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Relaxation for an optimal design problem in BD(Ω)

Published online by Cambridge University Press:  10 March 2022

Ana Cristina Barroso
Affiliation:
Departamento de Matemática and CMAFcIO, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edifício C6, Piso 1, 1749-016Lisboa, Portugal (acbarroso@ciencias.ulisboa.pt)
José Matias
Affiliation:
Departamento de Matemática and CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001Lisboa, Portugal (jose.c.matias@tecnico.ulisboa.pt)
Elvira Zappale
Affiliation:
Dipartimento di Scienze di Base ed Applicate per l'Ingegneria, Sapienza - Università di Roma, Via Antonio Scarpa, 16, 00161 Roma (RM), Italy CIMA, Universidade de Évora, Évora, Portugal (elvira.zappale@uniroma1.it)
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Abstract

We obtain a measure representation for a functional arising in the context of optimal design problems under linear growth conditions. The functional in question corresponds to the relaxation with respect to a pair $(\chi,u)$, where $\chi$ is the characteristic function of a set of finite perimeter and $u$ is a function of bounded deformation, of an energy with a bulk term depending on the symmetrized gradient as well as a perimeter term.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

1. Introduction

In optimal design one aims to find an optimal shape which minimizes a cost functional. The optimal shape is a subset $E$ of a bounded, open set $\Omega \subset \mathbb {R}^N$ which is described by its characteristic function $\chi : \Omega \to \{0,1\}$, $E = \{\chi = 1\}$, and, in the linear elasticity framework, the cost functional is usually a quadratic energy, so we are lead to the problem

(1.1)\begin{equation} \min_{(\chi,u)} \int_{\Omega}\chi(x) W_1({\mathcal{E}} u(x))+ (1 - \chi(x))W_0({\mathcal{E}} u(x))\,{\rm d}x, \end{equation}

where $W_0$ and $W_1$ are two elastic densities, with $W_0 \geq W_1$, and ${\mathcal {E}} u$ denotes the symmetrized gradient of the displacement $u$. We refer to the seminal papers [Reference Allaire and Lods1, Reference Kohn and Strang35Reference Kohn and Strang37, Reference Murat and Tartar43], among a wide literature (see, for instance, the recent contributions [Reference Babadjian, Iurlano and Rindler7, Reference Babadjian, Iurlano and Rindler8]).

However, as soon as plasticity comes into play, the observed stress–strain relation is no longer linear and, due to the linear growth of the stored elastic energy and to the lack of reflexivity of the space $L^1$, a suitable functional space is necessary to account for fields $u$ whose strains are measures. The space of special fields with bounded deformation, $BD(\Omega )$, was first proposed in [Reference Kohn and Strang36, Reference Kohn and Strang37, Reference Suquet47Reference Suquet50] and starting from these pioneering papers a vast literature developed.

Indeed, already in the case where $\chi \equiv \chi _\Omega$, the search for equilibria in the context of perfect plasticity leads naturally to the study of lower semicontinuity properties, and eventually relaxation, for energies of the type

(1.2)\begin{equation} \int_{\Omega}f({\mathcal{E}} u(x))\,{\rm d}x \end{equation}

where $f$ is the volume energy density. As mentioned above, $u$ belongs to the space $BD(\Omega )$ of functions of bounded deformation composed of integrable vector-valued functions for which all components $E_{ij}$, $i,j = 1,\ldots,N$, of the deformation tensor $Eu := {(Du + Du^T)}/{2}$ are bounded Radon measures and ${\mathcal {E}} u$ stands for the absolutely continuous part, with respect to the Lebesgue measure, of the symmetrized distributional derivative $Eu$.

Lower semicontinuity for (1.2) was established in [Reference Bellettini, Coscia and Dal Maso16] under convexity assumptions on $f$ and in [Reference Ebobisse27] for symmetric quasiconvex integrands, under linear growth conditions and for $u \in LD(\Omega )$, the subspace of $BD(\Omega )$ comprised of functions for which the singular part $E^su$ of the measure $Eu$ vanishes. For a symmetric quasiconvex density $f$ with an explicit dependence on the position in the body and satisfying superlinear growth assumptions, lower semicontinuity properties were established in [Reference Ebobisse28] for $u \in SBD(\Omega )$.

