_i^+,{{\hat{r}}}_i^+,{{\hat{L}}}_i^+)$, we finally conclude that the constant $c:=\max \{c_i^\pm : i=1,\dots ,N\}$ can be bounded uniformly in N, and hence, as $N=N(\alpha )$, uniformly in $\alpha$. In particular, in (5.4)–(5.6), we can replace $c_i^\pm$ with the uniform constant c.

Step 2.3: Conclusion. Thanks to b) and c) in Step 1, we have that

$$\begin{aligned} {\mathcal {H}}^{n-1}(J\cap B_i^\pm ) \le {\mathcal {H}}^{n-1}({\overline{B}}_i\cap (J\triangle \Gamma _i)) \le \alpha {\mathcal {H}}^{n-1}({\overline{B}}_i\cap J) \le \alpha (1+\alpha ) \gamma _{n-1}r_i^{n-1}, \end{aligned}$$

and hence from (5.6) and Step 2.2, ${\mathcal {H}}^{n-1}(\partial ^*\omega _i^\pm )\le c \, \alpha (1+\alpha ) \gamma _{n-1} r_i^{n-1}$. (Note that, in the case where $\omega _i^\pm =B_i^\pm$, we can simply let $v_i^\pm =0$; however by choosing $\alpha >0$ small enough we can assume with no loss of generality that this does not happen.)

It follows that on $\partial B_i$ the trace of each $v_i^\pm$ coincides with the trace of u, except on a set of total measure at most $2 c \,\alpha (1+\alpha )\gamma _{n-1}r_i^{n-1}$. Let now

$$\begin{aligned} v(x):= {\left\{ \begin{array}{ll} v_i^\pm (x) &{} if x\in B_i^\pm \,, i=1,\dots ,{N} \,, \\ u(x) &{} if x \in {{\mathbb {R}}}^n{\setminus } \bigcup _{i=1}^N B_i. \end{array}\right. } \end{aligned}$$

Then $v\in GSBD^p({{\mathbb {R}}}^n)$, and we have the following properties:

• 1) $J_v\cap B_i\subset \Gamma _i$ for each $i=1,\dots ,{N}$;

• 2) $\displaystyle \sum _{i=1}^{N} {\mathcal {H}}^{n-1}( J_v\cap \partial B_i) \le \alpha (1+2c){\mathcal {H}}^{n-1}(J)$;

• 3) ${\mathcal {H}}^{n-1}\left( J_v{\setminus }\Gamma \right) \le \alpha (1 + (1+2c){\mathcal {H}}^{n-1}(J))$.

Property 1) follows from the definition of v. For property 2), note that by c) and (5.6)

$$\begin{aligned} \sum _{i=1}^N {\mathcal {H}}^{n-1}\left( J_v\cap \partial B_i\right)&\le \sum _{i=1}^N{\mathcal {H}}^{n-1}\left( J_u\cap \partial B_i\right) + \sum _{i=1}^N{\mathcal {H}}^{n-1}\left( \partial B_i\cap (\partial ^*\omega ^+_i\cup \partial ^*\omega ^-_i)\right) \\&\qquad \le \sum _{i=1}^N{\mathcal {H}}^{n-1}\left( J\cap ({\overline{B}}_i{\setminus } \Gamma _i)\right) + \sum _{i=1}^N\left( {\mathcal {H}}^{n-1}(\partial ^*\omega ^+_i)+{\mathcal {H}}^{n-1}(\partial ^*\omega ^-_i)\right) \\&\qquad \le \alpha {\mathcal {H}}^{n-1}(J)+\sum _{i=1}^N\left( {\mathcal {H}}^{n-1}(\partial ^*\omega ^+_i)+{\mathcal {H}}^{n-1}(\partial ^*\omega ^-_i)\right) \nonumber \\&\qquad \le \alpha {\mathcal {H}}^{n-1}(J)+ 2 c\sum _{i=1}^N \left( {\mathcal {H}}^{n-1}(J\cap B_i^+)+ {\mathcal {H}}^{n-1}(J\cap B_i^-)\right) \nonumber \\&\qquad \le \alpha {\mathcal {H}}^{n-1}(J)+ 2 c\sum _{i=1}^N \left( {\mathcal {H}}^{n-1}((J\cap B_i){\setminus } \Gamma _i)\right) \nonumber \\&\qquad \le \alpha (1 +2c){\mathcal {H}}^{n-1}(J). \end{aligned}$$

