Integral representation for energies in linear elasticity with surface discontinuities

In this paper we prove an integral representation formula for a general class of energies defined on the space of generalized special functions of bounded deformation ($GSBD^p$) in arbitrary space dimensions. Functionals of this type naturally arise in the modeling of linear elastic solids with surface discontinuities including phenomena as fracture, damage, surface tension between different elastic phases, or material voids. Our approach is based on the global method for relaxation devised in Bouchitt\`e et al. '98 and a recent Korn-type inequality in $GSBD^p$ (Cagnetti-Chambolle-Scardia '20). Our general strategy also allows to generalize integral representation results in $SBD^p$, obtained in dimension two (Conti-Focardi-Iurlano '16), to higher dimensions, and to revisit results in the framework of generalized special functions of bounded variation ($GSBV^p$).


Introduction
Integral representation results are a fundamental tool in the abstract theory of variational limits by Γ -convergence or in relaxation problems (see [32]). The topic has attracted widespread attention in the mathematical community over the last decades, with applications in various contexts, such as homogenization, dimension reduction, or atomistic-to-continuum approximations. In this paper we contribute to this topic by proving an integral representation result for a general class of energies arising in the modeling of linear elastic solids with surface discontinuities.
Integral representation theorems have been provided with increasing generality, ranging from functionals defined on Sobolev spaces [1,17,18,19,33,48] to those defined on spaces of functions of bounded variation [12,22,30,14], in particular on the subspace SBV of special functions of bounded variation [13,15,16] and on piecewise constant functions [2]. In recent years, this analysis has been further improved to deal with functionals and variational limits on GSBV p (generalized special functions of bounded variation with p-integrable bulk density), which is the natural energy space for the variational description of many problems with free discontinuities, see among others [6,7,8,9,21,37,41]. A very general method for dealing with all the abovementioned classes of functionals, the so-called global method for relaxation, has been developed by Bouchitté, Fonseca, Leoni, and Mascarenhas in [13,14]. It essentially consists in comparing asymptotic Dirichlet problems on small balls with different boundary data depending on the local properties of the functions and allows to characterize energy densities in terms of cell formulas.
When coming to the variational description of rupture phenomena in general linearly elastic materials, however, the functional setting to be considered becomes weaker. Indeed, problems need to be formulated in suitable subspaces of functions of bounded deformation (BD functions) for which the distributional symmetrized gradient is a bounded Radon measure.
In the mathematical description of linear elasticity, the elastic properties are determined by the elastic strain. For a solid in a (bounded) reference configuration Ω ⊂ R d , whose displacement field with respect to the equilibrium is u : Ω → R d , the elastic strain is given by the symmetrized gradient e(u) = 1 2 (∇u + (∇u) T ). In standard models, the corresponding linear elastic energy is a suitable quadratic form of e(u), possibly depending on the material point, see e.g. [38,Section 2.1]. However, this is often generalized to the case of p-growth for a power p > 1 [45,Sections 10,11]. The presence of surface discontinuities is related to several dissipative phenomena, such as cracks, surface tension between different elastic phases, or internal cavities. In the energetic description, this is represented by a term concentrated on the jump set J u . This set is characterized by the property that for x ∈ J u , when blowing up around x, the jump set approximates a hyperplane with normal ν u (x) ∈ S d−1 and the displacement field is close to two suitable values u + (x), u − (x) ∈ R d on the two sides of the material with respect to this hyperplane. Prototypical examples of functionals described above are energies which are controlled from above and below by suitable multiples of where [u](x) = u + (x) − u − (x) denotes the jump opening, or which are controlled by multiples of Griffith's energy [44]ˆΩ Whereas in case (1) the energy space is be given by SBD p , a subspace of BD, problems with control of type (2) are naturally formulated on generalized special functions of bounded deformation GSBD p , introduced by Dal Maso [31]. (We refer to Section 3.1 for more details.) The only available integral representation result in this context is due to Conti, Focardi, and Iurlano [28] who considered