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BY 4.0 license Open Access Published online by De Gruyter November 7, 2020

Integral representation for energies in linear elasticity with surface discontinuities

Vito Crismale ORCID logo, Manuel Friedrich and Francesco Solombrino ORCID logo

Abstract

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 (GSBDp) 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 [G. Bouchitté, I. Fonseca and L. Mascarenhas, A global method for relaxation, Arch. Ration. Mech. Anal. 145 1998, 1, 51–98] and a recent Korn-type inequality in GSBDp, cf. [F. Cagnetti, A. Chambolle and L. Scardia, Korn and Poincaré–Korn inequalities for functions with a small jump set, preprint 2020]. Our general strategy also allows to generalize integral representation results in SBDp, obtained in dimension two [S. Conti, M. Focardi and F. Iurlano, Integral representation for functionals defined on SBDp in dimension two, Arch. Ration. Mech. Anal. 223 2017, 3, 1337–1374], to higher dimensions, and to revisit results in the framework of generalized special functions of bounded variation (GSBVp).

1 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, 49] 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 GSBVp (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 Ωd, whose displacement field with respect to the equilibrium is u:Ωd, the elastic strain is given by the symmetrized gradient e(u)=12(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, see [46, 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 setJu. This set is characterized by the property that for xJu, when blowing up around x, the jump set approximates a hyperplane with normal νu(x)𝕊d-1 and the displacement field is close to two suitable values u+(x), u-(x)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

(1)Ω|e(u)|pdx+JuΩ(1+|[u]|)dd-1,

where [u](x)=u+(x)-u-(x) denotes the jump opening, or which are controlled by multiples of Griffith’s energy [45]

(2)Ω|e(u)|pdx+d-1(JuΩ).

Whereas in case (1) the energy space is be given by SBDp, a subspace of BD, problems with control of type (2) are naturally formulated on generalized special functions of bounded deformationGSBDp, 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, 48], 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 SBVp. Then, by letting the perturbation parameter vanish and by truncating functions suitably, the representation can be extended to GSBVp. 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 GSBDp from the one in SBDp. 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 L1-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 [25, 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 PR(Ω) of piecewise rigid functions has been proved in [44].

In our main result (Theorem 2.1), we prove an integral representation for variational functionals

:GSBDp(Ω)×(Ω)[0,+)

((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(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:

  1. one first shows that, for fixed uGSBDp(Ω), the set function (u,) is asymptotically equivalent to its minimum 𝐦(u,) over competitors attaining the same boundary conditions as u on the boundaries of small balls centered in x0Ω with vanishing radii. With this we mean that the two quantities have the same Radon–Nikodym derivative with respect to μ:=dΩ+d-1JuΩ (Lemma 4.1),

  2. one then proves that the Radon–Nikodym derivative d𝐦(u,)dμ only depends on x0, the value u(x0), and the (approximate) gradient u(x0) at a Lebesgue point x0, while at a jump point x0 it is uniquely determined by the one-sided traces u+(x0), u-(x0) and the normal vector νu(x0) to Ju in x0 (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 uε, 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(x0)+u(x0)(-x0), or the two-valued function with values u-(x0) and u+(x0), 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 L1), and constitutes the counterpart of the SBV-Poincaré inequality [34] used in the SBV-case [13]. We also point out that our construction for approximating two-valued functions in Lemma 4.3 slightly differs from the ones in [13, 28] in order to fix a possible flaw contained in these proofs, see Remark 6.2 for details.

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

vδ:=viδχBiδ

approximate u in the topology of the convergence in measure, where Biδ is a fine cover of a given set with disjoint balls of radius smaller than δ and viδ denote minimizers for 𝐦(u,Biδ). 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 [50, Theorem 2.2]) shows that viδ is close to u on each Biδ. (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 viδ are close to u in Lp up to exceptional sets ωiδ whose volumes scale like δ((u,Biδ)+μ(Biδ)).

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 SBDp (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 SBDp and its generalized space.

For a related purpose, in Section 8 we discuss how our arguments can also provide a direct proof for integral representation results on GSBVp, if a local control on the full deformation gradient of the form

Ω|u|pdx+d-1(JuΩ)

is given, see Theorem 8.1. 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 GSBVp 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.

We close the introduction by mentioning that in a subsequent work [42] we use the present result to obtain integral representation of Γ-limits for sequences of energies in linear elasticity with surface discontinuities. There, we additionally characterize the bulk and surface densities as blow-up limits of cell minimization formulas where the minimization is not performed on GSBDp but more specifically on Sobolev functions (bulk density) and piecewise rigid functions [44] (surface density). The latter characterization particularly allows to identify integrands of relaxed functionals and to treat homogenization problems.

The paper is organized as follows. In Section 2 we present our main integral representation result in GSBDp. 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 SBDp-case. Finally, in Section 8 we explain how our method can be used to establish an integral representation result in GSBVp.

2 The integral representation result

In this section we present our main result. We start with some basic notation. Let Ωd be open, bounded with Lipschitz boundary. Let 𝒜(Ω) be the family of open subsets of Ω, and denote by (Ω) the family of Borel sets contained in Ω. For every xd and ε>0 we indicate by Bε(x)d the open ball with center x and radius ε. For x, yd, we use the notation xy for the scalar product and |x| for the Euclidean norm. Moreover, we let 𝕊d-1:={xd:|x|=1} and we denote by 𝕄d×d the set of d×d matrices. The m-dimensional Lebesgue measure of the unit ball in m is indicated by γm for every m. We denote by d and k the d-dimensional Lebesgue measure and the k-dimensional Hausdorff measure, respectively.

For definition and properties of the space GSBDp(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 3.5) and the (approximate) jump set is denoted by Ju with corresponding normal νu and one-sided limits u+ and u-. We also define e(u)=12(u+(u)T).

We consider functionals :GSBDp(Ω)×(Ω)[0,+) with the following general assumptions:

  1. (u,) is a Borel measure for any uGSBDp(Ω),

  2. (,A) is lower semicontinuous with respect to convergence in measure on Ω for any A𝒜(Ω),

  3. (,A) is local for any A𝒜(Ω), in the sense that if u,vGSBDp(Ω) satisfy u=v a.e. in A, then (u,A)=(v,A),

  4. there exist 0<α<β such that for any uGSBDp(Ω) and B(Ω) we have

    α(B|e(u)|pdx+d-1(JuB))(u,B)β(B(1+|e(u)|p)dx+d-1(JuB)).

We now formulate the main result of this article addressing integral representation of functionals satisfying (H1)(H4). To this end, we introduce some further notation: for every uGSBDp(Ω) and A𝒜(Ω) we define

(2.1)𝐦(u,A)=infvGSBDp(Ω){(v,A):v=u in a neighborhood of A}.

For x0Ω, u0d, and ξ𝕄d×d we introduce the functions x0,u0,ξ:dd by

(2.2)x0,u0,ξ(x)=u0+ξ(x-x0).

Moreover, for x0Ω, a,bd, and ν𝕊d-1 we introduce ux0,a,b,ν:dd by

(2.3)ux0,a,b,ν(x)={aif (x-x0)ν>0,bif (x-x0)ν<0.

In this paper, we will prove the following result.

Theorem 2.1 (Integral representation in GSBDp).

Let ΩRd be open, bounded with Lipschitz boundary and suppose that F:GSBDp(Ω)×B(Ω)[0,+) satisfies (H1)(H4). Then

(u,B)=Bf(x,u(x),u(x))dx+JuBg(x,u+(x),u-(x),νu(x))dd-1(x)

for all uGSBDp(Ω) and BB(Ω), where f is given by

(2.4)f(x0,u0,ξ)=lim supε0𝐦(x0,u0,ξ,Bε(x0))γdεd

for all x0Ω, u0Rd, ξMd×d, and g is given by

(2.5)g(x0,a,b,ν)=lim supε0𝐦(ux0,a,b,ν,Bε(x0))γd-1εd-1

for all x0Ω, a,bRd, and νSd-1.

Remark 2.2.

We proceed with some remarks on the result.

  1. In general, if f is not convex in ξ, in spite of the growth conditions (H4), 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.

  2. As is lower semicontinuous on W1,p with respect to weak convergence, the integrand f is quasiconvex [47]. Since is lower semicontinuous on piecewise rigid functions, the integrand g is BD-elliptic [43] (at least if one can ensure, for instance, that g has a continuous dependence in x). A fortiori, g is BV-elliptic [3].

  3. If the functional additionally satisfies (u+a,A)=(u,A) for all affine functions a:dd with e(a)=0, then there are two functions f:Ω×𝕄d×d[0,+) and g:Ω×d×𝕊d-1[0,+) such that

    (u,B)=Bf(x,e(u)(x))dx+JuBg(x,[u](x),νu(x))dd-1(x),

    where [u](x):=u+(x)-u-(x).

  4. A variant of the proof shows that, in the minimization problems (2.4)–(2.5), one may replace balls Bε(x0) by cubes Qεν(x0) with sidelength ε, centered at x0, and two faces orthogonal to ν=νu(x0).

  5. An analogous result holds on the space GSBVp(Ω;m) for m. We refer to Section 8 for details.

We will additionally discuss the minor modifications needed in order to deal with functionals

:SBDp(Ω)×(Ω)[0,+)

satisfying (H1)(H3) and

  1. there exist 0<α<β such that for any uSBDp(Ω) and B(Ω) we have

    α(B|e(u)|pdx+JuB(1+|[u]|)dd-1)(u,B)β(B(1+|e(u)|p)dx+JuB(1+|[u]|)dd-1).

