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BY 4.0 license Open Access Published 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 EMAIL 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 ( 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 [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 GSBD p , 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 SBD p , obtained in dimension two [S. Conti, M. Focardi and F. Iurlano, Integral representation for functionals defined on SBD p 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 ( GSBV p ).

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 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 d , whose displacement field with respect to the equilibrium is u : Ω 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 , 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 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 ) 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 ) | p d x + J u Ω ( 1 + | [ u ] | ) d d - 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 ) | p d x + d - 1 ( J u Ω ) .

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, 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 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 [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 P R ( 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

: GSBD p ( 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 u GSBD p ( 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 x 0 Ω with vanishing radii. With this we mean that the two quantities have the same Radon–Nikodym derivative with respect to μ := d Ω + d - 1 J u Ω (Lemma 4.1),

  2. one then proves that the Radon–Nikodym derivative d 𝐦 ( u , ) d μ 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 two-valued 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]. 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

v δ := v i δ χ B i δ

approximate u in the topology of the convergence in measure, where B i δ is a fine cover of a given set with disjoint balls of radius smaller than δ and v i δ denote minimizers for 𝐦 ( u , B i δ ) . In [28], the lower bound in (1) allows to control the distributional symmetrized gradient E u which along with a scaling argument and the classical Korn–Poincaré inequality in BD (see [50, Theorem 2.2]) shows that v i δ is close to u on each B i δ . (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 i δ are close to u in L p up to exceptional sets ω i δ whose volumes scale like δ ( ( u , B i δ ) + μ ( B i δ ) ) .

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, in Section 8 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

Ω | u | p d x + d - 1 ( J u Ω )

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 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.

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 GSBD p 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 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 Section 8 we explain how our method can be used to establish an integral representation result in GSBV p .

2 The integral representation result

In this section we present our main result. We start with some basic notation. Let Ω d 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 x d and ε > 0 we indicate by B ε ( x ) d the open ball with center x and radius ε. For x, y d , we use the notation x y for the scalar product and | x | for the Euclidean norm. Moreover, we let 𝕊 d - 1 := { x 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 m is indicated by γ m 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 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 3.5) 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 : GSBD p ( Ω ) × ( Ω ) [ 0 , + ) with the following general assumptions:

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

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

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

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

    α ( B | e ( u ) | p d x + d - 1 ( J u B ) ) ( u , B ) β ( B ( 1 + | e ( u ) | p ) d x + d - 1 ( J u B ) ) .

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 u GSBD p ( Ω ) and A 𝒜 ( Ω ) we define

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

For x 0 Ω , u 0 d , and ξ 𝕄 d × d we introduce the functions x 0 , u 0 , ξ : d d by

(2.2) x 0 , u 0 , ξ ( x ) = u 0 + ξ ( x - x 0 ) .

Moreover, for x 0 Ω , a , b d , and ν 𝕊 d - 1 we introduce u x 0 , a , b , ν : d d by

(2.3) u x 0 , a , b , ν ( x ) = { a if  ( x - x 0 ) ν > 0 , b if  ( x - x 0 ) ν < 0 .

In this paper, we will prove the following result.

Theorem 2.1 (Integral representation in GSBD p ).

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

( u , B ) = B f ( x , u ( x ) , u ( x ) ) d x + J u B g ( x , u + ( x ) , u - ( x ) , ν u ( x ) ) d d - 1 ( x )

for all u GSBD p ( Ω ) and B B ( Ω ) , where f is given by

(2.4) f ( x 0 , u 0 , ξ ) = lim sup ε 0 𝐦 ( x 0 , u 0 , ξ , B ε ( x 0 ) ) γ d ε d

for all x 0 Ω , u 0 R d , ξ M d × d , and g is given by

(2.5) g ( x 0 , a , b , ν ) = lim sup ε 0 𝐦 ( u x 0 , a , b , ν , B ε ( x 0 ) ) γ d - 1 ε d - 1

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.

  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 W 1 , 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 : d d with e ( a ) = 0 , then there are two functions f : Ω × 𝕄 d × d [ 0 , + ) and g : Ω × d × 𝕊 d - 1 [ 0 , + ) such that

    ( u , B ) = B f ( x , e ( u ) ( x ) ) d x + J u B g ( x , [ u ] ( x ) , ν u ( x ) ) d d - 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 B ε ( x 0 ) by cubes Q ε ν ( x 0 ) with sidelength ε, centered at x 0 , and two faces orthogonal to ν = ν u ( x 0 ) .