variational functionals controlled locally in terms of (1) in dimension d = 2. Let us mention that the behavior is quite different if linear growth on the symmetrized gradient is assumed (corresponding to p = 1), as suited for the description of plasticity. In that case, representation results in the framework of BD have been obtained, for instance, in [10,36] and [23] (see also [35,47], containing essential tools for the proof). The goal of the present article is twofold: we generalize the results of [28] for energies with control of type (1) to arbitrary space dimensions and, more importantly, we extend the theory to encompass also problems of the form (2), which are most relevant from an applicative viewpoint. Indeed, already in dimension two, the extension of [28] to the case where only a control of type (2) is available is no straightforward task. This is a fundamental difference with respect to the BV -theory where problems for generalized functions of bounded variation can be reconducted to SBV by a perturbation trick (see for instance [21]): one considers a small perturbation of the functional, depending on the jump opening, to represent functionals on SBV p . Then, by letting the perturbation parameter vanish and by truncating functions suitably, the representation can be extended to GSBV p . Unfortunately, the trick of reducing problem (2) to (1) is not expedient in the linearly elastic context and does not allow to deduce an integral representation result in GSBD p from the one in SBD p . This is mainly due to the fact that, given a control only on the symmetrized gradient, it is in principle not possible to use smooth truncations to decrease the energy up to a small error.
Let us also remark that, while in the majority of integral representation results in BV and BD the L 1 -topology was considered, this is not the right choice when only a lower bound of the form (2) is at hand. Indeed, in this case, the available compactness results [27,31] have been established with respect to the topology of the convergence in measure. This latter is also the topology where recently an integral representation result for the subspace P R(Ω) of piecewise rigid functions has been proved in [43].
In our main result (Theorem 2.1), we prove an integral representation for variational functionals F : GSBD p (Ω) × B(Ω) → [0, +∞) (B(Ω) denoting the Borel subsets of Ω) that satisfy the standard abstract conditions to be Borel measures in the second argument, lower semicontinuous with respect to convergence in measure, and local in the first argument. Moreover, we require control of type (2), localized to any B ∈ B(Ω).
Let us comment on the proof strategy. We follow the general approach of the global method for relaxation provided in [13,14] for variational functionals in BV . The proof strategy recovers the integral bulk and surface densities as blow-up limits of cell minimization formulas. The steps to be performed are the following: • one first shows that, for fixed u ∈ GSBD p (Ω), the set function F (u, ·) is asymptotically equivalent to its minimum m F (u, ·) over competitors attaining the same boundary conditions as u. With this we mean that the two quantities have the same Radon-Nikodym derivative with respect to µ := L d ⌊ Ω +H d−1 ⌊ Ju∩Ω (Lemma 4.1); • one then proves that the Radon-Nikodym derivative dmF (u,·) dµ only depends on x 0 , the value u(x 0 ), and the (approximate) gradient ∇u(x 0 ) at a Lebesgue point x 0 , while at a jump point x 0 it is uniquely determined by the one-sided traces u + (x 0 ), u − (x 0 ) and the normal vector ν u (x 0 ) to J u in x 0 (Lemmas 4.2 and 4.3).
When dealing with all of the abovementioned issues, a key ingredient is given by a Korn-type inequality for special functions of bounded deformation, established recently by Cagnetti, Chambolle, and Scardia [20], which generalizes a two-dimensional result in [28] (see also [39]) to arbitrary dimension. It provides a control of the full gradient in terms of the symmetrized gradient, up to an exceptional set whose perimeter has a surface measure comparable to that of the discontinuity set. In particular, this estimate is used to approximate the function u with functions u ε , which have Sobolev regularity in a ball (around a Lebesgue point), or in half-balls oriented by the jump normal (around a jump point), and which converge to the purely elastic competitor u(x 0 ) + ∇u(x 0 )(· − x 0 ), or the twovalued function with values u − (x 0 ) and u + (x 0 ), respectively. This is done in Lemmas 5.1 and 6.1, respectively, and is used for proving Lemmas 4.2 and 4.3. Let us mention that this application of the Korn-type inequality is similar to the one in dimension two [28] (with the topology of convergence in measure in place of L 1 ), and constitutes the counterpart of the SBV -Poincaré inequality [34] used in the SBV -case [13].