In this case, SBDp(Ω) (see Section 3.1) is the natural energy space for . Furthermore, sequences of competitors with bounded energy, which are converging in measure, are additionally L1-convergent if we assume (H4’), due to the classical Korn–Poincaré inequality in BD (see [50, Theorem II.2.2]). Hence, in this latter case, (H2) is equivalent to requiring lower semicontinuity with respect to the L1-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.

3 Preliminaries

We start this preliminary section by introducing some further notation. For Ed, ε>0, and x0d we set

(3.1)Eε,x0:=x0+ε(E-x0).

The diameter of E is indicated by diam(E). Given two sets E1,E2d, we denote their symmetric difference by E1E2. We write χE for the characteristic function of any Ed, 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 𝕄symd×d and 𝕄skewd×d, respectively.

3.1 BD and GBD functions

Let Ud be open. A function vL1(U;d) belongs to the space of functions of bounded deformation, denoted by BD(U), if the distribution Ev:=12(Dv+(Dv)T) is a bounded 𝕄symd×d-valued Radon measure on U, where Dv=(D1v,,Ddv) is the distributional differential. It is well known (see [4, 50]) that for vBD(U) the jump set Jv is countably d-1-rectifiable (in the sense of [5, Definition 2.57]), and that

Ev=Eav+Ecv+Ejv,

where Eav is absolutely continuous with respect to d, Ecv is singular with respect to d and such that |Ecv|(B)=0 if d-1(B)<, while Ejv is concentrated on Jv. The density of Eav with respect to d is denoted by e(v).

The space SBD(U) is the subspace of all functions vBD(U) such that Ecv=0. For p(1,), we define SBDp(U):={vSBD(U):e(v)Lp(U;𝕄symd×d),d-1(Jv)<}. For a complete treatment of BD and SBD functions, we refer to to [4, 11, 50].

The spaces GBD(U) of generalized functions of bounded deformation and GSBD(U)GBD(U) of generalized special functions of bounded deformation have been introduced in [31] (cf. [31, Definitions 4.1 and 4.2]), and are defined as follows.

Definition 3.1.

Let Ud be a bounded open set, and let v:Ud be measurable. We introduce the notation

Πξ:={yd:yξ=0},Byξ:={t:y+tξB} for any yd and Bd

for fixed ξ𝕊d-1, and for every tByξ we let

vyξ(t):=v(y+tξ),v^yξ(t):=vyξ(t)ξ.

Then vGBD(U) if there exists λvb+(U) such that v^yξBVloc(Uyξ) for d-1-a.e. yΠξ, and for every Borel set BU,

Πξ(|Dv^yξ|(ByξJv^yξ1)+0(ByξJv^yξ1))dd-1(y)λv(B),Jv^yξ1:={tJv^yξ:|[v^yξ]|(t)1}.

Moreover, the function v belongs to GSBD(U) if vGBD(U) and v^yξSBVloc(Uyξ) for every ξ𝕊d-1 and for d-1-a.e. yΠξ.

We recall that every vGBD(U) has an approximate symmetric gradiente(v)L1(U;𝕄symd×d) and an approximate jump setJv which is still countably d-1-rectifiable (cf. [31, Theorem 9.1, Theorem 6.2]).

The notation for e(v) and Jv, 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 d and d-1, respectively). For xJv there exist v+(x), v-(x)d and νv(x)𝕊d-1 such that

(3.2)limε0ε-dd({yBε(x):±(y-x)νv(x)>0}{|v-v±(x)|>ϱ})=0

for every ϱ>0, and the function [v]:=v+-v-:Jvd is measurable. For 1<p<, the space GSBDp(U) is given by

GSBDp(U):={vGSBD(U):e(v)Lp(U;𝕄symd×d),d-1(Jv)<}.

Any function vGSBD(U) with [v] integrable belongs to SBD(U), as follows from [27, Theorem 2.9] for 𝔸v=Ev (see [27, Remark 2.5]). This corresponds to the following proposition.

Proposition 3.2.

If vGSBDp(U) is such that [v]L1(Jv;Rd), then vSBDp(U).

If U has Lipschitz boundary, for each vGBD(U) the traces on U are well defined (see [31, Theorem 5.5]), in the sense that for d-1-a.e. xU there exists tr(v)(x)d such that

(3.3)limε0ε-dd(UBε(x){|v-tr(v)(x)|>ϱ})=0  for all ϱ>0.

3.2 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:dd is an infinitesimal rigid motion if a is affine with e(a)=12(a+(a)T)=0.

Theorem 3.3 (Korn inequality for functions with small jump set).

Let ΩRd be a bounded Lipschitz domain and let 1<p<+. Then there exists a constant c=c(Ω,p)>0 such that for all uGSBDp(Ω) there is a set of finite perimeter ωΩ with

(3.4)d-1(*ω)cd-1(Ju),d(ω)c(d-1(Ju))dd-1

and an infinitesimal rigid motion a such that

(3.5)u-aLp(Ωω)+u-aLp(Ωω)ce(u)Lp(Ω).

Moreover, there exists vW1,p(Ω;Rd) such that v=u on Ωω and

e(v)Lp(Ω)ce(u)Lp(Ω).

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], d(ω)c(d-1(Ju))dd-1 has not been shown, but it readily follows from d-1(*ω)cd-1(Ju) by the isoperimetric inequality.

Remark 3.4 (Almost Sobolev regularity, constants, and scaling invariance).

(i) More precisely, in [20] it is proved that there exists vW1,p(Ω;d) such that v=u on Ωω and e(v)Lp(Ω)ce(u)Lp(Ω), whence by Korn’s and Poincaré’s inequality in W1,p(Ω;d) we get

v-aLp(Ω)+v-aLp(Ω)ce(u)Lp(Ω)

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 (Ωk)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 3.3 we can choose a constant c uniformly for all Ωk, see [20, Remark 4.2].

(iii) Recall definition (3.1). Consider a bounded Lipschitz domain Ω, ε>0, and x0d. Then for each uGSBDp(Ωε,x0) we find ωΩε,x0 and a rigid motion a such that

d-1(*ω)Cd-1(Ju),d(ω)C(d-1(Ju))dd-1

and

ε-1u-aLp(Ωε,x0ω)+u-aLp(Ωε,x0ω)Ce(u)Lp(Ωε,x0),

where C=C(Ω,p)>0 is independent of ε. This follows by a standard rescaling argument.

From Theorem 3.3, one can also deduce that for uGSBDp(Ω) the approximate gradientu exists d-a.e. in Ω, see [20, Corollary 5.2].

Lemma 3.5 (Approximate gradient).

Let ΩRd be open, bounded with Lipschitz boundary, let 1<p<+, and uGSBDp(Ω). Then for Ld-a.e. x0Ω there exists a matrix in Md×d, denoted by u(x0), such that

limε0ε-dd({xBε(x0):|u(x)-u(x0)-u(x0)(x-x0)||x-x0|>ϱ})=0for all ϱ>0.

We point out that the result in Lemma 3.5 has already been obtained in [40] for p=2, as a consequence of the embedding GSBD2(Ω)(GBV(Ω))d, see [40, Theorem 2.9].

To control the affine mappings appearing in Theorem 3.3, we will make use of the following elementary lemma on affine mappings, see, e.g., [44, 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.)

Lemma 3.6.

Let 1p<+, let x0Rd, and let R,θ>0. Let a:RdRd be affine, defined by a(x)=Ax+b for xRd, and let EBR(x0)Rd with Ld(E)θLd(BR(x0)). Then there exists a constant c0>0 only depending on p and θ such that

aLp(BR(x0))γd1pRdpaL(BR(x0))c0aLp(E),|A|c0R-1-dpaLp(E).

We now proceed with another consequence of Theorem 3.3.

Corollary 3.7.

Let ΩRd be a bounded Lipschitz domain and let 1<p<+. Then there exists a constant C=C(Ω,p)>0 such that for all uGSBDp(Ω) with trace tr(u)=0 on Ω (see (3.3)) there is a set of finite perimeter ωRd with

(3.6)d-1(*ω)Cd-1(Ju),d(ω)C(d-1(Ju))dd-1

such that

(3.7)uLp(Ωω)+uLp(Ωω)Ce(u)Lp(Ω).

Proof.

We start by choosing a bounded Lipschitz domain Ωd with ΩΩ. Each uGSBDp(Ω) with tr(u)=0 on Ω can be extended to a function u~GSBDp(Ω) by u~=0 on ΩΩ such that Ju~=Ju. We first note that it is not restrictive to assume that

(3.8)d-1(Ju)(d(ΩΩ)2c)d-1d,

where c=c(Ω,p)>0 is the constant of Theorem 3.3. 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

d-1(*Ω)C(d(ΩΩ)2c)d-1d,d(Ω)Cd(ΩΩ)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 3.3 on u~GSBDp(Ω) and obtain a set ωΩd satisfying (3.4) as well as an infinitesimal rigid motion a such that

(3.9)u~-aLp(Ωω)+u~-aLp(Ωω)ce(u~)Lp(Ω)=ce(u)Lp(Ω).

In particular, u~=0 on ΩΩ implies

(3.10)aLp(Ω(Ωω))ce(u)Lp(Ω).

By (3.4) and (3.8) we get d(ω)12d(ΩΩ). In view of (3.10), we apply Lemma 3.6 on E=Ω(Ωω) with R=diam(Ω) and θ=12d(ΩΩ)/γdRd to get

aLp(Ω)caLp(Ω(Ωω))ce(u)Lp(Ω),

and, in a similar fashion, |a|ce(u)Lp(Ω), where c>0 depends on Ω, Ω, and p. Then (3.7) follows from (3.9), the triangle inequality, and the fact that u=u~ on Ω. ∎

We conclude this subsection with another important tool in the proof of the integral representation, namely a fundamental estimate in GSBDp.