In the case where the energy density takes the form $\|{\mathcal {E}} u\|^2$ or $\|{\mathcal {E}}^D u\|^2 + (\text {div}\, u)^2$ (where $A^D$ stands for the deviator of the $N \times N$ matrix $A$ given by $A^D := A - ({1}/{N})\text {tr} (A)I$), and the total energy also includes a surface term, a first relaxation result was proved in [Reference Braides, Defranceschi and Vitali18]. We also refer to [Reference Kosiba and Rindler39] for the relaxation in the case where there is no surface energy and to [Reference Jesenko and Schmidt33, Reference Matias, Morandotti and Santos40, Reference Mora42] for related models concerning evolutions and homogenization, among a wider list of contributions.

For general energy densities $f$, Barroso, Fonseca and Toader [Reference Barroso, Fonseca and Toader12] studied the relaxation of (1.2) for $u \in SBD(\Omega )$ under linear growth conditions but placing no convexity assumptions on $f$. They showed that the relaxed functional admits an integral representation where a surface energy term arises naturally. The global method for relaxation due to Bouchitté, Fonseca and Mascarenhas [Reference Bouchitté, Fonseca and Mascarenhas17] was used to characterize the density of this term, whereas the identification of the relaxed bulk energy term relied on the blow-up method [Reference Fonseca and Müller31] together with a Poincaré-type inequality.

Ebobisse and Toader [Reference Ebobisse and Toader29] obtained an integral representation result for general local functionals defined in $SBD(\Omega )$ which are lower semicontinuous with respect to the $L^1$ topology and satisfy linear growth and coercivity conditions. The functionals under consideration are restrictions of Radon measures and are assumed to be invariant with respect to rigid motions. Their work was extended to the space $SBD^p(\Omega )$, $p > 1$, which arises in connection with the study of fracture and damage models, by Conti, Focardi and Iurlano [Reference Conti, Focardi and Iurlano24] in the two-dimensional setting. A crucial and novel ingredient of their proof is the construction of a $W^{1,p}$ approximation of an $SBD^p$ function $u$ using finite-elements on a countable mesh which is chosen according to $u$ (recall that $SBD^p$ denotes the space of fields with bounded deformation such that the symmetrized gradient is the sum of an $L^p$ field and a measure supported on a set of finite $\mathcal {H}^{N-1}$ measure).

The analysis of an integral representation for a variational functional satisfying lower semicontinuity, linear growth conditions and the usual measure theoretical properties, was extended to the full space $BD(\Omega )$ by Caroccia, Focardi and Van Goethem [Reference Caroccia, Focardi and Van Goethem23]. In this work, the invariance of the studied functional with respect to rigid motions, required in [Reference Ebobisse and Toader29], is replaced by a weaker condition stating continuity with respect to infinitesimal rigid motions. Their result relies, as in papers mentioned above, on the global method for relaxation, as well as on the characterization of the Cantor part of the measure $Eu$, due to De Philippis and Rindler [Reference De Philippis and Rindler25], which extends to the $BD$ case the result of Alberti's rank-one theorem in $BV$.

In the study of the minimization problem (1.1) one usually prescribes the volume fraction of the optimal shape, leading to a constraint of the form $\displaystyle \frac {1}{{\mathcal {L}}^N(\Omega )}\int _\Omega \chi (x)\,{\rm d}x= \theta,\;\theta \in (0,1)$. It is sometimes convenient to replace this constraint by inserting, instead, a Lagrange multiplier in the modelling functional which, in the optimal design context, becomes

(1.3)\begin{equation} F(\chi,u;\Omega) : =\displaystyle \int_{\Omega}\chi(x) W_1({\mathcal{E}} u(x))+ (1 - \chi(x))W_0({\mathcal{E}} u(x)) + \int_\Omega k \chi(x)\,{\rm d}x. \end{equation}

Despite the fact that we have compactness for $u$ in $BD(\Omega )$ for functionals of the form (1.3), it is well known that the problem of minimizing (1.3) with respect to $(\chi,u)$, adding suitable forces and/or boundary conditions, is ill-posed, in the sense that minimizing sequences $\chi _n \in L^\infty (\Omega ;\{0,1\})$ tend to highly oscillate and develop microstructure, so that in the limit we may no longer obtain a characteristic function. To avoid this phenomenon, as in [Reference Ambrosio and Buttazzo2, Reference Kohn and Lin34], we add a perimeter penalization along the interface between the two zones $\{\chi = 0\}$ and $\{\chi = 1\}$ (see [Reference Carita and Zappale21] for the analogous analysis performed in $BV$, and [Reference Barroso and Zappale14, Reference Barroso and Zappale15, Reference Carita and Zappale20] for the Sobolev settings, also in the presence of a gap in the growth exponents).