Let us show property 3). By (5.2), 1) and 2) we have that

$$\begin{aligned} {\mathcal {H}}^{n-1}\left( J_v{\setminus } \Gamma \right)&\le {\mathcal {H}}^{n-1}\left( J{\setminus }\bigcup _{i=1}^N{\overline{B}}_i\right) + \sum _{i=1}^N {\mathcal {H}}^{n-1}\left( J_v\cap \partial B_i\right) \\&\le \alpha (1 + (1+2c){\mathcal {H}}^{n-1}(J)). \end{aligned}$$

Moreover, letting $\omega _B:= \bigcup _{i=1}^N(\omega _i^+\cup \omega _i^-)$, one has that $v=u$ ${\mathcal {L}}^n$-a.e. in ${{\mathbb {R}}}^n{\setminus }\omega _B$ and, by (5.6), ${\mathcal {H}}^{n-1}(\partial ^*\omega _B)\le 2c\,\alpha {\mathcal {H}}^{n-1}(J)$.

Step 3: Cleaning the jump set in the rest of the domain. We now pick $\delta >0$ with

$$\begin{aligned} 0< \delta \le 0.8 \alpha \Big (\min _{i=1,\dots ,N} r_i\Big )/(2\sqrt{n}), \end{aligned}$$

and consider the covering of ${{\mathbb {R}}}^n{\setminus }\bigcup _{i=1}^N B_i$ made of:

• the family ${\mathcal {Q}}_1$ of cubes $\delta z + [0,\delta ]^{n}$, $z\in {{\mathbb {Z}}}^n$, which intersect ${{\mathbb {R}}}^n{\setminus }\bigcup _{i=1}^N B_i$;

• the family ${\mathcal {Q}}_2$ of cubes $\delta z + [0,\delta ]^{n}$, $z\in {{\mathbb {Z}}}^n$, which are not in ${\mathcal {Q}}_1$, but intersect some cubes in ${\mathcal {Q}}_1$.

We set ${\mathcal {Q}} = {\mathcal {Q}}_1\cup {\mathcal {Q}}_2$. For each $q\in {\mathcal {Q}}$, we denote with $q\subset q'\subset q''$ the concentric cubes $q'$ and $q''$ with edges $(9/8)\delta$ and $(10/8)\delta$, respectively; we also denote with $\ell<\ell '<\ell ''$ the lengths of the edges of q, $q'$ and $q''$, respectively, so that in particular, $\ell '=(1-0.1)\ell ''$. For each i, letting $B'_i:=B_{(1-\alpha )r_i}(x_i)$, we observe that, since $\Gamma _i$ are equi-Lipschitz with constant less than $\frac{1}{2}$, one has

$$\begin{aligned} {\mathcal {H}}^{n-1}\big (\Gamma _i\cap (B_i{\setminus } B'_i)\big ) \le c\,\alpha r_i^{n-1} \le c\,\alpha {\mathcal {H}}^{n-1}(J\cap {\overline{B}}_i) \end{aligned}$$

(5.7)

for some constant $c=c(n)$, where in the last inequality we used property b). Hence, since by the definition of $\delta$ we have that $q''\cap B_i'=\emptyset$ for each $q\in {\mathcal {Q}}$ and for every i, we have that

$$\begin{aligned} J_v \cap \bigcup _{q\in {\mathcal {Q}}} q'' = \left( J_v {\setminus } \left( \bigcup _{i=1}^N {\overline{B}}_i\right) \right) \cup \left( \bigcup _{i=1}^N J_v \cap ( B_i {\setminus } B'_i)\right) \cup \left( \bigcup _{i=1}^N (J_v \cap \partial B_i)\right) . \end{aligned}$$

Then, recalling 1), and using (5.2), (5.7) and 2), we have

$$\begin{aligned} {\mathcal {H}}^{n-1}\bigg (J_v \cap \bigcup _{q\in {\mathcal {Q}}} q''\bigg )\le \alpha (1+ (1+3c){\mathcal {H}}^{n-1}(J)), \end{aligned}$$

(5.8)

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 $\varepsilon =1$ (which thus needs not be the $\varepsilon$ of the statement), and $\sigma =0.1$, and find constants $C=C(n,p)$ and $\tau =\tau (n,p)$ satisfying the thesis of the theorem.