  5. An analogous result holds on the space GSBV p ( 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

: SBD p ( Ω ) × ( Ω ) [ 0 , + )

satisfying (H1)(H3) and

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

    α ( B | e ( u ) | p d x + J u B ( 1 + | [ u ] | ) d d - 1 ) ( u , B ) β ( B ( 1 + | e ( u ) | p ) d x + J u B ( 1 + | [ u ] | ) d d - 1 ) .

In this case, SBD p ( 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 L 1 -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 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.

3 Preliminaries

We start this preliminary section by introducing some further notation. For E d , ε > 0 , and x 0 d we set

(3.1) E ε , x 0 := x 0 + ε ( E - x 0 ) .

The diameter of E is indicated by diam ( E ) . Given two sets E 1 , E 2 d , we denote their symmetric difference by E 1 E 2 . We write χ E for the characteristic function of any E 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 sym d × d and 𝕄 skew d × d , respectively.

3.1 BD and GBD functions

Let U d be open. A function v L 1 ( U ; d ) belongs to the space of functions of bounded deformation, denoted by BD ( U ) , if the distribution E v := 1 2 ( D v + ( D v ) T ) is a bounded 𝕄 sym d × d -valued Radon measure on U, where D v = ( D 1 v , , D d v ) is the distributional differential. It is well known (see [4, 50]) that for v BD ( U ) the jump set J v is countably d - 1 -rectifiable (in the sense of [5, Definition 2.57]), and that

E v = E a v + E c v + E j v ,

where E a v is absolutely continuous with respect to d , E c v is singular with respect to d and such that | E c v | ( B ) = 0 if d - 1 ( B ) < , while E j v is concentrated on J v . The density of E a v with respect to d is denoted by e ( v ) .

The space SBD ( U ) is the subspace of all functions v BD ( U ) such that E c v = 0 . For p ( 1 , ) , we define SBD p ( U ) := { v SBD ( U ) : e ( v ) L p ( U ; 𝕄 sym d × d ) , d - 1 ( J v ) < } . 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 U d be a bounded open set, and let v : U d be measurable. We introduce the notation

Π ξ := { y d : y ξ = 0 } , B y ξ := { t : y + t ξ B }  for any  y d  and  B d

for fixed ξ 𝕊 d - 1 , and for every t B y ξ we let

v y ξ ( t ) := v ( y + t ξ ) , v ^ y ξ ( t ) := v y ξ ( t ) ξ .

Then v GBD ( U ) if there exists λ v b + ( U ) such that v ^ y ξ BV loc ( U y ξ ) for d - 1 -a.e. y Π ξ , and for every Borel set B U ,

Π ξ ( | D v ^ y ξ | ( B y ξ J v ^ y ξ 1 ) + 0 ( B y ξ J v ^ y ξ 1 ) ) d d - 1 ( y ) λ v ( B ) , J v ^ y ξ 1 := { t J v ^ y ξ : | [ v ^ y ξ ] | ( t ) 1 } .

Moreover, the function v belongs to GSBD ( U ) if v GBD ( U ) and v ^ y ξ SBV loc ( U y ξ ) for every ξ 𝕊 d - 1 and for d - 1 -a.e. y Π ξ .

We recall that every v GBD ( U ) has an approximate symmetric gradient e ( v ) L 1 ( U ; 𝕄 sym d × d ) and an approximate jump set J v which is still countably d - 1 -rectifiable (cf. [31, Theorem 9.1, Theorem 6.2]).

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 d and d - 1 , respectively). For x J v there exist v + ( x ) , v - ( x ) d and ν v ( x ) 𝕊 d - 1 such that

(3.2) lim ε 0 ε - d d ( { y B ε ( x ) : ± ( y - x ) ν v ( x ) > 0 } { | v - v ± ( x ) | > ϱ } ) = 0

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

GSBD p ( U ) := { v GSBD ( U ) : e ( v ) L p ( U ; 𝕄 sym d × d ) , d - 1 ( J v ) < } .

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

Proposition 3.2.

If v GSBD p ( U ) is such that [ v ] L 1 ( J v ; R d ) , then v SBD p ( U ) .

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 d - 1 -a.e. x U there exists tr ( v ) ( x ) d such that

(3.3) lim ε 0 ε - d d ( U B ε ( 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 : d d is an infinitesimal rigid motion if a is affine with e ( a ) = 1 2 ( a + ( a ) T ) = 0 .

