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 () 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 , 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 , obtained in dimension two [S. Conti, M. Focardi and F. Iurlano, Integral representation for functionals defined on 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 ().
Integral representation results are a fundamental tool in the abstract theory of variational limits by Γ-convergence or in relaxation problems (see ). 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 of special functions of bounded variation [13, 15, 16] and on piecewise constant functions . In recent years, this analysis has been further improved to deal with functionals and variational limits on (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 ( 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 , whose displacement field with respect to the equilibrium is , the elastic strain is given by the symmetrized gradient . In standard models, the corresponding linear elastic energy is a suitable quadratic form of , 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 , 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. This set is characterized by the property that for , when blowing up around x, the jump set approximates a hyperplane with normal and the displacement field is close to two suitable values , on the two sides of the material with respect to this hyperplane.
Prototypical examples of functionals described above are energies which are controlled from above and below by suitable multiples of
where denotes the jump opening, or which are controlled by multiples of Griffith’s energy 
Whereas in case (1) the energy space is be given by , a subspace of , problems with control of type (2) are naturally formulated on generalized special functions of bounded deformation, introduced by Dal Maso . (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  who considered variational functionals controlled locally in terms of (1) in dimension . Let us mention that the behavior is quite different if linear growth on the symmetrized gradient is assumed (corresponding to ), as suited for the description of plasticity. In that case, representation results in the framework of have been obtained, for instance, in [10, 36] and  (see also [35, 48], containing essential tools for the proof).
The goal of the present article is twofold: we generalize the results of  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  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 -theory where problems for generalized functions of bounded variation can be reconducted to by a perturbation trick (see for instance ): one considers a small perturbation of the functional, depending on the jump opening, to represent functionals on . Then, by letting the perturbation parameter vanish and by truncating functions suitably, the representation can be extended to . 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 from the one in . 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 and the -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 of piecewise rigid functions has been proved in .
In our main result (Theorem 2.1), we prove an integral representation for variational functionals
( 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 .
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 . The proof strategy recovers the integral bulk and surface densities as blow-up limits of cell minimization formulas. The steps to be performed are the following:
one first shows that, for fixed , the set function is asymptotically equivalent to its minimum over competitors attaining the same boundary conditions as u on the boundaries of small balls centered in with vanishing radii. With this we mean that the two quantities have the same Radon–Nikodym derivative with respect to (Lemma 4.1),
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 , which generalizes a two-dimensional result in  (see also ) 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 , 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 , or the two-valued function with values and , 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  (with the topology of convergence in measure in place of ), and constitutes the counterpart of the -Poincaré inequality  used in the -case . 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.
approximate u in the topology of the convergence in measure, where is a fine cover of a given set with disjoint balls of radius smaller than δ and denote minimizers for . In , the lower bound in (1) allows to control the distributional symmetrized gradient which along with a scaling argument and the classical Korn–Poincaré inequality in (see [50, Theorem 2.2]) shows that is close to u on each . (In , the -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 are close to u in up to exceptional sets whose volumes scale like .
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 (see ) to arbitrary space dimensions, see Theorem 7.1. This is done by exploiting the stronger blow-up properties of functions. We note that, in principle, this result could be also obtained by adapting the arguments in  to higher dimension by employing the Korn inequality . 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 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 , if a local control on the full deformation gradient of the form
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 without necessity of the perturbation trick discussed before, relying on the result. Let us, however, mention that in  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  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 but more specifically on Sobolev functions (bulk density) and piecewise rigid functions  (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 . Section 3 is devoted to some preliminaries about the function space. In particular, we present the Korn-type inequality established in  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 -case. Finally, in Section 8 we explain how our method can be used to establish an integral representation result in .
2 The integral representation result
In this section we present our main result. We start with some basic notation. Let be open, bounded with Lipschitz boundary. Let be the family of open subsets of Ω, and denote by the family of Borel sets contained in Ω. For every and we indicate by the open ball with center x and radius ε. For x, , we use the notation for the scalar product and for the Euclidean norm. Moreover, we let and we denote by the set of matrices. The m-dimensional Lebesgue measure of the unit ball in is indicated by for every . We denote by and the d-dimensional Lebesgue measure and the k-dimensional Hausdorff measure, respectively.
For definition and properties of the space , , we refer the reader to . Some relevant properties are collected in Section 3 below. In particular, the approximate gradient is denoted by (it is well defined, see Lemma 3.5) and the (approximate) jump set is denoted by with corresponding normal and one-sided limits and . We also define .
We consider functionals with the following general assumptions:
is a Borel measure for any ,
is lower semicontinuous with respect to convergence in measure on Ω for any ,
is local for any , in the sense that if satisfy a.e. in A, then ,
there exist such that for any and we have
For , , and we introduce the functions by
Moreover, for , , and we introduce by
In this paper, we will prove the following result.