Let ${\mathcal {Q}}_g\subset {\mathcal {Q}}$ denote the set of cubes q such that ${\mathcal {H}}^{n-1}(J_v\cap q'')\le \tau \delta ^{n-1}$, let ${\mathcal {Q}}_b:= {\mathcal {Q}}{\setminus }{\mathcal {Q}}_g$, and ${\tilde{\omega }}:=\bigcup _{q\in {\mathcal {Q}}_b}q$. Since for $q\in {\mathcal {Q}}_b$ one has ${\mathcal {H}}^{n-1}(J_{ v}\cap q'') > \tau \delta ^{n-1}$, there can be only a finite number of such cubes. Moreover, thanks to (5.8) we have that

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {H}}^{n-1}(\partial ^*{\tilde{\omega }}) \le \frac{c(n)}{\tau }\alpha (1+ (1+3c){\mathcal {H}}^{n-1}(J)),\\ {\mathcal {L}}^n({\tilde{\omega }})\le \frac{c(n)}{\tau }\,\delta \,\alpha (1+ (1+3c){\mathcal {H}}^{n-1}(J)). \end{array}\right. } \end{aligned}$$

(5.9)

Now, let $q\in {\mathcal {Q}}_g$. By Theorem 3.2 there exist $w_q\in GSBD^p(q'')\cap W^{1,p}(q';{{\mathbb {R}}}^n)$ and $\omega _q\subset q''$, with $w_q=v$ in $q''{\setminus } \omega _q$, and

$$\begin{aligned}&\int _{q''}|e(w_q)|^pdx \le 2\int _{q''}|e(v)|^p dx,\nonumber \\&\int _{\omega _q}|e(w_q)|^pdx \le 2\int _{\omega _q}|e(v)|^p dx, \end{aligned}$$

(5.10)

$$\begin{aligned}&{\mathcal {H}}^{n-1}(\partial ^*\omega _q)\le C{\mathcal {H}}^{n-1}(J_v\cap q''), \end{aligned}$$

(5.11)

where (5.10) follows by Remark 3.5.

Possibly reducing $\tau$, we may assume that if $q'\cap \Gamma _i\ne \emptyset$ for some $i=1,\dots ,N$, then ${\mathcal {H}}^{n-1}(\Gamma _i\cap q'')\ge \tau \delta ^{n-1}$ (see point a) in Step 1), so that $q\not \in {\mathcal {Q}}_g$. It then follows that for any $q\in {\mathcal {Q}}_g$, when $q'\subset B_i$ for some i (or more precisely $q'\subset B_i^\pm$, since $q\in {\mathcal {Q}}_g$ is such that $q'$ does not intersect $\Gamma _i$), then $w_q=v$ in $q'$.

We now ‘glue’ the functions $w_q$ in order to find a global $W^{1,p}_{\text {loc}}$ function as in the claim of the theorem. To do so, we introduce a cut-off function $\psi \in C_c^\infty ((-9/16,9/16)^n;[0,1])$ with $\eta =1$ on $[-1/2,1/2]^n$. Then for each $q\in {\mathcal {Q}}_g$, with center $c_q$, we define $\psi _q(x):= \psi ((x-c_q)/\delta )\in C_c^\infty (q';[0,1])$, so that $\psi _q=1$ on q. We then let, for $x\in G:= \bigcup _{q\in {\mathcal {Q}}_g} q$, $\varphi _q(x):= \psi _q(x)/(\sum _{{{\hat{q}}}\in {\mathcal {Q}}_g} \psi _{{{\hat{q}}}}(x)) \in [0,1]$, and

$$\begin{aligned} w(x) := {\left\{ \begin{array}{ll} \sum _{q\in {\mathcal {Q}}_g} w_q(x)\varphi _q(x) &{} if x\in G\,,\\ 0 &{} if x\in {\tilde{\omega }}\,,\\ v(x) &{} if x\in {{\mathbb {R}}}^n{\setminus } (G\cup {\tilde{\omega }}). \end{array}\right. } \end{aligned}$$

(5.12)

By construction we have that $w\in W^{1,p}(B_R(0){\setminus } (\Gamma \cup \overline{{\tilde{\omega }}}); {{\mathbb {R}}}^n)$ for any $R>0$. Indeed, we observe that ${{\mathbb {R}}}^n{\setminus } (G\cup {\tilde{\omega }})\subset \bigcup _i B_i$, and hence (by the definition of v), in this set the function w is Sobolev outside $\Gamma$. Moreover, w does not jump on the intersection between the boundaries of G and ${{\mathbb {R}}}^n{\setminus } (G\cup {\tilde{\omega }})$. Indeed, if $q\in {\mathcal {Q}}_g$ is any cube touching the set ${{\mathbb {R}}}^n{\setminus } (G\cup {\tilde{\omega }})$, then it has to be that $q\in {\mathcal {Q}}_2$ and $q\subset B_i$ for some i (and therefore, as observed before, $w_q=v$ in $q'$).