Theorem 3.3 (Korn 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 ) > 0 such that for all u GSBD p ( Ω ) there is a set of finite perimeter ω Ω with

(3.4) d - 1 ( * ω ) c d - 1 ( J u ) , d ( ω ) c ( d - 1 ( J u ) ) d d - 1

and an infinitesimal rigid motion a such that

(3.5) u - a L p ( Ω ω ) + u - a L p ( Ω ω ) c e ( u ) L p ( Ω ) .

Moreover, 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 ) .

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 ( J u ) ) d d - 1 has not been shown, but it readily follows from d - 1 ( * ω ) c d - 1 ( J u ) 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 v W 1 , p ( Ω ; 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 W 1 , p ( Ω ; d ) we get

v - a L p ( Ω ) + v - a L p ( Ω ) c e ( u ) L p ( 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 ( 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 3.3 we can choose a constant c uniformly for all Ω k , see [20, Remark 4.2].

(iii) Recall definition (3.1). Consider a bounded Lipschitz domain Ω, ε > 0 , and x 0 d . Then for each u GSBD p ( Ω ε , x 0 ) we find ω Ω ε , x 0 and a rigid motion a such that

d - 1 ( * ω ) C d - 1 ( J u ) , d ( ω ) C ( d - 1 ( J u ) ) d d - 1

and

ε - 1 u - a L p ( Ω ε , x 0 ω ) + u - a L p ( Ω ε , x 0 ω ) C e ( u ) L p ( Ω ε , x 0 ) ,

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 u GSBD p ( Ω ) the approximate gradient u exists d -a.e. in Ω, see [20, Corollary 5.2].

Lemma 3.5 (Approximate gradient).

Let Ω R d be open, bounded with Lipschitz boundary, let 1 < p < + , and u GSBD p ( Ω ) . Then for L d -a.e. x 0 Ω there exists a matrix in M d × d , denoted by u ( x 0 ) , such that

lim ε 0 ε - d d ( { x B ε ( x 0 ) : | u ( x ) - u ( x 0 ) - u ( x 0 ) ( x - x 0 ) | | x - x 0 | > ϱ } ) = 0 for 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 GSBD 2 ( 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 1 p < + , let x 0 R d , and let R , θ > 0 . Let a : R d R d be affine, defined by a ( x ) = A x + b for x R d , and let E B R ( x 0 ) R d with L d ( E ) θ L d ( B R ( x 0 ) ) . Then there exists a constant c 0 > 0 only depending on p and θ such that

a L p ( B R ( x 0 ) ) γ d 1 p R d p a L ( B R ( x 0 ) ) c 0 a L p ( E ) , | A | c 0 R - 1 - d p a L p ( E ) .

We now proceed with another consequence of Theorem 3.3.

Corollary 3.7.

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

(3.6) d - 1 ( * ω ) C d - 1 ( J u ) , d ( ω ) C ( d - 1 ( J u ) ) d d - 1

such that

(3.7) u L p ( Ω ω ) + u L p ( Ω ω ) C e ( u ) L p ( Ω ) .

Proof.

We start by choosing a bounded Lipschitz domain Ω d with Ω Ω . Each u GSBD p ( Ω ) with tr ( u ) = 0 on Ω can be extended to a function u ~ GSBD p ( Ω ) by u ~ = 0 on Ω Ω such that J u ~ = J u . We first note that it is not restrictive to assume that

(3.8) d - 1 ( J u ) ( d ( Ω Ω ) 2 c ) d - 1 d ,

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 ( Ω Ω ) 2 c ) d - 1 d , d ( Ω ) C d ( Ω Ω ) 2 c

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 ~ GSBD p ( Ω ) and obtain a set ω Ω d satisfying (3.4) as well as an infinitesimal rigid motion a such that

(3.9) u ~ - a L p ( Ω ω ) + u ~ - a L p ( Ω ω ) c e ( u ~ ) L p ( Ω ) = c e ( u ) L p ( Ω ) .

In particular, u ~ = 0 on Ω Ω implies

(3.10) a L p ( Ω ( Ω ω ) ) c e ( u ) L p ( Ω ) .

By (3.4) and (3.8) we get d ( ω ) 1 2 d ( Ω Ω ) . In view of (3.10), we apply Lemma 3.6 on E = Ω ( Ω ω ) with R = diam ( Ω ) and θ = 1 2 d ( Ω Ω ) / γ d R d to get

a L p ( Ω ) c a L p ( Ω ( Ω ω ) ) c e ( u ) L p ( Ω ) ,

and, in a similar fashion, | a | c e ( u ) L p ( 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 GSBD p .