In contrast to [28], the Korn inequality is also used in the proof of Lemma 4.1: at this point, one needs to show that functions of the form v δ := v δ i χ B δ i approximate u in the topology of the convergence in measure, where B δ i is a fine cover of a given set with disjoint balls of radius smaller than δ and v δ i denote minimizers for m F (u, B δ i ). In [28], the lower bound in (1) allows to control the distributional symmetrized gradient Eu which along with a scaling argument and the classical Korn-Poincaré inequality in BD (see [49,Theorem 2.2]) shows that v δ i is close to u on each B δ i . (In [13], the SBV -Poincaré inequality is used.) Our weaker lower bound of the form (2), however, calls for novel arguments and we use the Korn-type inequality to show that v δ i are close to u in L p up to exceptional sets ω δ i whose volumes scale like δ(F (u, B δ i ) + µ(B δ i )). We also point out that, if instead a control of the type (1) is assumed, the arguments leading to Theorem 2.1 can be successfully adapted to extend the result for functionals on SBD p (see [28]) to arbitrary space dimensions, see Theorem 7.1. This is done by exploiting the stronger blow-up properties of SBD functions. We note that, in principle, this result could be also obtained by adapting the arguments in [28] to higher dimension by employing the Korn inequality [20]. We however preferred to give a self-contained proof of Theorem 7.1, which requires only slight modifications of the arguments used for Theorem 2.1 and nicely illustrates the differences between SBD p and its generalized space.
For a related purpose, we endow our paper with an appendix were we discuss how our arguments can also provide a direct proof for integral representation results on GSBV p , if a local control on the full deformation gradient of the form is given, see Theorem A.1. In particular, no perturbation or truncation arguments are needed in the proof. Therefore, we believe that this provides a new perspective and a slightly simpler approach to integral representation results in GSBV p without necessity of the perturbation trick discussed before, relying on the SBV result. Let us, however, mention that in [21] a more general growth condition from above is considered: dealing with such a condition would instead require a truncation method in the proof.
The paper is organized as follows. In Section 2 we present our main integral representation result in GSBD p . Section 3 is devoted to some preliminaries about the function space. In particular, we present the Korn-type inequality established in [20] and prove a fundamental estimate. Section 4 contains the general strategy and the proof of Lemma 4.1. The identifications of the bulk and surface density (Lemmas 4.2 and 4.3) are postponed to Sections 5 and 6, respectively. In Section 7 we describe the modifications necessary to obtain the SBD p -case. Finally, in Appendix A we explain how our method can be used to establish an integral representation result in GSBV p .

The integral representation result
In this section we present our main result. We start with some basic notation. Let Ω ⊂ R d be open, bounded with Lipschitz boundary. Let A(Ω) be the family of open subsets of Ω, and denote by B(Ω) the family of Borel sets contained in Ω. For every x ∈ R d and ε > 0 we indicate by B ε (x) ⊂ R d the open ball with center x and radius ε. For x, y ∈ R d , we use the notation x · y for the scalar product and |x| for the Euclidean norm. Moreover, we let S d−1 := {x ∈ R d : |x| = 1} and we denote by M d×d the set of d × d matrices. The m-dimensional Lebesgue measure of the unit ball in R m is indicated by γ m for every m ∈ N. We denote by L d and H k the d-dimensional Lebesgue measure and the k-dimensional Hausdorff measure, respectively.