Lemma 3.8 (Fundamental estimate in GSBDp).

Let ΩRd be open, bounded with Lipschitz boundary, and let 1<p<+. Let η>0 and let A,A,A′′A(Ω) with AA. For every functional F satisfying (H1), (H3), and (H4) and for every uGSBDp(A), vGSBDp(A′′) there exists a function φC(Rd;[0,1]) such that w:=φu+(1-φ)vGSBDp(AA′′) satisfies

(3.11){(i)(w,AA′′)(1+η)((u,A)+(v,A′′))+Mu-vLp((AA)A′′)p+ηd(AA′′),(ii)w=uon A  𝑎𝑛𝑑  w=von A′′A,

where M=M(A,A,A′′,p,η)>0 depends only on A,A,A′′,p,η, but is independent of u and v. Moreover, if for ε>0 and x0Rd we have Aε,x0,Aε,x0,Aε,x0′′Ω, then

(3.12)M(Aε,x0,Aε,x0,Aε,x0′′,p,η)=ε-pM(A,A,A′′,p,η),

where we used the notation introduced in (3.1).

The same statement holds if F satisfies (H4’), uSBDp(A), and vSBDp(A′′).

In the statement above, we intend that u-vLp((AA)A′′)p=+ if u-vLp((AA)A′′).

Proof.

The proof follows the lines of [16, Proposition 3.1]. Choose k such that

(3.13)kmax{3p-1βηα,βη}.

Let A1,,Ak+1 be open subsets of d with

AA1Ak+1A.

For i=1,,k let φiC0(Ai+1;[0,1]) with φi=1 in a neighborhood Vi of Ai¯.

Consider uGSBDp(A) and vGSBDp(A′′). We can clearly assume that u-vLp((AA)A′′) as otherwise the result is trivial. We define the function wi=φiu+(1-φi)vGSBDp(AA′′), where u and v are extended arbitrarily outside A and A′′, respectively. Letting Ti=A′′(Ai+1Ai¯) we get by (H1) and (H3)

(wi,AA′′)(u,(AA′′)Vi)+(v,A′′supp(φi))+(wi,Ti)
(3.14)(u,A)+(v,A′′)+(wi,Ti).

For the last term, we compute using (H4) ( denotes the symmetrized vector product)

(wi,Ti)βTi(1+|e(wi)|p)dx+βd-1(JwiTi)
βTi(1+|φie(u)+(1-φi)e(v)+φi(u-v)|p)+βd-1((JuJv)Ti)
βd(Ti)+3p-1βTi(|e(u)|p+|e(v)|p+|φi|p|u-v|p)+βd-1(JuTi)+βd-1(JvTi)
3p-1βα-1((u,Ti)+(v,Ti))+3p-1βϕipu-vLp(Ti)p+βd(Ti).

Notice that we can obtain the same estimate also if satisfies (H4’), uSBDp(A), and vSBDp(A′′). (We refer to [16, proof of Proposition 3.1] for details.) Consequently, recalling (3.13) and using (H1), we find i0{1,,k} such that

(wi0,Ti0)1ki=1k(wi,Ti)η((u,A)+(v,A′′))+Mu-vLp((AA)A′′)p+ηd((AA)A′′),

where M:=3p-1βk-1maxi=1,,kφip. This along with (3.14) concludes the proof of (3.11) by setting w=wi0. To see the scaling property (3.12), it suffices to use the cut-off functions φiεC0((Ai+1)ε,x0;[0,1]), i=1,,k, defined by φiε(x)=φi(x0+1ε(x-x0)) for x(Ai+1)ε,x0. This concludes the proof. ∎

4 The global method

This section is devoted to the proof of Theorem 2.1 which is based on three ingredients. First, we show that is equivalent to 𝐦 (see (2.1)) in the sense that the two quantities have the same Radon–Nikodym derivative with respect to μ:=dΩ+d-1JuΩ.

Lemma 4.1.

Suppose that F satisfies (H1)(H4). Let uGSBDp(Ω) and μ=LdΩ+Hd-1JuΩ. Then for μ-a.e. x0Ω we have

limε0(u,Bε(x0))μ(Bε(x0))=limε0𝐦(u,Bε(x0))μ(Bε(x0)).

We prove this lemma in the final part of this section. The second ingredient is that, asymptotically as ε0, the minimization problems 𝐦(u,Bε(x0)) and 𝐦(u¯x0bulk,Bε(x0)) coincide for d-a.e. x0Ω, where we write u¯x0bulk:=x0,u(x0),u(x0) for brevity, see (2.2).

Lemma 4.2.

Suppose that F satisfies (H1) and (H3)(H4) and let uGSBDp(Ω). Then for Ld-a.e. x0Ω we have

(4.1)limε0𝐦(u,Bε(x0))γdεd=lim supε0𝐦(u¯x0bulk,Bε(x0))γdεd.

We defer the proof of Lemma 4.2 to Section 5. The third ingredient is that, asymptotically as ε0, the minimization problems 𝐦(u,Bε(x0)) and 𝐦(u¯x0surf,Bε(x0)) coincide for d-1-a.e. x0Ju, where we write u¯x0surf:=ux0,u+(x0),u-(x0),νu(x0) for brevity, see (2.3).

Lemma 4.3.

Suppose that F satisfies (H1) and (H3)(H4) and let uGSBDp(Ω). Then for Hd-1-a.e. x0Ju we have

(4.2)limε0𝐦(u,Bε(x0))γd-1εd-1=lim supε0𝐦(u¯x0surf,Bε(x0))γd-1εd-1.

We defer the proof of Lemma 4.3 to Section 6, and now proceed to prove Theorem 2.1.

Proof of Theorem 2.1.

In view of (H4) on and of the Besicovitch derivation theorem (cf. [5, Theorem 2.22]), we need to show that for d-a.e. x0Ω one has

(4.3)d(u,)dd(x0)=f(x0,u(x0),u(x0)),

where f was defined in (2.4), and that for d-1-a.e. x0Ju one has

(4.4)d(u,)dd-1Ju(x0)=g(x0,u+(x0),u-(x0),νu(x0)),

where g was defined in (2.5).

By Lemma 4.1 and the fact that limε0(γdεd)-1μ(Bε(x0))=1 for d-a.e. x0Ω we deduce

d(u,)dd(x0)=limε0(u,Bε(x0))μ(Bε(x0))=limε0𝐦(u,Bε(x0))μ(Bε(x0))=limε0𝐦(u,Bε(x0))γdεd<

for d-a.e. x0Ω. Then (4.3) follows from (2.4) and Lemma 4.2. By Lemma 4.1 and the fact that

limε0(γd-1εd-1)-1μ(Bε(x0))=1

for d-1-a.e. x0Ju we deduce

d(u,)dd-1Ju(x0)=limε0(u,Bε(x0))μ(Bε(x0))=limε0𝐦(u,Bε(x0))μ(Bε(x0))=limε0𝐦(u,Bε(x0))γd-1εd-1<

for d-1-a.e. x0Ju. 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 H4) compared to [13, 14, 28]. We start with some notation. For δ>0 and A𝒜(Ω), we define

𝐦δ(u,A)=inf{i=1𝐦(u,Bi):BiA pairwise disjoint balls,diam(Bi)δ,μ(Ai=1Bi)=0},

where, as before, μ=dΩ+d-1JuΩ. As 𝐦δ(u,A) is decreasing in δ, we can also introduce

(4.5)𝐦*(u,A)=limδ0𝐦δ(u,A).

In the following lemma, we prove that and 𝐦* coincide under our assumptions.

Lemma 4.4.

Suppose that F satisfies (H1)(H4) and let uGSBDp(Ω). Then for all AA(Ω) there holds F(u,A)=mF*(u,A).

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 H4) compared to [28]. For each ball BA, 𝐦(u,B)(u,B) by definition. By (H1) we get 𝐦δ(u,A)(u,A) for all δ>0. This shows 𝐦*(u,A)(u,A), cf. (4.5).

We now address the reverse inequality. We fix A𝒜(Ω) and δ>0. Let (Biδ)i be balls as in the definition of 𝐦δ(u,A) such that

(4.6)i=1𝐦(u,Biδ)𝐦δ(u,A)+δ.

By the definition of 𝐦, we find viδGSBDp(Biδ) such that viδ=u in a neighborhood of Biδ and

(4.7)(viδ,Biδ)𝐦(u,Biδ)+δd(Biδ).

We define

(4.8)vδ,n:=i=1nviδχBiδ+uχN0δ,nfor n,vδ:=i=1viδχBiδ+uχN0δ,

where N0δ,n:=Ωi=1nBiδ and N0δ:=Ωi=1Biδ. By construction, we have that each vδ,n lies in GSBDp(Ω) and that supn(e(vδ,n)Lp(Ω)+d-1(Jvδ,n))<+ by (4.6)–(4.7) and (H4). Moreover, vδ,nvδ pointwise a.e. in Ω. Then [25, Theorem 1.1] yields vδGSBDp(Ω).

(vδ,A)=i=1(viδ,Biδ)+(u,N0δA)
i=1(𝐦(u,Biδ)+δd(Biδ))
(4.9)𝐦δ(u,A)+δ(1+d(A)),

where we also used the fact that μ(N0δA)=(u,N0δA)=0 by the definition of (Biδ)i and (H4). For later purpose, we also note by (H4) that this implies

(4.10)e(vδ)Lp(A)p+d-1(JvδA)α-1(𝐦δ(u,A)+δ(1+d(A))).