Lemma 3.8 (Fundamental estimate in GSBD p ).

Let Ω R d be open, bounded with Lipschitz boundary, and let 1 < p < + . Let η > 0 and let A , A , A ′′ A ( Ω ) with A A . For every functional F satisfying (H1), (H3), and (H4) and for every u GSBD p ( A ) , v GSBD p ( A ′′ ) there exists a function φ C ( R d ; [ 0 , 1 ] ) such that w := φ u + ( 1 - φ ) v GSBD p ( A A ′′ ) satisfies

(3.11) { (i) ( w , A A ′′ ) ( 1 + η ) ( ( u , A ) + ( v , A ′′ ) ) + M u - v L p ( ( A A ) A ′′ ) p + η d ( A A ′′ ) , (ii) w = u on  A    𝑎𝑛𝑑    w = v on  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 x 0 R d we have A ε , x 0 , A ε , x 0 , A ε , x 0 ′′ Ω , then

(3.12) M ( A ε , x 0 , A ε , x 0 , A ε , x 0 ′′ , p , η ) = ε - p M ( A , A , A ′′ , p , η ) ,

where we used the notation introduced in (3.1).

The same statement holds if F satisfies (H4’), u SBD p ( A ) , and v SBD p ( A ′′ ) .

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

Proof.

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

(3.13) k max { 3 p - 1 β η α , β η } .

Let A 1 , , A k + 1 be open subsets of d with

A A 1 A k + 1 A .

For i = 1 , , k let φ i C 0 ( A i + 1 ; [ 0 , 1 ] ) with φ i = 1 in a neighborhood V i of A i ¯ .

Consider u GSBD p ( A ) and v GSBD p ( A ′′ ) . We can clearly assume that u - v L p ( ( A A ) A ′′ ) as otherwise the result is trivial. We define the function w i = φ i u + ( 1 - φ i ) v GSBD p ( A A ′′ ) , where u and v are extended arbitrarily outside A and A ′′ , respectively. Letting T i = A ′′ ( A i + 1 A i ¯ ) we get by (H1) and (H3)

( w i , A A ′′ ) ( u , ( A A ′′ ) V i ) + ( v , A ′′ supp ( φ i ) ) + ( w i , T i )
(3.14) ( u , A ) + ( v , A ′′ ) + ( w i , T i ) .

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

( w i , T i ) β T i ( 1 + | e ( w i ) | p ) d x + β d - 1 ( J w i T i )
β T i ( 1 + | φ i e ( u ) + ( 1 - φ i ) e ( v ) + φ i ( u - v ) | p ) + β d - 1 ( ( J u J v ) T i )
β d ( T i ) + 3 p - 1 β T i ( | e ( u ) | p + | e ( v ) | p + | φ i | p | u - v | p ) + β d - 1 ( J u T i ) + β d - 1 ( J v T i )
3 p - 1 β α - 1 ( ( u , T i ) + ( v , T i ) ) + 3 p - 1 β ϕ i p u - v L p ( T i ) p + β d ( T i ) .

Notice that we can obtain the same estimate also if satisfies (H4’), 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 (H1), we find i 0 { 1 , , k } such that

( w i 0 , T i 0 ) 1 k i = 1 k ( w i , T i ) η ( ( u , A ) + ( v , A ′′ ) ) + M u - v L p ( ( A A ) A ′′ ) p + η d ( ( A A ) A ′′ ) ,

where M := 3 p - 1 β k - 1 max i = 1 , , k φ i p . This along with (3.14) concludes the proof of (3.11) by setting w = w i 0 . To see the scaling property (3.12), it suffices to use the cut-off functions φ i ε C 0 ( ( A i + 1 ) ε , x 0 ; [ 0 , 1 ] ) , i = 1 , , k , defined by φ i ε ( x ) = φ i ( x 0 + 1 ε ( x - x 0 ) ) for x ( A i + 1 ) ε , x 0 . 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 := d Ω + d - 1 J u Ω .

Lemma 4.1.

Suppose that F satisfies (H1)(H4). Let u GSBD p ( Ω ) and μ = L d Ω + H d - 1 J u Ω . Then for μ-a.e. x 0 Ω we have

lim ε 0 ( u , B ε ( x 0 ) ) μ ( B ε ( x 0 ) ) = lim ε 0 𝐦 ( u , B ε ( x 0 ) ) μ ( B ε ( x 0 ) ) .