Thus, with an abuse of notation (i.e. denoting $W_1+ k$, in (1.3), still by $W_1$), our aim in this paper is to study the energy functional given by

(1.4)\begin{equation} F(\chi,u;\Omega) : =\displaystyle \int_{\Omega}\chi(x) W_1({\mathcal{E}} u(x))+ (1 - \chi(x))W_0({\mathcal{E}} u(x))\,{\rm d}x + |D\chi|(\Omega), \end{equation}

where $u \in BD(\Omega )$, $\chi \in BV(\Omega ;\{0,1\})$ and the densities $W_i$, $i = 0,1$, are continuous functions satisfying the following linear growth conditions from above and below:

(1.5)\begin{equation} \exists \, \alpha, \beta > 0 \text{ such that }\alpha |\xi| \leq W_i(\xi) \leq \beta (1 + |\xi|),\quad \forall \xi \in {\mathbb{R}}^{N \times N}_s. \end{equation}

We point out that no convexity assumptions are placed on $W_i$, $i = 0,1$.

To simplify the notation, in the sequel, we let $f: \{0,1\} \times \mathbb {R}^{N \times N}_s \to [0,+\infty )$ be defined as

(1.6)\begin{equation} f\left( q,\xi\right):=q W_1(\xi)+ (1-q)W_0(\xi), \end{equation}

and for a fixed $q \in \{0,1\}$, we recall that the recession function of $f$, in its second argument, is given by

(1.7)\begin{equation} f^{\infty}(q,\xi) := \limsup_{t\rightarrow+\infty}\frac{f(q,t\xi)}{t}. \end{equation}

Since we place no convexity assumptions on $W_i$, we consider the relaxed localized functionals arising from the energy (1.4), defined, for an open subset $A \subset \Omega$, by

(1.8)\begin{align} \mathcal{F}\left(\chi,u;A\right)& :=\inf\left\{ \liminf_{n\rightarrow +\infty} F(\chi_n,u_n;A): u_{n} \in W^{1,1}(A;\mathbb{R}^{N}),\chi_{n} \in BV(A;\{0,1\}),\right.\nonumber\\ & \quad \vphantom{\liminf_{n\rightarrow +\infty}}\left.u_{n}\to u\text{ in }L^{1}(A;\mathbb{R}^{N}),\chi_{n}\to\chi\text{ in }L^1(A;\{0,1\})\vphantom{\liminf_{n\rightarrow +\infty}}\right\}, \end{align}

and

(1.9)\begin{align} \mathcal{F}_{LD}\left(\chi,u;A\right)& :=\inf\left\{\liminf_{n\rightarrow +\infty} F(\chi_n,u_n;A): u_{n} \in LD(A), \chi_{n} \in BV(A;\{0,1\}),\right.\nonumber\\ & \quad \vphantom{\liminf_{n\rightarrow +\infty}}\left.u_{n}\to u\text{ in }L^{1}(A;\mathbb{R}^{N}),\chi_{n}\to\chi\text{ in }L^1(A;\{0,1\})\vphantom{\liminf_{n\rightarrow +\infty}}\right\}, \end{align}

where $LD(\Omega ):=\{u \in BD(\Omega ): E^s u=0\}$.

Due to the expression of (1.4), and to the fact that $\chi _n \mathop {\rightharpoonup }\limits ^{\ast }\chi$ in $BV$ if and only if $\{\chi _n\}$ is uniformly bounded in $BV$ and $\chi _n \to \chi$ in $L^1$, it is equivalent to take $\chi _n \mathop {\rightharpoonup }\limits ^{\ast }\chi$ in $BV$ or $\chi _n \to \chi$ in $L^1$ in the definitions of the functionals (1.8) and (1.9), obtaining for each of them the same infimum regardless of the considered convergence.

As a simple consequence of the density of smooth functions in $LD(\Omega )$ we show in remark 3.5 that, under the above growth conditions on $W_0, W_1$,

\[ \mathcal{F}\left(\chi,u;A\right) = \mathcal{F}_{LD}\left(\chi,u;A\right),\;\text{for every} \; \chi \in BV(A;\left\{0,1\right\}), u \in BD(\Omega), A \in {\mathcal{O}}(\Omega). \]

We prove in proposition 3.8 that $\mathcal {F}\left (\chi,u;\cdot \right )$ is the restriction to the open subsets of $\Omega$ of a Radon measure, the main result of our paper concerns the characterization of this measure.