Theorem 2.1 (Integral representation in ).
for all and , where f is given by
for all , , , and g is given by
for all , , and .
We proceed with some remarks on the result.
As is lower semicontinuous on with respect to weak convergence, the integrand f is quasiconvex . Since is lower semicontinuous on piecewise rigid functions, the integrand g is -elliptic  (at least if one can ensure, for instance, that g has a continuous dependence in x). A fortiori, g is -elliptic .
If the functional additionally satisfies for all affine functions with , then there are two functions and such that
An analogous result holds on the space for . We refer to Section 8 for details.
We will additionally discuss the minor modifications needed in order to deal with functionals
there exist such that for any and we have
In this case, (see Section 3.1) is the natural energy space for . Furthermore, sequences of competitors with bounded energy, which are converging in measure, are additionally -convergent if we assume (H4’), due to the classical Korn–Poincaré inequality in (see [50, Theorem II.2.2]). Hence, in this latter case, (H2) is equivalent to requiring lower semicontinuity with respect to the -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.
We start this preliminary section by introducing some further notation. For , , and we set
The diameter of E is indicated by . Given two sets , we denote their symmetric difference by . We write for the characteristic function of any , which is 1 on E and 0 otherwise. If E is a set of finite perimeter, we denote its essential boundary by , see [5, Definition 3.60]. We denote the set of symmetric and skew-symmetric matrices by and , respectively.
3.1 BD and GBD functions
Let be open. A function belongs to the space of functions of bounded deformation, denoted by , if the distribution is a bounded -valued Radon measure on U, where is the distributional differential. It is well known (see [4, 50]) that for the jump set is countably -rectifiable (in the sense of [5, Definition 2.57]), and that
where is absolutely continuous with respect to , is singular with respect to and such that if , while is concentrated on . The density of with respect to is denoted by .
The spaces of generalized functions of bounded deformation and of generalized special functions of bounded deformation have been introduced in  (cf. [31, Definitions 4.1 and 4.2]), and are defined as follows.
Let be a bounded open set, and let be measurable. We introduce the notation
for fixed , and for every we let
Then if there exists such that for -a.e. , and for every Borel set ,
Moreover, the function v belongs to if and for every and for -a.e. .
We recall that every has an approximate symmetric gradient and an approximate jump set which is still countably -rectifiable (cf. [31, Theorem 9.1, Theorem 6.2]).
The notation for and , which is the same as that one in the case, is consistent: in fact, if v lies in , the objects coincide (up to negligible sets of points with respect to and , respectively). For there exist , and such that
for every , and the function is measurable. For , the space is given by
If is such that , then .
If U has Lipschitz boundary, for each the traces on are well defined (see [31, Theorem 5.5]), in the sense that for -a.e. there exists such that
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 for functions with small jump sets, see [20, Theorem 1.1, Theorem 1.2]. In the following, we say that is an infinitesimal rigid motion if a is affine with .
Theorem 3.3 (Korn inequality for functions with small jump set).
Let be a bounded Lipschitz domain and let . Then there exists a constant such that for all there is a set of finite perimeter with
and an infinitesimal rigid motion a such that
Moreover, there exists such that on and
Note that the result is indeed only relevant for functions with sufficiently small jump set, as otherwise one can choose , and (3.5) trivially holds. Note that, in , has not been shown, but it readily follows from by the isoperimetric inequality.
Remark 3.4 (Almost Sobolev regularity, constants, and scaling invariance).
(i) More precisely, in  it is proved that there exists such that on and , whence by Korn’s and Poincaré’s inequality in we get
(ii) Given a collection of bounded Lipschitz domains 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 , see [20, Remark 4.2].
(iii) Recall definition (3.1). Consider a bounded Lipschitz domain Ω, , and . Then for each we find and a rigid motion a such that
where is independent of ε. This follows by a standard rescaling argument.
Lemma 3.5 (Approximate gradient).
Let be open, bounded with Lipschitz boundary, let , and . Then for -a.e. there exists a matrix in , denoted by , such that
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.)
Let , let , and let . Let be affine, defined by for , and let with . Then there exists a constant only depending on p and θ such that
We now proceed with another consequence of Theorem 3.3.
Let be a bounded Lipschitz domain and let . Then there exists a constant such that for all with trace on (see (3.3)) there is a set of finite perimeter with
We start by choosing a bounded Lipschitz domain with . Each with on can be extended to a function by on such that . We first note that it is not restrictive to assume that
for a sufficiently large constant depending only on Ω and .
In particular, on implies
We conclude this subsection with another important tool in the proof of the integral representation, namely a fundamental estimate in .