For definition and properties of the space GSBD p (Ω), 1 < p < ∞, we refer the reader to [31]. Some relevant properties are collected in Section 3 below. In particular, the approximate gradient is denoted by ∇u (it is well-defined, see Lemma 3.4) and the (approximate) jump set is denoted by J u with corresponding normal ν u and one-sided limits u + and u − . We also define e(u) = 1 2 (∇u+(∇u) T ). We consider functionals F : GSBD p (Ω)×B(Ω) → [0, +∞) with the following general assumptions: is a Borel measure for any u ∈ GSBD p (Ω), (H 2 ) F (·, A) is lower semicontinuous with respect to convergence in measure on Ω for any A ∈ A(Ω), ) there exist 0 < α < β such that for any u ∈ GSBD p (Ω) and B ∈ B(Ω) we have We now formulate the main result of this article addressing integral representation of functionals F satisfying (H 1 )-(H 4 ). To this end, we introduce some further notation: for every u ∈ GSBD p (Ω) and A ∈ A(Ω) we define For x 0 ∈ Ω, u 0 ∈ R d , and ξ ∈ M d×d we introduce the functions ℓ x0,u0,ξ : In this paper, we will prove the following result.
for all u ∈ GSBD p (Ω) and B ∈ B(Ω), where f is given by for all x 0 ∈ Ω, u 0 ∈ R d , ξ ∈ M d×d , and g is given by for all x 0 ∈ Ω, a, b ∈ R d , and ν ∈ S d−1 .
Remark 2.2. We proceed with some remarks on the result. (i) In general, if f is not convex in ξ, in spite of the growth conditions (H 4 ), the functional may fully depend on ∇u and not just on the symmetric part e(u). We refer to [28,Remark 4.14] for an example in this direction.
(ii) As F is lower semicontinuous on W 1,p , the integrand f is quasiconvex [46]. Since F is lower semicontinuous on piecewise rigid functions, the integrand g is BD-elliptic [42] (at least if one can ensure, for instance, that g has a continuous dependence in x). A fortiori, g is BV -elliptic [3].
(iv) A variant of the proof shows that, in the minimization problems (2.4)-(2.5), one may replace balls B ε (x 0 ) by cubes Q ν ε (x 0 ) with sidelength ε, centered at x 0 , and two faces orthogonal to ν = ν u (x 0 ). (v) An analogous result holds on the space GSBV p (Ω; R m ) for m ∈ N. We refer to Appendix A for details.
We will additionally discuss the minor modifications needed in order to deal with functionals (H ′ 4 ) there exist 0 < α < β such that for any u ∈ SBD p (Ω) and B ∈ B(Ω) we have In this case, SBD p (Ω) (see Subsection 3.1) is the natural energy space for F . Furthermore, sequences of competitors with bounded energy, which are converging in measure, are additionally L 1 -convergent if we assume (H ′ 4 ), due to the classical Korn-Poincaré inequality in BD (see [49,Theorem 2.2]). Hence, in this latter case, (H 2 ) is equivalent to requiring lower semicontinuity with respect to the L 1 -convergence. The statement of the result in this setting, as well as of the changes needed in the proofs, will be given in Section 7.

Preliminaries
We start this preliminary section by introducing some further notation. For E ⊂ R d , ε > 0, and The diameter of E is indicated by diam(E). Given two sets E 1 , E 2 ⊂ R d , we denote their symmetric difference by E 1 △E 2 . We write χ E for the characteristic function of any E ⊂ R d , which is 1 on E and 0 otherwise. If E is a set of finite perimeter, we denote its essential boundary by ∂ * E, see [5,Definition 3.60]. We denote the set of symmetric and skew-symmetric matrices by M d×d sym and M d×d skew , respectively. . . , D d v) is the distributional differential. It is well known (see [4,49]) that for v ∈ BD(U ) the jump set J v is countably (H d−1 , d−1) rectifiable, and that For a complete treatment of BD and SBD functions, we refer to to [4,11,49].