Theorem 1.1 Let $f:\{0,1\} \times \mathbb {R}^{N \times N}_s\to [0, + \infty )$ be a continuous function as in (1.6), where $W_0$ and $W_1$ satisfy (1.5), and consider $F:BV(\Omega ;\{0,1\})\times BD(\Omega )\times \mathcal {O}(\Omega )$ defined in (1.4). Then

(1.10)\begin{align} \mathcal{F}\left(\chi,u;A\right)& =\int_A SQf(\chi(x),{\mathcal{E}} u(x))\,{\rm d}x\notag\\ & \quad +\int_{A \cap (J_\chi \cup J_u)} g(x,\chi^+(x),\chi^-(x),u^+(x),u^-(x),\nu(x))\,{\rm d}{\mathcal{H}}^{N-1}(x)\nonumber\\ & \quad + \int_A(SQf)^{\infty}(\chi(x), \frac{{\rm d} E^c u}{{\rm d} |E^c u|}(x))\,{\rm d}|E^c u|(x), \end{align}

where $SQf$ is the symmetric quasiconvex envelope of $f$ and $(SQf)^{\infty }$ is its recession function $($cf. § 2.3 and (1.7), respectively$)$. The relaxed surface energy density is given by

\[ g(x_0,a,b,c,d,\nu) := \limsup_{\varepsilon \to 0^+}\frac{m(\chi_{a,b,\nu}({\cdot}{-} x_0),u_{c,d,\nu}({\cdot}{-} x_0);Q_\nu(x_0,{\varepsilon}))} {{\varepsilon}^{N-1}} \]

where $Q_{\nu }(x_0,{\varepsilon })$ stands for an open cube with centre $x_0$, sidelength ${\varepsilon }$ and two of its faces parallel to the unit vector $\nu$,

\begin{align*} & m(\chi,u;V) := \inf\left\{\mathcal{F} (\theta,v;V) :\theta \in BV(\Omega;\{0,1\}), v \in BD(\Omega),\right.\\ & \left.\theta = \chi \text{ on } \partial V, v = u \text{ on } \partial V\right\}, \end{align*}

for any $V$ open subset of $\Omega$ with Lipschitz boundary, and, for $(a,b,c,d,\nu ) \in \{0,1\} \times \{0,1\} \times \mathbb {R}^{N} \times \mathbb {R}^{N} \times S^{N-1},$ the functions $\chi _{a,b,\nu }$ and $u_{c,d,\nu }$ are defined as

\[ \chi_{a,b,\nu}(y) := \begin{cases} a, & {\rm if\ }\; y \cdot \nu > 0\\ b, & {\rm if\ }\; y \cdot \nu < 0 \end{cases}\quad{\rm and}\quad u_{c,d,\nu}(y) := \begin{cases} c, & {\rm if\ }\; y \cdot \nu > 0\\ d, & {\rm if\ }\; y \cdot \nu < 0. \end{cases} \]

For the notation regarding the jump sets $J_\chi$, $J_u$ and the corresponding vectors $\chi ^+(x)$, $\chi ^-(x)$, $\nu _\chi (x)$, $u^+(x)$, $u^-(x)$ and $\nu _u(x)$ we refer to § 2.1, 2.2 and 4.3.

The above expression for the relaxed surface energy density arises as an application of the global method for relaxation [Reference Bouchitté, Fonseca and Mascarenhas17]. However, as we will see in § 4.3, in the case where $f$ satisfies the additional hypothesis (3.9), this density can be described more explicitly, leading to an integral representation for (1.8), in the $BD$ setting, entirely similar to the one in $BV$, obtained in [Reference Carita and Zappale21], when $W_0$ and $W_1$ depend on the whole gradient $\nabla u$. Indeed, under this assumption, we show that

\[ g(x_0,a,b,c,d,\nu) = K(a,b,c,d,\nu) \]

where

(1.11)\begin{align} & K(a,b,c,d,\nu)\notag\\ & \quad :=\inf\left\{\int_{Q_{\nu}}(SQf)^{\infty}(\chi(x),{\mathcal{E}} u(x))\,{\rm d}x+|D\chi|(Q_{\nu}):\left(\chi,u\right) \in\mathcal{A}(a,b,c,d,\nu)\right\}, \end{align}