We now claim that vδu in measure on A. To this end, we apply Remark 3.4 (iii) and Corollary 3.7 on each Biδ for the function u-viδ and we get sets of finite perimeter ωiδBiδ such that

(4.11){(i)(d(ωiδ))d-1dCd-1((Juvδ)Biδ),(ii)u-viδLp(Biδωiδ)pCδp(e(u)Lp(Biδ)p+e(vδ)Lp(Biδ)p)

for a constant C>0 only depending on p. Here, we used that diam(Biδ)δ and (u-vδ)BiδGSBDp(Biδ) with trace zero on Biδ. We define ψ:[0,+)[0,+) by ψ(t)=min{tp,1} and observe that vδu in measure on A is equivalent to Aψ(|u-vδ|)dx0 as δ0. In view of (4.8), we compute

(4.12)Aψ(|u-vδ|)dx=i=1Biδψ(|u-viδ|)dxi=1(u-viδLp(Biδωiδ)p+d(ωiδ)).

By (4.11) (ii) and the fact that the balls (Biδ)i are pairwise disjoint we get

(4.13)i=1u-viδLp(Biδωiδ)pCδp(e(u)Lp(A)p+e(vδ)Lp(A)p).

As ωiδBδi and diam(Biδ)δ, we further get by (4.11) (i)

(4.14)i=1d(ωiδ)γd1dδi=1(d(ωiδ))d-1dγd1/dCδd-1((JuJvδ)A).

Now, combining (4.12)–(4.14) and using (4.10), we find Aψ(|u-vδ|)dx0 as δ0. With this, using (H2), (4.5), and (4.9) we get the required inequality 𝐦*(u,A)(u,A) in the limit as δ0. This concludes the proof. ∎

Proof of Lemma 4.1.

The statement follows by repeating exactly the arguments in [28, proofs of Lemma 4.2 and Lemma 4.3]. We report a sketch for the reader’s convenience.

The definition of 𝐦 gives readily that for every x0Ω,

lim supε0𝐦(u,Bε(x0))μ(Bε(x0))lim supε0(u,Bε(x0))μ(Bε(x0)).

The converse inequality follows by proving that, for every t>0, the set

Et:={xΩ:lim infε0(u,Bε(x))-𝐦(u,Bε(x))μ(Bε(x))>t}

satisfies μ(Et)=0. To this end, we fix t>0. We introduce the family of balls (depending on t)

Xδ:={Bε(x):ε<δ,Bε(x)¯Ω,μ(Bε(x))=0,(u,Bε(x))>𝐦(u,Bε(x))+tμ(Bε(x))},

and we show that

U*:=δ>0{xΩ:Bε(x)Xδ for some ε>0}

satisfies

(4.15)EtU*,μ(U*)=0.

Then μ(Et)=0 indeed holds true and the proof is concluded.

We now confirm (4.15). The inclusion EtU* follows from the definition of Et which permits to find, for any xEt, ε<δ such that (u,Bε(x))>𝐦(u,Bε(x))+tμ(Bε(x)). Note that, due to the left continuity of ε𝐦(u,Bε(x)) (cf. [28, Lemma 4.2]) one can also ensure the additional property μ(Bε(x))=0 by slightly varying ε. In order to prove that μ(U*)=0, one fixes a compact set KU*, two positive numbers δ<η, and defines

Uη:={Bε(x):Bε(x)Xη},Yδ:={Bε(x):ε<δ,Bε(x)¯UηK,μ(Bε(x))=0}.

By recalling also the definition of U*, we see that Xδ and Yδ are fine covers of K and UηK, respectively. Thus there exists countable many pairwise disjoint BiXδ, B^jYδ, and a set N with μ(N)=0 such that Uη=iBijB^jN. In view of assumptions (H1), the definitions of Xδ, Yδ (in particular the balls have radii smaller than δ) give that

(u,Uη)i𝐦(u,Bi)+j𝐦(u,B^j)+tμ(iBi)𝐦δ(u,Uη)+tμ(K).

Passing to the limit in δ, (4.5) and Lemma 4.4 imply

(u,Uη)𝐦*(u,Uη)+tμ(K)=(u,Uη)+tμ(K),

so that μ(K)=0. Then μ(U*)=0 by the regularity of μ. ∎

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.

5 The bulk density

This section is devoted to the proof of Lemma 4.2. We start by analyzing the blow-up at points with approximate gradient. The latter exists for d-a.e. point in Ω by Lemma 3.5.

Lemma 5.1 (Blow-up at points with approximate gradient).

Let uGSBDp(Ω). Let θ(0,1). Then for Ld-a.e. x0Ω there exists a family uεGSBDp(Bε(x0)) such that

(5.1){(i)uε=u in a neighborhood of Bε(x0),limε0ε-(d+1)d({uεu})=0,(ii)limε0ε-(d+p)B(1-θ)ε(x0)|uε(x)-u(x0)-u(x0)(x-x0)|pdx=0,(iii)limε0ε-dBε(x0)|e(uε)(x)-e(u)(x0)|pdx=0,(iv)limε0ε-dd-1(Juε)=0.

Proof.

Let x0Ω be such that

(5.2){(i)limε0ε-dBε(x0)|e(u)(x)-e(u)(x0)|pdx=0,(ii)limε0ε-dd-1(JuBε(x0))=0,(iii)limε0ε-dd({xBε(x0):|u(x)-u(x0)-u(x0)(x-x0)||x-x0|>ϱ})=0 for all ϱ>0.

These properties hold for d-a.e. x0Ω by Lemma 3.5 and the facts that |e(u)|pL1(Ω) and Ju is countably d-1-rectifiable. We use again the notation u¯x0bulk=x0,u0,u(x0)=u(x0)+u(x0)(-x0) for brevity, see (2.2).

Fix θ>0. We apply Theorem 3.3 and Remark 3.4 (i) for the function u-u¯x0bulk on the set B(1-θ)ε(x0) to obtain a set of finite perimeter ωεB(1-θ)ε(x0), a function vεW1,p(B(1-θ)ε(x0);d) with vε=u-u¯x0bulk in B(1-θ)ε(x0)ωε, and an infinitesimal rigid motion aε such that

(5.3){(i)d-1(*ωε)cd-1(JuBε(x0)),d(ωε)c(d-1(JuBε(x0)))dd-1,(ii)vε-aεLp(B(1-θ)ε(x0))cεe(u-u¯x0bulk)Lp(Bε(x0)),(iii)e(vε)Lp(B(1-θ)ε(x0))ce(u-u¯x0bulk)Lp(Bε(x0)),

where c>0 depends only on p, cf. also Remark 3.4 (iii). We directly note by (5.2) (ii) and (5.3) (i) that

(5.4)limε0ε-d2d-1d(ωε)=0.

We define uεGSBDp(Bε(x0)) as

(5.5)uε:=uχBε(x0)B(1-θ)ε(x0)+(vε+u¯x0bulk)χB(1-θ)ε(x0),

and proceed by confirming the properties stated in (5.1). Notice that, by construction, uε=u in Bε(x0)ωε. First, (5.1) (i) follows directly from the fact that ωεB(1-θ)ε(x0), as well as (5.4)–(5.5). Moreover, (5.3) (i) and (5.2) (ii) imply (5.1) (iv). As for (5.1) (iii), we notice that by (5.3) (iii) and (5.2) (i) we have

limε0ε-dB(1-θ)ε(x0)|e(vε)(x)|pdx=0.

Since, by a direct computation, e(uε)(x)-e(u)(x0)=e(vε)(x) for xB(1-θ)ε(x0), see (5.5), in combination with (5.2) (i) we obtain (5.1) (iii). It therefore remains to prove (5.1) (ii).

To this end, fix ϱ>0 and define ω^ε:={xBε(x0):|u(x)-u¯x0bulk(x)|>ϱε}. In view of (5.2) (iii) and (5.4), we can choose ε0>0 sufficiently small such that for all 0<εε0 we have

(5.6)d(ωεω^ε)12d(B(1-θ)ε(x0)).

By the definition of ω^ε and the fact that vε=u-u¯x0bulk in B(1-θ)ε(x0)ωε, we have |vε(x)|ϱε for all xB(1-θ)ε(x0)(ωεω^ε). Hence, (5.3) (ii) and the triangle inequality give

aεLp(B(1-θ)ε(x0)(ωεω^ε))pCεpe(u-u¯x0bulk)Lp(Bε(x0))p+Cd(Bε(x0))ϱpεp,

where C>0 depends only on p. By (5.6) and Lemma 3.6 we get

aεLp(B(1-θ)ε(x0))pCεpe(u-u¯x0bulk)Lp(Bε(x0))p+Cd(Bε(x0))ϱpεp.

Therefore, by using also (5.2) (i), we derive lim supε0ε-(d+p)aεLp(B(1-θ)ε(x0))pCγdϱp. As ϱ>0 was arbitrary, we get

(5.7)limε0ε-(d+p)B(1-θ)ε(x0)|aε|pdx=0.

Now, (5.2) (i) and (5.3) (ii) give that

limε0ε-(d+p)vε-aεLp(B(1-θ)ε(x0))pcε-de(u-u¯x0bulk)Lp(Bε(x0))p=0.

As uε-u¯x0bulk=vε in B(1-θ)ε(x0), this shows (5.1) (ii) by (5.7). ∎

We are now in a position to prove Lemma 4.2.

Proof of Lemma 4.2.

It suffices to prove (4.1) for points x0Ω where the statement of Lemma 5.1 holds and we have limε0ε-dμ(Bε(x0))=γd. This holds true for d-a.e. x0Ω. Then also limε0ε-d𝐦(u,Bε(x0)) exists, see Lemma 4.1. As before, we write u¯x0bulk=u(x0)+u(x0)(-x0) for shorthand.