Let $\omega _G:=\bigcup _{q\in {\mathcal {Q}}_g} \omega _q$ and ${\hat{\omega }}:=\omega _B\cup \omega _G$. Then, $w=u$ in ${{\mathbb {R}}}^n{\setminus } ({\tilde{\omega }}\cup {\hat{\omega }})$, since $w=v$ outside $\omega _G\cup {\tilde{\omega }}$, and $v=u$ outside $\omega _B$. Hence $e(w)=e(u)$ in that set.

Step 3.1: Traces of w on $\Gamma$. We now compare the traces of w and of u on the two sides of $\Gamma$. We have already observed that $w=u$ in ${{\mathbb {R}}}^n{\setminus } ({\tilde{\omega }}\cup {\hat{\omega }})$, where ${\hat{\omega }} = \omega _B\cup \omega _G$.

Note that, since $q''\cap B_i'=\emptyset$ for every $q\in {\mathcal {Q}}$ and for every $i=1,\dots ,N$, $(\overline{{\tilde{\omega }}}\cup \overline{\omega _G}) \cap \bigcup _i B_i'=\emptyset$. Hence the exceptional sets ${\tilde{\omega }}$ and $\omega _G$ affect the traces of w only on a subset of $\Gamma$ of small (in terms of $\alpha$) ${\mathcal {H}}^{n-1}$-measure, by (5.7), namely

$$\begin{aligned} {\mathcal {H}}^{n-1}(\Gamma \cap \{w^\pm \ne v^\pm \})\le c\alpha {\mathcal {H}}^{n-1}(J). \end{aligned}$$

(5.13)

For the set $\omega _B$ we observe that, by the definition of v and by (5.4), for every $i=1,\dots ,N$, $v^\pm (x)=u^\pm (x)$ for ${\mathcal {H}}^{n-1}$-a.e. $x\in \Gamma _i{\setminus } (\partial ^*\omega ^+_i\cup \partial ^*\omega _i^-)$. Hence

$$\begin{aligned} v^\pm (x)=u^\pm (x) \text { for } {\mathcal {H}}^{n-1}\text {- a.e. } x\in \Gamma {\setminus } \bigcup _{i=1}^N (\partial ^*\omega ^+_i\cup \partial ^*\omega _i^-). \end{aligned}$$

By the estimates of ${\mathcal {H}}^{n-1}(\partial ^*\omega ^\pm _i)$ at the end of Step 2 it follows that the traces of u and v on the two sides of $\Gamma$ can only differ on a small (in terms of $\alpha$) portion of $\Gamma$, namely

$$\begin{aligned} {\mathcal {H}}^{n-1}(\Gamma \cap \{v^\pm \ne u^\pm \})\le 2c\alpha {\mathcal {H}}^{n-1}(J). \end{aligned}$$

(5.14)

In conclusion, $w^\pm = u^\pm$ in $\Gamma$, up to a set of small (in terms of $\alpha$) ${\mathcal {H}}^{n-1}$ measure. More precisely,

$$\begin{aligned} w^\pm (x)=u^\pm (x) \text { for } {\mathcal {H}}^{n-1}\text {- a.e. } x\in \Gamma {\setminus } \left( \overline{{\tilde{\omega }}}\cup {\hat{\omega }}^{(1)}\cup \partial ^*{\hat{\omega }}\right) , \end{aligned}$$

(5.15)

and from (5.13) and (5.14)

$$\begin{aligned} {\mathcal {H}}^{n-1}(\Gamma \cap \{w^\pm \ne u^\pm \})\le 3c\alpha {\mathcal {H}}^{n-1}(J). \end{aligned}$$

(5.16)