and, for $(a,b,c,d,\nu ) \in \{0,1\} \times \{0,1\} \times \mathbb {R}^{N} \times \mathbb {R}^{N} \times S^{N-1},$ the set of admissible functions is

\begin{align*} \mathcal{A}(a,b,c,d,\nu)& :=\left\{ \left(\chi,u\right)\in BV_{\rm loc}\left(S_{\nu};\{0,1\}\right) \times W^{1,1}_{\rm loc}\left(S_{\nu};\mathbb{R}^{N}\right):\right.\nonumber\\ & (\chi(y),u(y)) = (a,c) \text{ if } y\cdot\nu=\frac{1}{2},\ (\chi(y),u(y)) = (b,d) \text{ if } y\cdot\nu={-}\frac{1}{2},\nonumber\\ & \left.(\chi, u)\text{ are 1-periodic in the directions of }\nu_{1},\dots,\nu_{N-1}\right\}, \end{align*}

$\left \{\nu _{1},\nu _{2},\dots,\nu _{N-1},\nu \right \}$ is an orthonormal basis of $\mathbb {R}^{N}$ and $S_\nu$ is the strip given by

\[ S_\nu = \left\{x \in {\mathbb{R}}^N : |x \cdot \nu| < \frac{1}{2}\right\}. \]

As an application of the result of Caroccia, Focardi and Van Goethem, obtained in the abstract variational functional setting in [Reference Caroccia, Focardi and Van Goethem23], the authors proved an integral representation for the relaxed functional, defined in $BD(\Omega ) \times \mathcal {O}(\Omega )$,

\[ \mathcal{F}_{0}(u; A) := \inf \left\{\liminf_{n \to + \infty}F_0(u_n;A) : u_n \in W^{1, 1}(A; {\mathbb{R}}^N), u_n \to u \; {\rm in} \; L^1(A;{\mathbb{R}}^N) \right\}, \]

where

\[ F_0(u;A) :=\begin{cases} \int_A f_{0} (x,u(x),{\mathcal{E}} u(x))\,{\rm d}x, & {\rm if}\ u \in W^{1,1}(\Omega;{\mathbb{R}}^N)\\ +\infty, & {\rm otherwise} \end{cases} \]

and the density $f_0$ satisfies linear growth conditions from above and below

\[ \frac{1}{C}|A| \leq f_0(x,u,A) \leq C\left(1 + |A|\right),\ \forall (x,u,A) \in \Omega \times {\mathbb{R}}^N \times {\mathbb{R}}^{N \times N}_s, \]

as well as a continuity condition with respect to $(x,u)$. This generalizes to the full space $BD(\Omega )$, and to the case of densities $f_0$ depending explicitly on $(x,u)$, the results obtained in [Reference Barroso, Fonseca and Toader12]. We will make use of their work in § 4.2 to prove both lower and upper bounds for the density of the Cantor part of the measure $\mathcal {F}(\chi,u;\cdot )$, by means of an argument based on Chacon's biting lemma which allows us to fix $\chi$ at an appropriately chosen point $x_0$, as in [Reference Matias, Morandotti and Zappale41].

The contents of this paper are organized as follows. In § 2 we fix our notation and provide some results pertaining to $BV$ and $BD$ functions and notions of quasiconvexity which will be used in the sequel. Section 3 contains some auxiliary results which are needed to prove our main theorem. In particular, in proposition 3.8 we show that $\mathcal {F}(\chi,u;\cdot )$ is the restriction to the open subsets of $\Omega$ of a Radon measure $\mu$. Section 4 is dedicated to the proof of our main theorem, which characterizes this measure. In each of § 4.1, 4.2 and 4.3 we prove lower and upper bounds of the densities of $\mu$ with respect to the bulk and Cantor parts of $Eu$, as well as with respect to a surface measure which is concentrated on the union of the jump sets of $\chi$ and $u$.

The fact that our functionals have an explicit dependence on the $\chi$ field prevented us from applying existing results (such as [Reference Arroyo-Rabasa, De Philippis and Rindler5, Reference Breit, Diening and Gmeineder19]) directly and required us to obtain direct proofs.

2. Preliminaries

In this section we fix notations and quote some definitions and results that will be used in the sequel.

Throughout the text $\Omega \subset \mathbb {R}^{N}$ will denote an open, bounded set with Lipschitz boundary.