Step 1 (Inequality “” in (4.1)). We fix η>0 and θ>0. Choose zεGSBDp(B(1-3θ)ε(x0)) with zε=u¯x0bulk in a neighborhood of B(1-3θ)ε(x0) and

(5.8)(zε,B(1-3θ)ε(x0))𝐦(u¯x0bulk,B(1-3θ)ε(x0))+εd+1.

We extend zε to a function in GSBDp(Bε(x0)) by setting zε=u¯x0bulk outside B(1-3θ)ε(x0). Let (uε)ε be the family given by Lemma 5.1. We apply Lemma 3.8 on zε (in place of u) and uε (in place of v) for η as above and the sets

(5.9)A=B1-2θ(x0),A=B1-θ(x0),A′′=B1(x0)B1-4θ(x0)¯.

By (3.11)–(3.12) there exist functions wεGSBDp(Bε(x0)) such that wε=uε on Bε(x0)B(1-θ)ε(x0) and

(5.10)(wε,Bε(x0))(1+η)((zε,Aε,x0)+(uε,Aε,x0′′))+Mεpzε-uεLp((AA)ε,x0)p+d(Bε(x0))η,

where M>0 depends on θ and η, but is independent of ε. Here and in the following, we use notation (3.1). In particular, we have wε=uε=u in a neighborhood of Bε(x0) by (5.1) (i). By (5.1) (ii), (5.9), and the fact that zε=u¯x0bulk outside B(1-3θ)ε(x0) we find

(5.11)limε0ε-(d+p)zε-uεLp((AA)ε,x0)p=limε0ε-(d+p)uε-u¯x0bulkLp(B(1-θ)ε(x0))p=0.

This along with (5.10) shows that there exists a sequence (ρε)ε(0,+) with ρε0 such that

(5.12)(wε,Bε(x0))(1+η)((zε,Aε,x0)+(uε,Aε,x0′′))+εdρε+γdεdη.

On the one hand, by using that zε=u¯x0bulk on Bε(x0)B(1-3θ)ε(x0)Aε,x0′′, (H1), (H4), and (5.8) we compute

lim supε0(zε,Aε,x0)εdlim supε0(zε,B(1-3θ)ε(x0))εd+lim supε0(u¯x0bulk,Aε,x0′′)εd
lim supε0𝐦(u¯x0bulk,B(1-3θ)ε(x0))εd+βγd[1-(1-4θ)d](1+|e(u)(x0)|p)
(5.13)(1-3θ)dlim supε0𝐦(u¯x0bulk,Bε(x0))(ε)d+βγd[1-(1-4θ)d](1+|e(u)(x0)|p),

where in the last step we substituted (1-3θ)ε by ε. On the other hand, by (H4) and (5.9) we also find

(uε,Aε,x0′′)βAε,x0′′(1+|e(uε)|p)+βd-1(JuεAε,x0′′)
γdεdβ[1-(1-4θ)d](1+2p-1|e(u)(x0)|p)+2p-1βe(uε)-e(u)(x0)Lp(Bε(x0))p+βd-1(Juε).

By (5.1) (iii)–(iv) this implies

(5.14)lim supε0(uε,Aε,x0′′)εdβγd[1-(1-4θ)d](1+2p-1|e(u)(x0)|p).

Recall that wε=u in a neighborhood of Bε(x0). This along with (5.12)–(5.14) and ρε0 yields

limε0𝐦(u,Bε(x0))γdεdlim supε0(wε,Bε(x0))γdεd
(1+η)(1-3θ)dlim supε0𝐦(u¯x0bulk,Bε(x0))γdεd
+2(1+η)β[1-(1-4θ)d](1+2p-1|e(u)(x0)|p)+η.

Passing to η,θ0, we obtain inequality “” in (4.1).

Step 2 (Inequality “” in (4.1)). We fix η,θ>0 and let (uε)ε be again the family from Lemma 5.1. By (5.1) (i) and Fubini’s Theorem, for each ε>0 we can find sε(1-4θ,1-3θ)ε such that

(5.15){(i)limε0ε-dd-1({uuε}Bsε(x0))=0,(ii)d-1((JuJuε)Bsε(x0))=0for all ε>0.

We consider zεGSBDp(Bsε(x0)) such that zε=u in a neighborhood of Bsε(x0), and

(5.16)(zε,Bsε(x0))𝐦(u,Bsε(x0))+εd+1.

We extend zε to a function in GSBDp(Bε(x0)) by setting

(5.17)zε=uεin Bε(x0)Bsε(x0).

We apply Lemma 3.8 on zε (in place of u) and u¯x0bulk (in place of v) for the sets indicated in equation (5.9). By (3.11)–(3.12) there exist functions wεGSBDp(Bε(x0)) such that wε=u¯x0bulk on Bε(x0)B(1-θ)ε(x0) and

(wε,Bε(x0))(1+η)((zε,Aε,x0)+(u¯x0bulk,Aε,x0′′))+Mεpzε-u¯x0bulkLp((AA)ε,x0)p+d(Bε(x0))η.

By (5.17) and the choice of sε we get that zε=uε outside B(1-3θ)ε(x0). Thus, similar to Step 1, cf. (5.11) and (5.12), we find a sequence (ρε)ε(0,+) with ρε0 such that

(5.18)(wε,Bε(x0))(1+η)((zε,Aε,x0)+(u¯x0bulk,Aε,x0′′))+εdρε+γdεdη.

Let us now estimate the terms in (5.18). We get by (H1), (H4), (5.16)–(5.17), and the choice of sε that

(5.19)(zε,Aε,x0)𝐦(u,Bsε(x0))+εd+1+βd-1(({uuε}JuJuε)Bsε(x0))+(uε,Aε,x0′′).

Therefore, by (5.14), (5.15), and the fact that sε(1-3θ)ε we derive

lim supε0(zε,Aε,x0)εd(sεε)dlim supε0𝐦(u,Bsε(x0))sεd+βγd[1-(1-4θ)d](1+2p-1|e(u)(x0)|p)
(5.20)(1-3θ)dlim supε0𝐦(u,Bε(x0))εd+βγd[1-(1-4θ)d](1+2p-1|e(u)(x0)|p).

Estimating (u¯x0bulk,Aε,x0′′) as in (5.13), with (5.18)–(5.20) and ρε0, we then obtain

lim supε0(wε,Bε(x0))εd(1+η)(1-3θ)dlim supε0𝐦(u,Bε(x0))εd
+2(1+η)βγd[1-(1-4θ)d](1+2p-1|e(u)(x0)|p)+γdη.

Passing to η,θ0 and recalling that wε=u¯x0bulk in a neighborhood of Bε(x0), we derive

lim supε0𝐦(u¯x0bulk,Bε(x0))γdεdlim supε0(wε,Bε(x0))γdεd
lim supε0𝐦(u,Bε(x0))γdεd=limε0𝐦(u,Bε(x0))γdεd.

This shows inequality “” in (4.1) and concludes the proof. ∎

6 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 x0Ju we adopt the notation u¯x0surf for the function ux0,u+(x0),u-(x0),νu(x0), see (2.3), with νu(x0)𝕊d-1 and u±(x0)d being the approximate normal to Ju and the traces on both sides of Ju at x0, respectively. Recall also notation (3.1).

Lemma 6.1 (Blow-up at jump points).

Let uGSBDp(Ω) and let θ(0,1). For Hd-1-a.e. x0Ju there exists a family uεGSBDp(Bε(x0)) such that

(6.1){(i)uε=u in a neighborhood of Bε(x0)limε0ε-dd({uεu})=0,(ii)limε0ε-(d-1+p)B(1-θ)ε(x0)|uε(x)-u¯x0surf|pdx=0,(iii)limε0ε-(d-1)d-1(JuεEε,x0)=d-1(Π0E)for all Borel sets EB1(x0),(iv)limε0ε-(d-1)Bε(x0)|e(uε)(x)|pdx=0,

where Π0 denotes the hyperplane passing through x0 with normal νu(x0).

Proof.

We start by using the fact that Ju is countably d-1-rectifiable and the blow-up properties of GSBDp functions. Arguing as in, e.g., [24, proof of Theorem 2], [26, proof of Theorem 1.1], [28, Lemma 3.4], we infer that for d-1-a.e. x0Ju there exist ε¯>0, νu(x0)𝕊d-1, u±(x0)d, and a hypersurface Γ which is a graph of a function h defined on Π0, being C1 and Lipschitz, such that x0Γ, Π0 is tangent to Γ in x0, ΓBε(x0) separates Bε(x0) in two open connected components BεΓ,±(x0) for each ε<ε¯, and

(6.2){(i)limε0ε-(d-1)d-1((JuΓ)Bε(x0))=0,limε0ε-(d-1)d-1(ΓEε,x0)=d-1(Π0E)for all Borel sets EB1(x0),(ii)limε0ε-(d-1)Bε(x0)|e(u)|pdx=0,(iii)limε0ε-dd({xBε(x0):|u-u¯x0surf|>ϱ})=0for all ϱ>0.

In particular, (ii) follows from the fact that |e(u)|pL1(Ω) and (iii) from (3.2). Then, since Π0 is tangent to Γ in x0, ΓBε(x0) is the graph of a Lipschitz function hε defined on a subset of Π0, with Lipschitz constant Lε such that limε0Lε=0. Therefore, it holds that

(6.3)limε0ε-dd(BεΓ,±(x0)Bε±(x0))=0,

where Bε±(x0):={yBε(x0):±(y-x0)νu(x0)>0}. By this and Fubini’s Theorem, for each ε>0 we can find sε(1-θ,1-θ2)ε such that

(6.4)limε0ε-(d-1)d((BεΓ,±(x0)Bε±(x0))Bsε(x0))=0.