Step 3.2: Estimate of ${\hat{\omega }}$. We now estimate the exceptional set ${\hat{\omega }}$, both in perimeter and in volume. Note that, by (5.11) and (5.8), ${\mathcal {H}}^{n-1}(\partial ^*\omega _G)\le C\sum _{q\in {\mathcal {Q}}_g} {\mathcal {H}}^{n-1}(J_v\cap q'')\le c\alpha (1+{\mathcal {H}}^{n-1}(J))$, for some constant $c=c(n,p)$. We also remark that for each q, one has ${\mathcal {L}}^n(\omega _{q})\le c\delta {\mathcal {H}}^{n-1}(\partial ^*\omega _q)$ for a dimensional constant c, so in particular

$$\begin{aligned} {\mathcal {L}}^n(\omega _G)\le c\delta \alpha (1+{\mathcal {H}}^{n-1}(J)), \end{aligned}$$

(5.17)

with $c=c(n,p)$.

Combining these estimates with the bound on ${\mathcal {H}}^{n-1}(\partial ^*\omega _B)$ at the end of Step 2, we have that

$$\begin{aligned} {\mathcal {H}}^{n-1}(\partial ^*{\hat{\omega }})\le c\alpha (1+{\mathcal {H}}^{n-1}(J)), \end{aligned}$$

(5.18)

for a constant $c=c(n,p)$.

Step 3.3: $L^p$-estimate of e(w). We start by estimating $\int _{\omega _G} |e(w)|^pdx$. From (5.12) we have, for $x\in G$,

$$\begin{aligned} e(w)(x) = \sum _{q\in {\mathcal {Q}}_g} \big (e(w_q)(x)\varphi _q(x) + w_q(x)\odot \nabla \varphi _q(x)\big ). \end{aligned}$$

(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) separately. For the first term we have that

$$\begin{aligned} \int _{\omega _G} \bigg |\sum _{q\in {\mathcal {Q}}_g} e(w_q)(x)\varphi _q(x)\bigg |^p dx&\le \sum _{{{\hat{q}}}\in {\mathcal {Q}}_g} \int _{\omega _{{{\hat{q}}}}} \bigg |\sum _{q\in {\mathcal {Q}}_g} e(w_q)(x)\varphi _q(x)\bigg |^p dx \nonumber \\&\le c\sum _{{{\hat{q}}}\in {\mathcal {Q}}_g} \sum _{ \begin{array}{c} q\in {\mathcal {Q}}_g\\ q'\cap {{\hat{q}}}'\ne \emptyset \end{array} }\int _{\omega _{{{\hat{q}}}}\cap q'} |e(w_q)(x)|^p dx. \end{aligned}$$

(5.20)

For fixed ${{\hat{q}}}$ and q with $q'\cap {{\hat{q}}}'\ne \emptyset$ we estimate

$$\begin{aligned} \int _{\omega _{{{\hat{q}}}}\cap q'} |e(w_q)(x)|^p dx&= \int _{(\omega _{{{\hat{q}}}}\cap \omega _{q})\cap q'} |e(w_q)(x)|^p dx + \int _{(\omega _{{{\hat{q}}}}{\setminus } \omega _{q})\cap q'} |e(v)(x)|^p dx\\&\le \int _{\omega _{q}} |e(w_q)(x)|^p dx + \int _{\omega _{{{\hat{q}}}}} |e(v)(x)|^p dx\\&\le 2\int _{\omega _{q}} |e(v)(x)|^p dx + \int _{\omega _{{{\hat{q}}}}} |e(v)(x)|^p dx, \end{aligned}$$

where in the last step we used (5.10). Since the cubes $q'$ have finite overlap, from (5.20) we conclude that

$$\begin{aligned} \int _{\omega _G} \bigg |\sum _{q\in {\mathcal {Q}}_g} e(w_q)(x)\varphi _q(x)\bigg |^p dx \le c \int _{\omega _G} |e(v)|^p\,dx. \end{aligned}$$

Therefore,

$$\begin{aligned} \int _{\omega _G} \bigg |\sum _{q\in {\mathcal {Q}}_g} e(w_q)(x)\varphi _q(x)\bigg |^p dx&\le c \int _{\omega _G} |e(v)|^p\,dx \nonumber \\&=c \bigg (\int _{\omega _G{\setminus } \cup _i B_i} |e(v)|^p\,dx + \sum _{i=1}^N \int _{\omega _G \cap B_i}|e(v)|^p dx \bigg ) \nonumber \\&\le c\bigg (\int _{\omega _G} |e(u)|^p dx + \sum _{i=1}^N \int _{B_i}|e(u)|^p dx\bigg ), \end{aligned}$$

(5.21)

where in the last inequality we have used the definition of v, and in particular the fact that $v=u$ outside $\cup _iB_i$, and the estimate of $e(v_i^\pm )$ in terms of e(u) (see Step 2).