The notation for e(v) and J v , which is the same as that one in the SBD case, is consistent: in fact, if v lies in SBD(U ), the objects coincide (up to negligible sets of points with respect to L d and for every ̺ > 0, and the function integrable belongs to SBD(U ), as follows from [26,Theorem 2.9] for Av = Ev (see [26,Remark 2.5]). This corresponds to the following proposition.
If U has Lipschitz boundary, for each v ∈ GBD(U ) the traces on ∂U are well defined (see [31,Theorem 5.5]), in the sense that for Korn's inequality and fundamental estimate. In this subsection we discuss two important tools which will be instrumental for the proof of Theorem 2.1. We start by the following Korn and Korn-Poincaré inequalities in GSBD for functions with small jump sets, see [20, Theorem 1.1, Theorem 1.2]. In the following, we say that a : and an infinitesimal rigid motion a such that Note that the result is indeed only relevant for functions with sufficiently small jump set, as otherwise one can choose ω = Ω, and (3.5) trivially holds. Note that, in [20], 1) has not been shown, but it readily follows from H d−1 (∂ * ω) ≤ cH d−1 (J u ) by the isoperimetric inequality. Remark 3.3 (Almost Sobolev regularity, constants, and scaling invariance). (i) More precisely, in [20] it is proved that there exists v ∈ W 1,p (Ω; R d ) such that v = u on Ω \ ω and e(v) L p (Ω) ≤ c e(u) L p (Ω) , whence by Korn's and Poincaré's inequality in for an infinitesimal rigid motion a. This directly implies (3.5), see [20,Theorem 4.1,Theorem 4.4].
(ii) Given a collection of bounded Lipschitz domains (Ω k ) k which are related through bi-Lipschitzian homeomorphisms with Lipschitz constants of both the homeomorphism itself and its inverse bounded uniformly in k, in Theorem 3.2 we can choose a constant c uniformly for all Ω k , see [20,Remark 4.2].
To control the affine mappings appearing in Theorem 3.2, we will make use of the following elementary lemma on affine mappings, see, e.g., [43,Lemma 3.4] or [29,Lemmas 4.3] for similar statements. (It is obtained by the equivalence of norms in finite dimensions and by standard rescaling arguments.) Then, there exists a constant c 0 > 0 only depending on p and θ such that We now proceed with another consequence of Theorem 3.2.
Let Ω ⊂ R d be a bounded Lipschitz domain and let 1 < p < +∞. Then there exists such that Proof. We start by choosing a bounded Lipschitz domain Ω ′ ⊂ R d with Ω ⊂⊂ Ω ′ . Each u ∈ GSBD p (Ω) with tr(u) = 0 on ∂Ω can be extended to a functionũ ∈ GSBD p (Ω ′ ) byũ = 0 on Ω ′ \ Ω such that Jũ = J u . We first note that it is not restrictive to assume that where c = c(Ω ′ , p) > 0 is the constant of Theorem 3.2. In fact, otherwise we could take ω = Ω and the statement would be trivially satisfied since (3.7) is clearly trivial and for (3.6) we use that 2c for a sufficiently large constant C > 0 depending only on Ω and Ω ′ . Now, consider a function u satisfying (3.8). We apply Theorem 3.2 onũ ∈ GSBD p (Ω ′ ) and obtain a set ω ⊂ Ω ′ ⊂ R d satisfying (3.4) as well as an infinitesimal rigid motion a such that (3.9) In particular,ũ = 0 on Ω ′ \ Ω implies In view of (3.10), we apply Lemma 3.
We conclude this subsection with another important tool in the proof of the integral representation, namely a fundamental estimate in GSBD p .
where we used the notation introduced in (3.1).