For any ε>0, we apply Theorem 3.3 and Remark 3.4 (i) on u in the two connected components BsεΓ,±(x0) for ε<ε¯. This gives two functions vε±W1,p(BsεΓ,±(x0);d), two sets of finite perimeter ωε±BsεΓ,±(x0), and two infinitesimal rigid motions aε± such that

(6.5){(i)vε±=uin BsεΓ,±(x0)ωε±,(ii)d-1(*ωε±)cd-1(JuBεΓ,±(x0)),d(ωε±)c(d-1(JuBεΓ,±(x0)))dd-1,(iii)vε±-aε±Lp(BsεΓ,±(x0))cεe(u)Lp(BεΓ,±(x0)),(iv)vε±-aε±Lp(BsεΓ,±(x0))ce(u)Lp(BεΓ,±(x0)),

where c>0 is independent of ε. (See Remark 3.4 (ii)–(iii) and recall that the Lipschitz constant of hε vanishes as ε0.) By the Sobolev extension theorem we extend vε± to v^ε±W1,p(Bsε(x0);d), and (6.5) (iii)–(iv) along with the linearity of the extension operator yield

(6.6)ε-1v^ε±-aε±Lp(Bsε(x0))+v^ε±-aε±Lp(Bsε(x0))ce(u)Lp(BεΓ,±(x0)),

where, as before, the constant is independent of ε. (Here, we used again the properties of the functions hε recalled below (6.2).) We define uεGSBDp(Bε(x0)) as

(6.7)uε:={v^ε+in Bsε+(x0),v^ε-in Bsε-(x0),uin Bε(x0)Bsε(x0),

where Bsε±(x0) is defined below (6.3). We now prove the properties in (6.1). First, by definition we have that uε=u in a neighborhood of Bε(x0). By B(1-θ)εΓ,±(x0)Γ=, (6.2) (i), (6.3), and (6.5) (i)–(ii) we obtain limε0ε-dd({uεu})=0. This concludes (6.1) (i). Moreover, (6.2) (ii) and (6.6) imply (6.1) (iv). By the definition of uε and (6.5) (i) it holds that

Juε(Π0Bsε(x0)¯)(Ju(Bε(x0)Bsε(x0)))*ωε+*ωε-((BεΓ,±(x0)Bε±(x0))Bsε(x0)).

We now show (6.1) (iii). Concerning the ”” inequality, for a fixed Borel set EB1(x0) we have to estimate the measure of the intersection with Eε,x0 and any of the five sets in the right-hand side above: it holds that

d-1(Π0Bsε(x0)¯Eε,x0)=εd-1d-1(Π0Bsε/ε(x0)¯E)

for any ε>0 by rescaling, that

limε0|ε-(d-1)d-1(Ju(Eε,x0Bsε(x0)))-d-1(Π0(EBsε/ε(x0)))|=0

by (6.2) (i), while the last three terms are estimated by (6.5) (ii) and (6.4). To see the converse, we first apply [25, Theorem 1.1] to the functions uε(x0+ε), which converge in measure to u¯x0surf in B1(0) by (6.2), (6.3), and (6.5) (ii). Then we scale back to Bε(x0). Hence, (6.1) (iii) holds.

It remains to prove (6.1) (ii). We notice that this easily follows from

(6.8)limε0ε-(d-1+p)Bsε±(x0)|aε±-u¯x0surf|pdx=0.

In fact, (6.2) (ii) and (6.6) give that

(6.9)limε0ε-(d-1+p)Bsε±(x0)|v^ε±-aε±|pdx=0.

Then (6.8), (6.9), the triangle inequality, sε(1-θ)ε, and (6.7) imply (6.1) (ii).

Therefore, let us now confirm (6.8). We only address the “+” case, for the “-” case is analogous. We first observe that by (6.2) (iii), (6.3), and a diagonal argument, we may find a sequence (ϱε)ε(0,+) with limε0ϱε=0 such that the sets

ω^ε+:={xBε(x0):|u(x)-u+(x0)|>ϱε}BsεΓ,+(x0)

satisfy

(6.10)limε0ε-dd(ω^ε+)=0.

In view of (6.5) (i), (iii), we have that

(6.11)u-aε+Lp(BsεΓ,+(x0)ωε+)cεe(u)Lp(BεΓ,+(x0)).

Then, by (6.11), the definition of ω^ε+, and the triangle inequality we get that

(6.12)u+(x0)-aε+Lp(BsεΓ,+(x0)(ωε+ω^ε+))cεe(u)Lp(BεΓ,+(x0))+γd1pεdpϱε.

By (6.2) (i), (6.3), (6.5) (ii), and (6.10) we obtain d(ωε+ω^ε+)12d(BsεΓ,+(x0)) for ε sufficiently small. Then, by Lemma 3.6, we have that γdεdpu+(x0)-aε+L(Bε(x0)) is less or equal than the right-hand side of (6.12), up to multiplication with a constant. This along with (6.2) (ii), p1, and the fact that ϱε0 implies

(6.13)limε0u+(x0)-aε+L(Bε(x0))=0.

Let us consider Aε+𝕄skewd×d and bε+d such that aε+(x)=Aε+(x-x0)+bε+. Then (6.8) follows by

(6.14)

(6.14a)limε0ε|Aε+|p=0,
(6.14b)limε0ε1-pp|bε+-u+(x0)|=0.

So we are left to prove (6.14) which corresponds to [28, equations (3.18)–(3.19)]. The proof goes in the same way with slight modifications that we indicate below. For fixed δ>0 small, by (6.2) (ii) there exists ε^>0, depending on δ, such that

(6.15)ε-(d-1)Bε(x0)|e(u)|pdxδpfor all εε^.

For ε~<ε<ε^, we set εk:=min{2kε~,ε} and adopt the notation k in place of εk in the subscripts. We then obtain

(6.16)ak+-ak+1+L(Bεk+(x0))cγd-1pεk-dpak+-ak+1+Lp(BsεkΓ,+(x0)(ωk+ωk+1+))cδεkp-1p.

In fact, the first inequality follows from (6.5) (ii) and Lemma 3.6, and the second one from (6.11), (6.15), and the triangle inequality. Similarly, employing (6.5) (i), (iv) in place of (6.11), and recalling aεk+=Ak+, we obtain

(6.17)|Ak+-Ak+1+|cδεk-1p.

At this stage, (6.14) follow exactly as in [28]: for k^ being the first index such that εk^=ε, recalling ε~=ε0 and summing (6.17) gives

ε~|Aε~+|pε~(|Ak^+|+k=0k^-1|Ak+-Ak+1+|)pcδp+cε~|Ak^+|p.

The right-hand side vanishes as ε~0 and δ0, and this proves (6.14a). Moreover, summing (6.16) (and since |bk+-bk+1+|ak+-ak+1+L(Bεk+(x0))), we obtain

|bε~+-bε+|cδεp-1p

for all 0<ε~ε. By passing to the limit as ε~0 together with (6.13), we get

ε1-pp|u+(x0)-bε+|cδ

for ε<ε^=ε^(δ). Thus, (6.14b) follows by the arbitrariness of δ>0, concluding the proof. ∎

Remark 6.2 (Construction of uε).

We point out that our definition of uε in (6.7) differs from the corresponding constructions in [13, Lemma 3] and [28, Lemma 3.4] in order to fix a possible flaw contained in these proofs. Roughly speaking, in our notation, in [13, 28], uε on Bsε(x0) is defined as

(6.18)uε={vε+in BsεΓ,+(x0),vε-in BsεΓ,-(x0).

Then, instead of (6.8), one needs to check

limε0ε-(d-1+p)BsεΓ±(x0)|aε±-u¯x0surf|pdx=0.

This, however, is in general false if lim infε0ε-(d-1+p)d(BεΓ,±(x0)Bε±(x0))>0 (which is clearly possible). Let us also remark that, in contrast to our construction, (6.18) allows to prove an estimate of the form

(6.19)limε0ε-(d-1)d-1(JuεJu)=0,

see [13, equation (24)] and [28, Lemma 3.4 (i)]. It is not clear to us if it is possible that for uε satisfying the fundamental blow-up property (6.1) (ii) one may still have an estimate of the form (6.19). The latter, however, is not needed for our proofs.

We now proceed with the proof of Lemma 4.3.

Proof of Lemma 4.3.

The proof follows the same strategy of the proof of Lemma 4.2. We fix x0Ju such that the statement of Lemma 6.1 holds at x0 and limε0ε-(d-1)μ(Bε(x0))=γd-1. This is possible for d-1-a.e. x0Ju. Then also limε0ε-(d-1)𝐦(u,Bε(x0)) exists, see Lemma 4.1. We prove (4.2) for x0 of this type.

Step 1 (Inequality “” in (4.2)). We fix η,θ>0, and consider zεGSBDp(B(1-3θ)ε(x0)) with zε=u¯x0surf in a neighborhood of B(1-3θ)ε(x0) and

(6.20)(zε,B(1-3θ)ε(x0))𝐦(u¯x0surf,B(1-3θ)ε(x0))+εd.

We extend zε to a function in GSBDp(Bε(x0)) by setting zε=u¯x0surf outside B(1-3θ)ε(x0). Let (uε)ε be the family given by Lemma 6.1. As in the proof of Lemma 4.2, we apply Lemma 3.8 on zε (in place of u) and uε (in place of v) for η fixed above and the sets

A=B1-2θ(x0),A=B1-θ(x0),A′′=B1(x0)B1-4θ(x0)¯.