We will use the following notations:

  • ${\mathcal {B}}(\Omega )$, ${\mathcal {O}}(\Omega )$ and ${\mathcal {O}}_{\infty }(\Omega )$ represent the families of all Borel, open and open subsets of $\Omega$ with Lipschitz boundary, respectively;

  • $\mathcal {M} (\Omega )$ is the set of finite Radon measures on $\Omega$;

  • $\left |\mu \right |$ stands for the total variation of a measure $\mu \in \mathcal {M} (\Omega )$;

  • $\mathcal {L}^{N}$ and $\mathcal {H}^{N-1}$ stand for the $N$-dimensional Lebesgue measure and the $\left (N-1\right )$-dimensional Hausdorff measure in $\mathbb {R}^N$, respectively;

  • the symbol ${\rm d} x$ will also be used to denote integration with respect to $\mathcal {L}^{N}$;

  • the set of symmetric $N \times N$ matrices is denoted by $\mathbb{R}_s^{N\times N}$;

  • given two vectors $a, b \in {\mathbb {R}}^N$, $a \odot b$ is the symmetric $N \times N$ matrix defined by $a \odot b : = \dfrac {a \otimes b + b \otimes a}{2}$, where $\otimes$ indicates tensor product;

  • $B(x, {\varepsilon })$ is the open ball in ${\mathbb {R}}^N$ with centre $x$ and radius ${\varepsilon }$, $Q(x,{\varepsilon })$ is the open cube in ${\mathbb {R}}^N$ with two of its faces parallel to the unit vector $e_N$, centre $x$ and sidelength ${\varepsilon }$, whereas $Q_{\nu }(x,{\varepsilon })$ stands for a cube with two of its faces parallel to the unit vector $\nu$; when $x=0$ and ${\varepsilon } = 1$, $\nu =e_N$ we simply write $B$ and $Q$;

  • $S^{N-1} := \partial B$ is the unit sphere in ${\mathbb {R}}^N$;

  • $C_c^{\infty }(\Omega ; {\mathbb {R}}^N)$ and $C_{\rm per}^{\infty }(Q; {\mathbb {R}}^N)$ are the spaces of ${\mathbb {R}}^N$-valued smooth functions with compact support in $\Omega$ and smooth and $Q$-periodic functions from $Q$ to ${\mathbb {R}}^N$, respectively;

  • by $\displaystyle \lim _{\delta, n}$ we mean $\displaystyle \lim _{\delta \to 0^+} \lim _{n\to +\infty }$, $\displaystyle \lim _{k, n}$ means $\displaystyle \lim _{k \to +\infty } \lim _{n\to +\infty }$;

  • $C$ represents a generic positive constant that may change from line to line.

2.1 BV functions and sets of finite perimeter

In the following we give some preliminary notions regarding functions of bounded variation and sets of finite perimeter. For a detailed treatment we refer to [Reference Ambrosio, Fusco and Pallara4].

Given $u \in L^1(\Omega ; {\mathbb {R}}^d)$ we let $\Omega _u$ be the set of Lebesgue points of $u$, i.e. $x\in \Omega _u$ if there exists $\widetilde u(x)\in {\mathbb {R}}^d$ such that

\[ \lim_{\varepsilon\to 0^+} \frac{1}{{\varepsilon}^N}\int_{B(x,\varepsilon)}|u(y)-\widetilde u(x)|\,{\rm d}y=0, \]

$\widetilde u(x)$ is called the approximate limit of $u$ at $x$. The Lebesgue discontinuity set $S_u$ of $u$ is defined as $S_u := \Omega {\setminus} \Omega _u$. It is known that ${\mathcal {L}}^{N}(S_u) = 0$ and the function $x \in \Omega \mapsto \widetilde u(x)$, which coincides with $u$ ${\mathcal {L}} ^N$-a.e. in $\Omega _u$, is called the Lebesgue representative of $u$.