In the statement above, we intend that Proof. The proof follows the lines of [16, where u and v are extended arbitrarily outside A and A ′′ , respectively. Letting (3.14) For the last term, we compute using (H 4 ) (⊙ denotes the symmetrized vector product) Notice that we can obtain the same estimate also if F satisfies (H ′ 4 ), u ∈ SBD p (A), and v ∈ SBD p (A ′′ ). (We refer to [16, Proof of Proposition 3.1] for details.) Consequently, recalling (3.13) and using (H 1 ) we find i 0 ∈ {1, . . . , k} such that This along with (3.14) concludes the proof of (3.11) by setting w = w i0 . To see the scaling property (3.12), it suffices to use the cut-off functions . This concludes the proof.

The global method
This section is devoted to the proof of Theorem 2.1 which is based on three ingredients. First, we show that F is equivalent to m F (see (2.1)) in the sense that the two quantities have the same Radon-Nikodym derivative with respect to µ : We prove this lemma in the final part of this section. The second ingredient is that, asymptotically as ε → 0, the minimization problems We defer the proof of Lemma 4.2 to Section 5. The third ingredient is that, asymptotically as ε → 0, the minimization problems m F (u, B ε (x 0 )) and We defer the proof of Lemma 4.3 to Section 6, and now proceed to prove Theorem 2.1.
Proof of Theorem 2.1. We need to show that for L d -a.e. x 0 ∈ Ω one has where f was defined in (2.4), and that for H d−1 -a.e. x 0 ∈ J u one has where g was defined in (2.5).

By Lemma 4.1 and the fact that lim
for L d -a.e. x 0 ∈ Ω. Then, (4.3) follows from (2.4) and Lemma 4.2. By Lemma 4.1 and the fact that e. x 0 ∈ J u . Now, (4.4) follows from (2.5) and Lemma 4.3.
In the remaining part of the section we prove Lemma 4.1. We basically follow the lines of [13,14,28], with the difference that the required compactness results are more delicate due to the weaker growth condition from below (see (H 4 )) compared to [13,14,28]. We start with some notation. For δ > 0 and A ∈ A(Ω), we define Proof In view of Lemma 4.4, in order to see that F and m F have the same Radon-Nikodym derivative with respect to µ, it remains to show the following. Proof. We follow the lines of the proof of [28, Lemma 4.1] focusing on the necessary adaptions due to the weaker growth condition from below (see (H 4 )) compared to [28].
We now address the reverse inequality. We fix A ∈ A(Ω) and δ > 0. Let (B δ i ) i be balls as in the definition of m δ By the definition of m F , we find v δ i ∈ GSBD p (B δ i ) such that v δ i = u in a neighborhood of ∂B δ i and 8) where N δ,n By construction, we have that each v δ,n lies in GSBD p (Ω) and that sup n∈N ( e(v δ,n ) L p (Ω) +H d−1 (J v δ,n )) < +∞ by (4.6)-(4.7) and (H 4 ). Moreover, v δ,n → v δ pointwise a.e. in Ω. Then, [27,Theorem 1.1] yields v δ ∈ GSBD p (Ω).
where we also used the fact that µ(N δ 0 ∩ A) = F (u, N δ 0 ∩ A) = 0 by the definition of (B δ i ) i and (H 4 ). For later purpose, we also note by (H 4 ) that this implies We now claim that v δ → u in measure on A. To this end, we apply Remark 3.3(iii) and Corollary 3.6 on each B δ i for the function u − v δ i and we get sets of finite perimeter ω δ i ⊂ B δ i such that for a constant C > 0 only depending on p. Here, we used that diam(B δ i ) ≤ δ and the fact that In view of (4.8), we computê By (4.11)(ii) and the fact that the balls (B δ i ) i are pairwise disjoint we get As ω δ i ⊂ B i δ and diam(B δ i ) ≤ δ, we further get by (4.11)(i) (4.14) Now, combining (4.12)-(4.14) and using (4.10), we find´A ψ(|u − v δ |) dx → 0 as δ → 0. With this, using (H 2 ), (4.5), and (4.9) we get the required inequality m * F (u, A) ≥ F (u, A) in the limit as δ → 0. This concludes the proof. To conclude the proof of Theorem 2.1, it remains to prove Lemmas 4.2 and 4.3. This is the subject of the following two sections.