Recalling notation (3.1) and (3.11)–(3.12), we find a function wεGSBDp(Bε(x0)) such that wε=uε on Bε(x0)B(1-θ)ε(x0) and

(6.21)(wε,Bε(x0))(1+η)((zε,Aε,x0)+(uε,Aε,x0′′))+Mεpzε-uεLp((AA)ε,x0)p+d(Bε(x0))η,

where M>0 depends on θ and η, but is independent of ε. In particular, we have wε=uε=u in a neighborhood of Bε(x0) by (6.1) (i). By (6.1) (ii) and the fact that zε=u¯x0surf outside B(1-3θ)ε(x0) we find

limε0ε-(d-1+p)zε-uεLp((AA)ε,x0)p=limε0ε-(d-1+p)uε-u¯x0surfLp(B(1-θ)ε(x0))p=0.

Inserting this in (6.21), we find that, for a suitable sequence (ρε)ε(0,+) with ρε0,

(6.22)(wε,Bε(x0))(1+η)((zε,Aε,x0)+(uε,Aε,x0′′))+εd-1ρε+γdεdη.

We evaluate the first terms on the right-hand side of (6.22): since zε=u¯x0bulk on Bε(x0)B(1-3θ)ε(x0)Aε,x0′′, by (H1), (H4), and (6.20) we have

lim supε0(zε,Aε,x0)εd-1lim supε0(zε,B(1-3θ)ε(x0))εd-1+lim supε0(u¯x0surf,Aε,x0′′)εd-1
lim supε0𝐦(u¯x0surf,B(1-3θ)ε(x0))εd-1+βd-1(A′′Π0)
(6.23)(1-3θ)d-1lim supε0𝐦(u¯x0surf,Bε(x0))εd-1+βγd-1(1-(1-4θ)d-1),

where, as in Lemma 6.1, we denote by Π0 the hyperplane passing through x0 with normal νu(x0). By (H4) and (6.1) (iii), (iv) we get

lim supε0(uε,Aε,x0′′)βεd-1lim supε0Aε,x0′′(1+|e(uε)|p)dxεd-1+lim supε0d-1(JuεAε,x0′′)εd-1
(6.24)=γd-1(1-(1-4θ)d-1).

Collecting (6.22), (6.23), (6.24), and recalling ρε0, as well as the fact that wε=u in a neighborhood of Bε(x0), we obtain

limε0𝐦(u,Bε(x0))γd-1εd-1lim supε0(wε,Bε(x0))γd-1εd-1(1+η)(1-3θ)d-1lim supε0𝐦(u¯x0surf,Bε(x0))γd-1εd-1+2(1+η)β(1-(1-4θ)d-1).

Passing to the limit as η,θ0, we conclude inequality “” in (4.2).

Step 2 (Inequality “” in (4.2)). We fix η,θ>0 and let, as in Step 1, (uε)ε be the family given by Lemma 6.1. By (6.1) (i) and Fubini’s Theorem, for each ε>0 we can find sε(1-4θ,1-3θ)ε such that

(6.25){(i)limε0ε-(d-1)d-1({uuε}Bsε(x0))=0,(ii)d-1((JuJuε)Bsε(x0))=0for all ε>0.

We consider zεGSBDp(Bsε(x0)) such that zε=u in a neighborhood of Bsε(x0), and

(6.26)(zε,Bsε(x0))𝐦(u,Bsε(x0))+εd.

We extend zε to a function in GSBDp(Bε(x0)) by setting

(6.27)zε=uεin Bε(x0)Bsε(x0).

We apply Lemma 3.8 for zε (in place of u), u¯x0surf (in place of v), and for the sets A, A, B as in Step 1, in correspondence to ε. By (3.11)–(3.12), there exists wεGSBDp(Bε(x0)) such that wε=u¯x0surf on Bε(x0)B(1-θ)ε(x0), and

(wε,Bε(x0))(1+η)((zε,Aε,x0)+(u¯x0surf,Aε,x0′′))+Mεpzε-u¯x0surfLp((AA)ε,x0)p+d(Bε(x0))η.

We observe that zε=uε outside B(1-3θ)ε(x0), by (6.27) and the choice of sε. Then, as done in Step 1, we may employ (6.1) (ii). This gives us a sequence (ρε)ε(0,+) with ρε0 as ε0 such that

(6.28)(wε,Bε(x0))(1+η)((zε,Aε,x0)+(u¯x0surf,Aε,x0′′))+εd-1ρε+γdεdη.

We estimate the first terms in (6.28). We get by (H1), (H4), (6.26)–(6.27), and the choice of sε that

(6.29)(zε,Aε,x0)𝐦(u,Bsε(x0))+εd+βd-1(({uuε}JuJuε)Bsε(x0))+(uε,Aε,x0′′).

By (6.24), (6.25), and the fact that sε(1-3θ)ε we thus deduce that

lim supε0(zε,Aε,x0)εd-1lim supε0𝐦(u,Bsε(x0))εd-1+βγd-1(1-(1-4θ)d-1)
(6.30)(1-3θ)d-1lim supε0𝐦(u,Bε(x0))εd-1+βγd-1(1-(1-4θ)d-1),

and, similarly to (6.23),

(6.31)lim supε0(u¯x0surf,Aε,x0′′)εd-1βγd-1(1-(1-4θ)d-1).

Collecting (6.28), (6.30), (6.31) and using ρε0, we derive

lim supε0(wε,Bε(x0))εd-1(1+η)((1-3θ)d-1lim supε0𝐦(u,Bε(x0))εd-1+2βγd-1(1-(1-4θ)d-1)).

Finally, recalling that wε=u¯x0surf in a neighborhood of Bε(x0), and using the arbitrariness of η, θ>0, we obtain

lim supε0𝐦(u¯x0surf,Bε(x0))γd-1εd-1lim supε0(wε,Bε(x0))γd-1εd-1lim supε0𝐦(u,Bε(x0))γd-1εd-1=limε0𝐦(u,Bε(x0))γd-1εd-1.

This shows “” in (4.2) and concludes the proof. ∎

7 The SBDp case

This section is devoted to the analysis of the integral representation result for :SBDp(Ω)×(Ω)[0,+) satisfying (H1)(H3) and (H4’). This case has been addressed, for d=2, in [28]. On the one hand, the arguments there could be now generalized to general dimension by virtue of Theorem 3.3. On the other hand, as we are going to show, the result can also be obtained with minor changes of our more general strategy.

We start by pointing out that, under (H4’), only competitors in SBDp may have finite energy. In fact, in view of Proposition 3.2, in the present setting definition (2.1) reads as

(7.1)𝐦(u,A)=infvSBDp(Ω){(v,A):v=u in a neighborhood of A}.

Then the following integral representation result holds.

Theorem 7.1 (Integral representation in SBDp).

Let ΩRd be open, bounded with Lipschitz boundary and suppose that F:SBDp(Ω)×B(Ω)[0,+) satisfies (H1)(H3) and (H4’). Then

(u,B)=Bf(x,u(x),u(x))dx+JuBg(x,u+(x),u-(x),νu(x))dd-1(x)

for all uSBDp(Ω) and BB(Ω), where f is given by

f(x0,u0,ξ)=lim supε0𝐦(x0,u0,ξ,Bε(x0))γdεd

for all x0Ω, u0Rd, ξMd×d, and x0,u0,ξ as in (2.2), and g is given by

g(x0,a,b,ν)=lim supε0𝐦(ux0,a,b,ν,Bε(x0))γd-1εd-1

for all x0Ω, a,bRd, νSd-1, and ux0,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.

Lemma 7.2.

Suppose that F satisfies (H1)(H3) and (H4’). Let uSBDp(Ω) and let μ=LdΩ+Hd-1JuΩ. Then for μ-a.e. x0Ω we have

limε0(u,Bε(x0))μ(Bε(x0))=limε0𝐦(u,Bε(x0))μ(Bε(x0)).

Proof.

One can follow the same argument used to prove Lemma 4.1 through Lemma 4.4.

First, we remark that [28, Lemmas 4.2 and 4.3] is proved under the assumptions (H4’) and uSBDp(Ω), hence it can be used to derive (4.15).

Concerning Lemma 4.4, as the lower bound of (H4’) is stronger than the one of (H4), the GSBD compactness result [25, Theorem 1.1] is still applicable. First, this shows vδGSBDp(Ω). Additionally, (4.9), (H4’), and Proposition 3.2 imply that the function vδ belongs indeed to SBDp(Ω). The rest of the proof remains unchanged, upon noticing that, under assumption (H4’), (4.10) still holds. ∎

We now address the adaptions necessary for the bulk density. When uSBDp(Ω), we show that the approximating sequence constructed in Lemma 5.1 satisfies some additional properties.

Lemma 7.3.

Let uSBDp(Ω). Let θ(0,1). For Ld-a.e. x0Ω there exists a family uεSBDp(Bε(x0)) such that (5.1) holds, and additionally

(7.2){(i)limε0ε-(d+1)Bε(x0)|uε-u|dx=0,(ii)limε0ε-dJuε|[uε]|dd-1=0.

Proof.

Since uSBD(Ω), for d-a.e. x0 it holds that

(7.3)limε0ε-dJuBε(x0)|[u]|dd-1=0,

and (see [4, Theorem 7.4]) that

(7.4)limε0ε-(d+1)Bε(x0)|u-u¯x0bulk|dx=0,

where for brevity we again let u¯x0bulk=x0,u(x0),u(x0). Hence, with Fubini’s Theorem we can fix sε(1-θ,1-θ2)ε so that d-1(JuBsε(x0))=0 and

(7.5)limε0ε-dBsε(x0)|u-u¯x0bulk|dd-1=0.