Lemma 3.8 (Fundamental estimate in ).
where depends only on , but is independent of u and v. Moreover, if for and we have , then
where we used the notation introduced in (3.1).
The same statement holds if satisfies (H4’), , and .
In the statement above, we intend that if .
The proof follows the lines of [16, Proposition 3.1]. Choose such that
Let be open subsets of with
For let with in a neighborhood of .
Consider and . We can clearly assume that as otherwise the result is trivial. We define the function , where u and v are extended arbitrarily outside A and , respectively. Letting we get by (H1) and (H3)
For the last term, we compute using (H4) ( denotes the symmetrized vector product)
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 (see (2.1)) in the sense that the two quantities have the same Radon–Nikodym derivative with respect to .
We prove this lemma in the final part of this section. The second ingredient is that, asymptotically as , the minimization problems and coincide for -a.e. , where we write for brevity, see (2.2).
Proof of Theorem 2.1.
where f was defined in (2.4), and that for -a.e. one has
where g was defined in (2.5).
By Lemma 4.1 and the fact that for -a.e. we deduce
for -a.e. we deduce
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 and , we define
where, as before, . As is decreasing in δ, we can also introduce
In the following lemma, we prove that and coincide under our assumptions.
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 . For each ball , by definition. By (H1) we get for all . This shows , cf. (4.5).
We now address the reverse inequality. We fix and . Let be balls as in the definition of such that
By the definition of , we find such that in a neighborhood of and
for a constant only depending on p. Here, we used that and with trace zero on . We define by and observe that in measure on A is equivalent to as . In view of (4.8), we compute
By (4.11) (ii) and the fact that the balls are pairwise disjoint we get
As and , we further get by (4.11) (i)
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 gives readily that for every ,
The converse inequality follows by proving that, for every , the set
satisfies . To this end, we fix . We introduce the family of balls (depending on t)
and we show that
Then indeed holds true and the proof is concluded.
We now confirm (4.15). The inclusion follows from the definition of which permits to find, for any , such that . Note that, due to the left continuity of (cf. [28, Lemma 4.2]) one can also ensure the additional property by slightly varying ε. In order to prove that , one fixes a compact set , two positive numbers , and defines
By recalling also the definition of , we see that and are fine covers of K and , respectively. Thus there exists countable many pairwise disjoint , , and a set N with such that . In view of assumptions (H1), the definitions of , (in particular the balls have radii smaller than δ) give that
so that . Then by the regularity of μ. ∎
5 The bulk density
Lemma 5.1 (Blow-up at points with approximate gradient).
Let . Let . Then for -a.e. there exists a family such that
Let be such that
We define as
and proceed by confirming the properties stated in (5.1). Notice that, by construction, in . First, (5.1) (i) follows directly from the fact that , 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
By the definition of and the fact that in , we have for all . Hence, (5.3) (ii) and the triangle inequality give
Therefore, by using also (5.2) (i), we derive As was arbitrary, we get
We are now in a position to prove Lemma 4.2.
Proof of Lemma 4.2.
Step 1 (Inequality “” in (4.1)). We fix and . Choose with in a neighborhood of and
where depends on θ and η, but is independent of ε. Here and in the following, we use notation (3.1). In particular, we have in a neighborhood of by (5.1) (i). By (5.1) (ii), (5.9), and the fact that outside we find
This along with (5.10) shows that there exists a sequence with such that
By (5.1) (iii)–(iv) this implies
Passing to , we obtain inequality “” in (4.1).
We consider such that in a neighborhood of , and
We extend to a function in by setting
Passing to and recalling that in a neighborhood of , we derive
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 we adopt the notation for the function , see (2.3), with and being the approximate normal to and the traces on both sides of at , respectively. Recall also notation (3.1).
Lemma 6.1 (Blow-up at jump points).
Let and let . For -a.e. there exists a family such that
where denotes the hyperplane passing through with normal .
We start by using the fact that is countably -rectifiable and the blow-up properties of 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 -a.e. there exist , , , and a hypersurface Γ which is a graph of a function h defined on , being and Lipschitz, such that , is tangent to Γ in , separates in two open connected components for each , and
In particular, (ii) follows from the fact that and (iii) from (3.2). Then, since is tangent to Γ in , is the graph of a Lipschitz function defined on a subset of , with Lipschitz constant such that . Therefore, it holds that
where . By this and Fubini’s Theorem, for each we can find such that
where is independent of ε. (See Remark 3.4 (ii)–(iii) and recall that the Lipschitz constant of vanishes as .) By the Sobolev extension theorem we extend to , and (6.5) (iii)–(iv) along with the linearity of the extension operator yield
where, as before, the constant is independent of ε. (Here, we used again the properties of the functions recalled below (6.2).) We define as
where is defined below (6.3). We now prove the properties in (6.1). First, by definition we have that in a neighborhood of . By , (6.2) (i), (6.3), and (6.5) (i)–(ii) we obtain . This concludes (6.1) (i). Moreover, (6.2) (ii) and (6.6) imply (6.1) (iv). By the definition of and (6.5) (i) it holds that
We now show (6.1) (iii). Concerning the ”” inequality, for a fixed Borel set we have to estimate the measure of the intersection with and any of the five sets in the right-hand side above: it holds that
for any by rescaling, that
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 , which converge in measure to in by (6.2), (6.3), and (6.5) (ii). Then we scale back to . Hence, (6.1) (iii) holds.