The surface density
This section is devoted to the proof of Lemma 4.3. We start by analyzing the blow-up at jump points. In the following, for any x 0 ∈ J u we adopt the notationū surf x0 for the function u x0,u + (x0),u − (x0),νu(x0) , see (2.3), with ν u (x 0 ) ∈ S d−1 and u ± (x 0 ) ∈ R d being the approximate normal to J u and the traces on both sides of J u at x 0 , respectively. Recall also notation (3.1). Lemma 6.1 (Blow-up at jump points). Let u ∈ GSBD p (Ω) and θ ∈ (0, 1). For H d−1 -a.e. x 0 ∈ J u there exists a family u ε ∈ GSBD p (B ε (x 0 )) such that where Π 0 denotes the hyperplane passing through x 0 with normal ν u (x 0 ).
We now proceed with the proof of Lemma 4.3.
We prove (4.2) for x 0 of this type.
We start by pointing out that, under (H for all u ∈ SBD p (Ω) and B ∈ B(Ω), where f is given by for all x 0 ∈ Ω, u 0 ∈ R d , ξ ∈ M d×d , and ℓ x0,u0,ξ as in (2.2), and g is given by for all x 0 ∈ Ω, a, b ∈ R d , ν ∈ S d−1 , and u x0,a,b,ν as in (2.3).
The remainder of this section is devoted to the proof of Theorem 7.1 which follows along the lines of the proof of Theorem 2.1 devised in Section 4. First, the analogue of Lemma 4.1 holds essentially with the same proof.
Proof. One can follow the same argument used to prove Lemma 4.1 through Lemmas 4.4 and 4.5.
First, we remark that [28,Lemma 4.2] is proved under the assumptions (H ′ 4 ) and u ∈ SBD p (Ω), hence it can be used in Lemma 4.4.
We now address the adaptions necessary for the bulk density. When u ∈ SBD p (Ω), we show that the approximating sequence constructed in Lemma 5.1 satisfies some additional properties. Lemma 7.3. Let u ∈ SBD p (Ω). Let θ ∈ (0, 1). For L d -a.e. x 0 ∈ Ω there exists a family u ε ∈ SBD p (B ε (x 0 )) such that (5.1) holds, and additionally Proof. Since u ∈ SBD(Ω), for L d -a.e. x 0 it holds that and (see [4,Theorem 7.4]) that where for brevity we again letū bulk x0 = ℓ x0,u(x0),∇u(x0) . Hence, with Fubini's Theorem we can fix We can now perform the same construction as in (5.5) with s ε in place of (1 − θ)ε. Notice that in this case u ε ∈ SBD p (B ε (x 0 )). By arguing exactly as in the proof of Lemma 5.1, we derive (5.1)(i),(iii),(iv) while (ii) holds in B sε (x 0 ) and a fortiori in B (1−θ)ε (x 0 ). In particular, this in combination with Hölder's inequality, (7.4), To see (7.2)(ii), observe that, since s ε /ε is bounded from above and from below, (5.1)(ii),(iii), the fact that u ε ⌊ Bs ε (x0) ∈ W 1,p (B sε (x 0 ); R d ) (see (5.5)), and Korn's inequality imply that Hence, by the trace inequality and by scaling back to B sε (x 0 ), we obtain With this, (7.5), and the fact that s ε /ε is bounded from below we then have Hence, we get by construction of u ε and (7.3) that which concludes the proof.
Proof. We argue as in the proof of Lemma 4.2.