We now estimate the second term of the sum in (5.19). For $x\in G$ we define ${\mathcal {Q}}_g^x:=\{q\in {\mathcal {Q}}_g: \varphi _q(x)>0\}$, and denote $N^x_{{\mathcal {Q}}}:=\#{\mathcal {Q}}_g^x$ (which, as already observed, is uniformly bounded by a quantity depending only on the dimension, namely $2^n$). Using that $\sum _{q\in {\mathcal {Q}}_g^x} \nabla \varphi _q(x)=0$, one has

$$\begin{aligned} \sum _{q\in {\mathcal {Q}}_g} w_q(x)\odot \nabla \varphi _q(x)&= \sum _{q\in {\mathcal {Q}}_g^x}\bigg (w_{q}(x)- \tfrac{1}{N^x_{{\mathcal {Q}}}}\sum _{{{\hat{q}}}\in {\mathcal {Q}}_g^x}w_{{{\hat{q}}}}(x)\bigg )\odot \nabla \varphi _{q}(x) \\&= \frac{1}{N^x_{{\mathcal {Q}}}}\sum _{q,{{\hat{q}}}\in {\mathcal {Q}}_g^x} (w_{q}(x)-w_{{{\hat{q}}}}(x))\odot \nabla \varphi _{q}(x). \end{aligned}$$

Since $q,{{\hat{q}}}\in {\mathcal {Q}}_g^x\Rightarrow x\in q'\cap {{\hat{q}}}'$, to bound the $L^p$ norm of the above expression, it is enough to estimate

$$\begin{aligned} \int _{q'\cap {{\hat{q}}}'} |w_{q}-w_{{{\hat{q}}}}|^p|\nabla \varphi _{q}|^p dx \end{aligned}$$

(5.22)

for any pair of neighbouring cubes $q,{{\hat{q}}}\in {\mathcal {Q}}^x_g$. Note that $w_{q}-w_{{{\hat{q}}}}\in W^{1,p}(q'\cap {{\hat{q}}}';{{\mathbb {R}}}^n)$ and $w_{q}-w_{{{\hat{q}}}}=0$ in $(q'\cap {{\hat{q}}}'){\setminus } (\omega _{q}\cup \omega _{{{\hat{q}}}})$, since both functions coincide with v. Moreover, since ${\mathcal {L}}^n(q'\cap {{\hat{q}}}')\ge \delta ^n/8^n$ and ${\mathcal {L}}^n(\omega _{q}\cup \omega _{{{\hat{q}}}})\le C\tau ^{n/(n-1)}\delta ^n$ for some dimensional constant C, provided $\tau$ is chosen small enough one can ensure that

$$\begin{aligned} {\mathcal {L}}^n\big (\{ x\in q'\cap {{\hat{q}}}': w_{q}-w_{{{\hat{q}}}}= 0\}\big ) \ge \frac{1}{2}{\mathcal {L}}^n(q'\cap {{\hat{q}}}'). \end{aligned}$$

One can then easily deduce from Lemma 4.4 that, for some constant c (depending on p and on the dimension):

$$\begin{aligned} \int _{q'\cap {{\hat{q}}}'} |w_{q}-w_{{{\hat{q}}}}|^p dx \le c\delta ^p \int _{q'\cap {{\hat{q}}}'} |e(w_{q}-w_{{{\hat{q}}}})|^p dx \le c\delta ^p \int _{\omega _{q}\cup \omega _{{{\hat{q}}}}} |e(w_{q}-w_{{{\hat{q}}}})|^p dx. \end{aligned}$$

Since $|\nabla \varphi _q|\le C/\delta$ in each cube, we can estimate (5.22) as

$$\begin{aligned} \int _{q'\cap {{\hat{q}}}'} |w_{q}-w_{{{\hat{q}}}}|^p|\nabla \varphi _{q}|^p dx&\le c \int _{(q'\cap {{\hat{q}}}')\cap (\omega _{q}\cup \omega _{{{\hat{q}}}})} |e(w_{q}-w_{{{\hat{q}}}})|^p dx\\&\le c \left( \int _{q'\cap (\omega _{q}\cup \omega _{{{\hat{q}}}})} |e(w_{q})|^p dx + \int _{{{\hat{q}}}'\cap (\omega _{q}\cup \omega _{{{\hat{q}}}})} |e(w_{{{\hat{q}}}})|^p dx\right) \\&\le c\int _{\omega _{q}\cup \omega _{{{\hat{q}}}}} |e(v)|^p dx. \end{aligned}$$