It remains to prove (6.1) (ii). We notice that this easily follows from
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 with such that the sets
In view of (6.5) (i), (iii), we have that
Then, by (6.11), the definition of , and the triangle inequality we get that
By (6.2) (i), (6.3), (6.5) (ii), and (6.10) we obtain for ε sufficiently small. Then, by Lemma 3.6, we have that is less or equal than the right-hand side of (6.12), up to multiplication with a constant. This along with (6.2) (ii), , and the fact that implies
Let us consider and such that . Then (6.8) follows by
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 small, by (6.2) (ii) there exists , depending on δ, such that
For , we set and adopt the notation k in place of in the subscripts. We then obtain
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 , we obtain
for all . By passing to the limit as together with (6.13), we get
for . Thus, (6.14b) follows by the arbitrariness of , concluding the proof. ∎
Remark 6.2 (Construction of ).
We point out that our definition of 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], on is defined as
Then, instead of (6.8), one needs to check
This, however, is in general false if (which is clearly possible). Let us also remark that, in contrast to our construction, (6.18) allows to prove an estimate of the form
see [13, equation (24)] and [28, Lemma 3.4 (i)]. It is not clear to us if it is possible that for 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 such that the statement of Lemma 6.1 holds at and . This is possible for -a.e. . Then also exists, see Lemma 4.1. We prove (4.2) for of this type.
Step 1 (Inequality “” in (4.2)). We fix , and consider with in a neighborhood of and
We extend to a function in by setting outside . Let be the family given by Lemma 6.1. As in the proof of Lemma 4.2, we apply Lemma 3.8 on (in place of u) and (in place of v) for η fixed above and the sets
Inserting this in (6.21), we find that, for a suitable sequence with ,
Passing to the limit as , we conclude inequality “” in (4.2).
We consider such that in a neighborhood of , and
We extend to a function in by setting
and, similarly to (6.23),
Finally, recalling that in a neighborhood of , and using the arbitrariness of η, , we obtain
This shows “” in (4.2) and concludes the proof. ∎
7 The case
This section is devoted to the analysis of the integral representation result for satisfying (H1)–(H3) and (H4’). This case has been addressed, for , in . 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.
Then the following integral representation result holds.
Theorem 7.1 (Integral representation in ).
for all and , where f is given by
for all , , , and as in (2.2), and g is given by
for all , , , and 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.
Concerning Lemma 4.4, as the lower bound of (H4’) is stronger than the one of (H4), the compactness result [25, Theorem 1.1] is still applicable. First, this shows . Additionally, (4.9), (H4’), and Proposition 3.2 imply that the function belongs indeed to . 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 , we show that the approximating sequence constructed in Lemma 5.1 satisfies some additional properties.
Let . Let . For -a.e. there exists a family such that (5.1) holds, and additionally
Since , for -a.e. it holds that
and (see [4, Theorem 7.4]) that
where for brevity we again let . Hence, with Fubini’s Theorem we can fix so that and
We can now perform the same construction as in (5.5) with in place of . Notice that in this case . By arguing exactly as in the proof of Lemma 5.1, we derive (5.1) (i), (iii), (iv) while (ii) holds in and a fortiori in . In particular, this in combination with Hölder’s inequality, (7.4), and in yields (7.2) (i).
Hence, by the trace inequality and by scaling back to , we obtain
With this, (7.5), and the fact that is bounded from below we then have
Hence, we get by the construction of and (7.3) that
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 .
We argue as in the proof of Lemma 4.2.
For the ”” inequality, take 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
where and indicate the inner and outer traces at , 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
Let and . For -a.e. there exists a family such that (6.1) holds, and additionally
where denotes the hyperplane passing through with normal .
Since , for -a.e. it holds that
Hence, by Fubini’s Theorem we find such that (6.4) holds, we have
Hence, by the trace inequality and by scaling back to , we obtain
Since is bounded from below, we then get that
where and indicate the inner and outer traces at , respectively. Hence, by construction of in (6.7) and since , we obtain