We prove this lemma in the final part of this section. The second ingredient is that, asymptotically as ε 0 , the minimization problems 𝐦 ( u , B ε ( x 0 ) ) and 𝐦 ( u ¯ x 0 bulk , B ε ( x 0 ) ) coincide for d -a.e. x 0 Ω , where we write u ¯ x 0 bulk := x 0 , u ( x 0 ) , u ( x 0 ) for brevity, see (2.2).

Lemma 4.2.

Suppose that F satisfies (H1) and (H3)(H4) and let u GSBD p ( Ω ) . Then for L d -a.e. x 0 Ω we have

(4.1) lim ε 0 𝐦 ( u , B ε ( x 0 ) ) γ d ε d = lim sup ε 0 𝐦 ( u ¯ x 0 bulk , B ε ( x 0 ) ) γ d ε d .

We defer the proof of Lemma 4.2 to Section 5. The third ingredient is that, asymptotically as ε 0 , the minimization problems 𝐦 ( u , B ε ( x 0 ) ) and 𝐦 ( u ¯ x 0 surf , B ε ( x 0 ) ) coincide for d - 1 -a.e. x 0 J u , where we write u ¯ x 0 surf := u x 0 , u + ( x 0 ) , u - ( x 0 ) , ν u ( x 0 ) for brevity, see (2.3).

Lemma 4.3.

Suppose that F satisfies (H1) and (H3)(H4) and let u GSBD p ( Ω ) . Then for H d - 1 -a.e. x 0 J u we have

(4.2) lim ε 0 𝐦 ( u , B ε ( x 0 ) ) γ d - 1 ε d - 1 = lim sup ε 0 𝐦 ( u ¯ x 0 surf , B ε ( x 0 ) ) γ d - 1 ε d - 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. x 0 Ω one has

(4.3) d ( u , ) d d ( x 0 ) = f ( x 0 , u ( x 0 ) , u ( x 0 ) ) ,

where f was defined in (2.4), and that for d - 1 -a.e. x 0 J u one has

(4.4) d ( u , ) d d - 1 J u ( x 0 ) = g ( x 0 , u + ( x 0 ) , u - ( x 0 ) , ν u ( x 0 ) ) ,

where g was defined in (2.5).

By Lemma 4.1 and the fact that lim ε 0 ( γ d ε d ) - 1 μ ( B ε ( x 0 ) ) = 1 for d -a.e. x 0 Ω we deduce

d ( u , ) d d ( x 0 ) = lim ε 0 ( u , B ε ( x 0 ) ) μ ( B ε ( x 0 ) ) = lim ε 0 𝐦 ( u , B ε ( x 0 ) ) μ ( B ε ( x 0 ) ) = lim ε 0 𝐦 ( u , B ε ( x 0 ) ) γ d ε d <

for 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

lim ε 0 ( γ d - 1 ε d - 1 ) - 1 μ ( B ε ( x 0 ) ) = 1

for d - 1 -a.e. x 0 J u we deduce

d ( u , ) d d - 1 J u ( x 0 ) = lim ε 0 ( u , B ε ( x 0 ) ) μ ( B ε ( x 0 ) ) = lim ε 0 𝐦 ( u , B ε ( x 0 ) ) μ ( B ε ( x 0 ) ) = lim ε 0 𝐦 ( u , B ε ( x 0 ) ) γ d - 1 ε d - 1 <

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

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

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

(4.5) 𝐦 * ( u , A ) = lim δ 0 𝐦 δ ( 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 u GSBD p ( Ω ) . Then for all A A ( Ω ) there holds F ( u , A ) = m F * ( 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 B A , 𝐦 ( u , B ) ( u , B ) by definition. By (H1) we get 𝐦 δ ( 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 A 𝒜 ( Ω ) and δ > 0 . Let ( B i δ ) i be balls as in the definition of 𝐦 δ ( u , A ) such that

(4.6) i = 1 𝐦 ( u , B i δ ) 𝐦 δ ( u , A ) + δ .

By the definition of 𝐦 , we find v i δ GSBD p ( B i δ ) such that v i δ = u in a neighborhood of B i δ and

(4.7) ( v i δ , B i δ ) 𝐦 ( u , B i δ ) + δ d ( B i δ ) .

We define

(4.8)