Hence we have that

$$\begin{aligned} \int _{\omega _G} \bigg |\sum _{q\in {\mathcal {Q}}_g} w_q(x)\odot \nabla \varphi _q(x)\bigg |^p dx \le c \int _{\omega _G} |e(v)|^p \,dx \end{aligned}$$

which, together with (5.21), gives, from (5.19),

$$\begin{aligned} \int _{\omega _G} |e(w)|^p dx \le c\int _{\omega _G\cup \left( \bigcup _{i=1}^N B_i\right) } |e(u)|^p dx. \end{aligned}$$

(5.23)

Finally, we estimate $\int _{{{\mathbb {R}}}^n{\setminus } (\Gamma \cup {\tilde{\omega }})} |e(w)|^p dx$. We have

$$\begin{aligned}&\int _{{{\mathbb {R}}}^n{\setminus } (\Gamma \cup {\tilde{\omega }})} |e(w)|^p dx \le \int _{{{\mathbb {R}}}^n{\setminus } (\Gamma \cup {\tilde{\omega }}\cup {\hat{\omega }})} |e(w)|^p dx + \int _{\big ({{\mathbb {R}}}^n{\setminus } (\Gamma \cup {\tilde{\omega }})\big )\cap {\hat{\omega }}} |e(w)|^p dx\nonumber \\&\qquad \le \int _{{{\mathbb {R}}}^n} |e(u)|^p dx + \int _{({{\mathbb {R}}}^n{\setminus } {\tilde{\omega }})\cap \omega _G} |e(w)|^p dx + \int _{\big ({{\mathbb {R}}}^n{\setminus } ({\tilde{\omega }}\cup \omega _G)\big )\cap \omega _B} |e(w)|^p dx, \end{aligned}$$

(5.24)

since $w=u$ in ${{\mathbb {R}}}^n{\setminus } ({\tilde{\omega }}\cup {\hat{\omega }})$. Using that $w=v$ outside ${\tilde{\omega }}\cup \omega _G$ we have

$$\begin{aligned} \int _{\big ({{\mathbb {R}}}^n{\setminus } ({\tilde{\omega }}\cup \omega _G)\big )\cap \omega _B} |e(w)|^p dx = \int _{\big ({{\mathbb {R}}}^n{\setminus } ({\tilde{\omega }}\cup \omega _G)\big )\cap \omega _B} |e(v)|^p dx \le c \sum _{i=1}^N \int _{B_i}|e(u)|^p dx, \end{aligned}$$

(5.25)

where the last inequality follows from the definition of $\omega _B$, the fact that $v=v_i^\pm$ in $B_i^\pm$, and the bound (5.5), for $i=1,\dots ,N$.

In conclusion, from (5.24), (5.23) and (5.25) it follows that

$$\begin{aligned} \int _{{{\mathbb {R}}}^n{\setminus } (\Gamma \cup {\tilde{\omega }})} |e(w)|^p dx \le \int _{{{\mathbb {R}}}^n} |e(u)|^p dx + c\int _{\omega _G\cup \left( \bigcup _{i=1}^N B_i\right) } |e(u)|^p dx. \end{aligned}$$

As a consequence, if we recall point e) of the construction of the $B_i$’s and (5.17) above, if $\rho >0$ and $\delta >0$ are chosen small enough, one can ensure that

$$\begin{aligned} \int _{{{\mathbb {R}}}^n{\setminus } (\Gamma \cup {\tilde{\omega }})} |e(w)|^p dx \le (1+\alpha )\int _{{{\mathbb {R}}}^n} |e(u)|^p dx . \end{aligned}$$

(5.26)

(Note that for (5.26) to hold true, the choice of $\rho >0$ and $\delta >0$ makes them dependent on the function u, but $\alpha$ is independent of u.) By choosing $\alpha$ sufficiently small in (5.3), (5.9), (5.15), (5.16), (5.18) and (5.26) the conclusion follows. $\square$