For the "≤" inequality, take u ε satisfying (5.1) and (7.2), and perform the same construction as in Lemma 4.2. (Observe that the fundamental estimate also holds in this case, see Lemma 3.7.) Notice that, in this case, we have by (H ′ 4 ) and (5.9) that Thus, using (5.1)(iii),(iv) and (7.2)(ii), we still get (5.14), and may deduce inequality "≤" in (7.6). For the "≥" inequality, we start by observing that, if u ε also satisfies (7.2), in addition to (5.15) we may require that where u − ε and u + indicate the inner and outer traces at ∂B sε (x 0 ), respectively. We also have that (5.14) holds, as seen in the previous step. We then perform the same construction as in Lemma 4.2. In this case, inequality (5.19) is replaced by so that, using (5.14), (5.15), (7.7), and the fact that s ε ≤ (1 − 3θ)ε we are still in a position to deduce (5.20). The rest of the argument remains unchanged and we obtain inequality "≥" in (7.6).
Hence, by the trace inequality and by scaling back to B ± sε (x 0 ), we obtain Since s ε /ε is bounded from below, we then get that Given a Borel set E ⊂ B 1 (x 0 ), we define E ε = E ∩ B sε ε (x 0 ) for every ε > 0. Then by (6.7) and (7.11) we get where u − ε and u + indicate the inner and outer traces at ∂B sε (x 0 ), respectively. Hence, by construction of u ε in (6.7) and since u ∈ SBD(Ω), we obtain Combining with (7.12), this concludes the proof of (7.8)(ii).
With this lemma at hand, we can address the equivalence of minimization problems for the surface scaling. Proof. We argue as in the proof of Lemma 4.3.
Proof of Theorem 7.1. The result follows from Lemmas 7.2, 7.4, and 7.6, arguing exactly as in the proof of Theorem 2.1.
We point out that integral representation results in GSBV p have been used in several contributions, see e.g. [6,7,8,9,21,37]. They all rely on [13] along with a perturbation and truncation argument as follows: first, one considers the regularization F σ (u) := F (u) + σ´J u |[u]| dH d−1 , restricted to u ∈ SBV p (Ω; R m ). Then, the assumptions of the integral representation result in SBV p [13] are satisfied and one obtains a representation of F σ . In a second step, this representation is extended to GSBV p by a truncation argument which allows to approximate GSBV p functions by SBV p functions. Eventually, by sending σ → 0, an integral representation result for the original functional can be obtained. We refer to [21, Theorem 4.3, Theorem 5.1] for details on this procedure. (We notice that in [21] a more general growth condition from above is allowed in the surface energy density, cf. [21, assumption (1.4)], analogous to the one in (H ′ 4 )). With our result at hand, this method can be considerably simplified since no perturbation and truncation arguments are needed.
For the proof we need the following Poincaré-type inequality, which can be directly deduced from Theorem 3.2.
Theorem A.2 (Poincaré inequality for functions with small jump set). Let Ω ⊂ R d be a bounded Lipschitz domain and let 1 < p < +∞. Then there exists a constant c = c(Ω, p, m) > 0 such that for all u ∈ GSBV p (Ω; R m ) there is a set of finite perimeter ω ⊂ Ω with for L d -a.e. x 0 ∈ Ω (as ∇u ∈ L p (Ω; M m×d )) and using (A.4), we further get With (5.5), and using again (A.5), we deduce that (5.1)(iii) can be improved to This adaption is enough to redo the proof of Lemma 4.2 in the present situation. Lemma 4.3: Here, we can follow the proof of Lemma 4.3 for each u ∈ GSBV p (Ω; R m ) with (H 4 ) instead of (H 4 ), and Theorem A.2 in place of Theorem 3.2. The family (u ε ) ε defined in Lemma 6.1 needs to satisfy u ε ∈ GSBV p (B ε (x 0 ); R m ) and (6.1)(iv) needs to be improved to First, we use (6.7) to see that u ε ∈ GSBV p (B ε (x 0 ); R m ). Arguing as in the proof of (6.1)(iv), with Theorem A.2 at hand, we obtain This along with lim ε→0 ε −(d−1)´B ε(x0) |∇u| p dx = 0 for H d−1 -a.e. x 0 ∈ J u and (6.7) concludes the proof of (A.6).