We can now perform the same construction as in (5.5) with sε in place of (1-θ)ε. Notice that in this case uεSBDp(Bε(x0)). By arguing exactly as in the proof of Lemma 5.1, we derive (5.1) (i), (iii), (iv) while (ii) holds in Bsε(x0) and a fortiori in B(1-θ)ε(x0). In particular, this in combination with Hölder’s inequality, (7.4), and u=uε in Bε(x0)Bsε(x0) yields (7.2) (i).

To see (7.2) (ii), observe that, since sεε is bounded from above and from below, (5.1) (ii), (iii), the fact that uεBsε(x0)W1,p(Bsε(x0);d) (see (5.5)), and Korn’s inequality imply that

uε(x0+sε)-u¯x0bulk(x0+sε)sε0in W1,p(B1(0);d).

Hence, by the trace inequality and by scaling back to Bsε(x0), we obtain

limε0sε-(d-1)Bsε(x0)1sε|uε-u¯x0bulk|dd-1=0.

With this, (7.5), and the fact that sεε is bounded from below we then have

limε0ε-dBsε(x0)|uε-u|dd-1=0.

Hence, we get by the construction of uε and (7.3) that

limε0ε-dJuε|[uε]|dd-1=limε0ε-dJu(Bε(x0)Bsε(x0))|[u]|dd-1=0,

which concludes the proof. ∎

With the above lemma at our disposal, we can deduce the asymptotic equivalence of the minimization problems (7.1) for u and u¯x0bulk=x0,u(x0),u(x0).

Lemma 7.4.

Suppose that F satisfies (H1), (H3), and (H4’). Let uSBDp(Ω). Then for Ld-a.e. x0Ω we have

(7.6)limε0𝐦(u,Bε(x0))γdεd=lim supε0𝐦(u¯x0bulk,Bε(x0))γdεd.

Proof.

We argue as in the proof of Lemma 4.2.

For the ”” inequality, take uε 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 3.8.) Notice that, in this case, we have by (H4’) and (5.9) that

(uε,Aε,x0′′)βAε,x0′′(1+|e(uε)|p)dx+βJuεAε,x0′′(1+|[uε]|)dd-1.

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 uε also satisfies (7.2), in addition to (5.15) we may require that

(7.7)limε0ε-dBsε(x0)|u+-uε-|dd-1=0,

where uε- and u+ indicate the inner and outer traces at Bsε(x0), 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

(zε,Aε,x0)𝐦(u,Bsε(x0))+εd+1+βd-1(({uuε}JuJuε)Bsε(x0))+βBsε(x0)|u+-uε-|dd-1+(uε,Aε,x0′′),

so that, using (5.14), (5.15), (7.7), and the fact that sε(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). ∎

Similar changes have to be performed also for the surface density. We first deduce the analogue of Lemma 6.1. We again set u¯x0surf=ux0,u+(x0),u-(x0),νu(x0) for brevity, see (2.3). We also recall the notation in (3.1).

Lemma 7.5.

Let uSBDp(Ω) and θ(0,1). For Hd-1-a.e. x0Ju there exists a family uεSBDp(Bε(x0)) such that (6.1) holds, and additionally

(7.8){(i)limε0ε-dBε(x0)|uε-u|dx=0,(ii)limε0ε-(d-1)JuεEε,x0|[uε]|dd-1=|[u¯x0surf]|d-1(Π0E)for all Borel sets EB1(x0),

where Π0 denotes the hyperplane passing through x0 with normal νu(x0).

Proof.

Since uSBD(Ω), for d-1-a.e. x0Ju it holds that

(7.9)limε0ε-dBε(x0)|u-u¯x0surf|dx=0.

Hence, by Fubini’s Theorem we find sε(1-θ,1-θ2)ε such that (6.4) holds, we have

d-1(JuBsε(x0))=0

and

(7.10)limε0ε-(d-1)Bsε(x0)|u-u¯x0surf|dd-1=0.

We can now perform the same construction as in (6.7) and derive (6.1). In particular, this in combination with Hölder’s inequality, p>1, (7.9), and u=uε in Bε(x0)Bsε(x0) yields (7.8) (i).

To see (7.8) (ii), observe that, since sεε is bounded from above and from below, (6.1) (ii), (iv), p>1, the fact that uεBsε±(x0)W1,p(Bsε±(x0);d) (see (6.7)), and Korn’s inequality imply that

sε-1(Bsε±(x0)-x0)|uε(x0+sεy)-u¯x0surf|pdy+sε-1(Bsε±(x0)-x0)|yuε(x0+sεy)|pdy0.

Hence, by the trace inequality and by scaling back to Bsε±(x0), we obtain

limε0sε-(d-1)Bsε±(x0)|uε-u±(x0)|dd-1=0.

Since sεε is bounded from below, we then get that

(7.11)limε0ε-(d-1)Bsε±(x0)|uε-u±(x0)|dd-1=0.

Given a Borel set EB1(x0), we define Eε=EBsεε(x0) for every ε>0. Then by (6.7) and (7.11) we get

(7.12)limε0(ε-(d-1)JuεEε,x0ε|[uε]|dd-1-|[u¯x0surf]|d-1(Π0Eε))=0.

By (7.10) and (7.11) we also have

limε0ε-(d-1)Bsε(x0)|uε--u+|dd-1=0,

where uε- and u+ indicate the inner and outer traces at Bsε(x0), respectively. Hence, by construction of uε in (6.7) and since uSBD(Ω), we obtain

limε0(ε-(d-1)Juε(EEε)ε,x0|[uε]|dd-1-|[u¯x0surf]|d-1(Π0(EEε)))=limε0(ε-(d-1)Ju(EEε)ε,x0|[u]|dd-1-|[u¯x0surf]|d-1(Π0(EEε)))=0.

By 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.

Lemma 7.6.

Suppose that F satisfies (H1), (H3), and (H4’). Let uSBDp(Ω). Then for Hd-1-a.e. x0Ju we have

(7.13)limε0𝐦(u,Bε(x0))γd-1εd-1=lim supε0𝐦(u¯x0surf,Bε(x0))γd-1εd-1.

Proof.

We argue as in the proof of Lemma 4.3.

For the “” inequality, take uε satisfying (6.1) and (7.8), and perform the same construction as in Lemma 4.3. Estimates (6.22) and (6.23) continue to hold, provided one replaces β with the larger constant β(1+|[u¯x0surf]|). Then, with (H4’), (6.1) (iii), (iv), and (7.8) (ii) we get

(7.14)lim supε0(uε,Aε,x0′′)εd-1β(1+|[u¯x0surf]|)γd-1(1-(1-4θ)d-1),

which is the analogue of (6.24). This is enough to derive inequality “” in (7.13).

For the reverse one, take again uε satisfying (6.1) and (7.8). Then, by (7.8) (i), in addition to (6.25) we may also require that

(7.15)limε0ε-(d-1)Bsε(x0)|u+-uε-|dd-1=0,

where uε- and u+ indicate the inner and outer traces at Bsε(x0), respectively. We perform the same construction as in Lemma 4.3. In this case, inequality (6.29) is replaced by

(zε,Aε,x0)𝐦(u,Bsε(x0))+εd+βd-1(({uuε}JuJuε)Bsε(x0))+βBsε(x0)|u+-uε-|dd-1+(uε,Aε,x0′′),

so that, using (6.25), (7.14), (7.15), and the fact that sε(1-3θ)ε we deduce that

lim supε0(zε,Aε,x0)εd-1(1-3θ)d-1lim supε0𝐦(u,Bε(x0))εd-1+β(1+|[u¯x0surf]|)γd-1(1-(1-4θ)d-1).

Then one concludes exactly in the same way, upon replacing β in equation (6.31) with the larger constant β(1+|[u¯x0surf]|). ∎

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. ∎

8 The GSBVp case

In this section we briefly remark that the strategy devised in this paper allows also to establish an integral representation result in the space GSBVp(Ω;m) for m. We consider functionals

:GSBVp(Ω;m)×(Ω)[0,+)

with the following general assumptions:

  1. (u,) is a Borel measure for any uGSBVp(Ω;m),

  2. (,A) is lower semicontinuous with respect to convergence in measure on Ω for any A𝒜(Ω),

  3. (,A) is local for any A𝒜(Ω), in the sense that if u,vGSBVp(Ω;m) satisfy u=v a.e. in A, then (u,A)=(v,A),

  4. there exist 0<α<β such that for any uGSBVp(Ω;m) and B(Ω) we have

    α(B|u|pdx+d-1(JuB))(u,B)β(B(1+|u|p)dx+d-1(JuB)).

In this setting, we replace definition (2.1) by

𝐦(u,A)=infvGSBVp(Ω;m){(v,A):v=u in a neighborhood of A}.

Moreover, as in (2.2)–(2.3), we define the functions x0,u0,ξ(x)=u0+ξ(x-x0) and ux0,a,b,ν(x)=a on the set {(x-x0)ν>0} and ux0,a,b,ν(x)=b on the set {(x-x0)ν<0} for x0Ω, u0m, ξ𝕄m×d, a,bm, and ν𝕊d-1.

Theorem 8.1 (Integral representation in GSBVp).

Let ΩRd be open, bounded with Lipschitz boundary, let mN, and suppose that F:GSBVp(Ω;Rm)×B(Ω)[0,+) satisfies ($\mathbb{H}$1)($\mathbb{H}$4). Then

(u,B)=Bf(x,u(x),u(x))dx