The jump set of the function $u$, denoted by $J_u$, is the set of points $x\in \Omega {\setminus} \Omega _u$ for which there exist $a,\,b\in {\mathbb {R}}^d$ and a unit vector $\nu \in S^{N-1}$, normal to $J_u$ at $x$, such that $a\neq b$ and

\begin{align*} & \lim_{\varepsilon \to 0^+} \frac{1}{\varepsilon^N} \int_{\{ y \in B(x,\varepsilon) : (y-x)\cdot\nu > 0 \}} | u(y) - a|\,{\rm d}y = 0,\\ & \lim_{\varepsilon \to 0^+} \frac{1}{\varepsilon^N} \int_{\{ y \in B(x,\varepsilon) : (y-x)\cdot\nu < 0 \}} | u(y) - b|\,{\rm d}y = 0. \end{align*}

The triple $(a,b,\nu )$ is uniquely determined by the conditions above, up to a permutation of $(a,b)$ and a change of sign of $\nu$, and is denoted by $(u^+ (x),u^- (x),\nu _u (x)).$ The jump of $u$ at $x$ is defined by $[u](x) : = u^+(x) - u^-(x).$

We recall that a function $u\in L^{1}(\Omega ;{\mathbb {R}}^{d})$ is said to be of bounded variation, and we write $u\in BV(\Omega ;{\mathbb {R}}^{d})$, if all its first-order distributional derivatives $D_{j}u_{i}$ belong to $\mathcal {M}(\Omega )$ for $1\leq i\leq d$ and $1\leq j\leq N$.

The matrix-valued measure whose entries are $D_{j}u_{i}$ is denoted by $Du$ and $|Du|$ stands for its total variation. The space $BV(\Omega ; {\mathbb {R}}^d)$ is a Banach space when endowed with the norm

\[ \|u\|_{BV(\Omega ; {\mathbb{R}}^d)} = \|u\|_{L^1(\Omega ; {\mathbb{R}}^d)} + |Du|(\Omega ) \]

and we observe that if $u\in BV(\Omega ;\mathbb {R}^{d})$ then $u\mapsto |Du|(\Omega )$ is lower semicontinuous in $BV(\Omega ;\mathbb {R}^{d})$ with respect to the $L_{\mathrm {loc}}^{1}(\Omega ;\mathbb {R}^{d})$ topology.

By the Lebesgue decomposition theorem, $Du$ can be split into the sum of two mutually singular measures $D^{a}u$ and $D^{s}u$, the absolutely continuous part and the singular part, respectively, of $Du$ with respect to the Lebesgue measure $\mathcal {L}^N$. By $\nabla u$ we denote the Radon–Nikodým derivative of $D^{a}u$ with respect to $\mathcal {L}^N$, so that we can write

\[ Du= \nabla u \mathcal{L}^N \lfloor \Omega + D^{s}u. \]

If $u \in BV(\Omega )$ it is well known that $S_u$ is countably $(N-1)$-rectifiable, see [Reference Ambrosio, Fusco and Pallara4], and the following decomposition holds

\[ Du= \nabla u \mathcal{L}^N \lfloor \Omega + [u] \otimes \nu_u {\mathcal{H}}^{N-1}\lfloor S_u + D^cu, \]

where $D^cu$ is the Cantor part of the measure $Du$.

If $\Omega$ is an open and bounded set with Lipschitz boundary then the outer unit normal to $\partial \Omega$ (denoted by $\nu$) exists ${\mathcal {H}}^{N-1}$-a.e. and the trace for functions in $BV(\Omega ;{\mathbb {R}}^d)$ is defined.

Theorem 2.1 (Approximate differentiability)

If $u \in BV(\Omega ; {\mathbb {R}}^d),$ then for $\mathcal {L}^N$-a.e. $x \in \Omega$

(2.1)\begin{equation} \lim_{\varepsilon \rightarrow 0^+} \frac{1}{\varepsilon^{N+1}}\int_{Q(x, \varepsilon)} |u(y) - u(x) - \nabla u(x).(y-x)|\,{\rm d}y = 0. \end{equation}

Definition 2.2 Let $E$ be an $\mathcal {L}^{N}$-measurable subset of $\mathbb {R}^{N}$. For any open set $\Omega \subset \mathbb {R}^{N}$ the perimeter of $E$ in $\Omega$, denoted by $P(E;\Omega )$, is given by

(2.2)\begin{equation} P(E;\Omega):=\sup\left\{\int_{E} \mathrm{div}\,\varphi(x)\,{\rm d}x:\varphi\in C^{1}_{c}(\Omega;\mathbb{R}^{N}), \|\varphi\|_{L^{\infty}}\leq1\right\}. \end{equation}

We say that $E$ is a set of finite perimeter in $\Omega$ if $P(E;\Omega ) <+ \infty.$

Recalling that if $\mathcal {L}^{N}(E \cap \Omega )$ is finite, then $\chi _{E} \in L^{1}(\Omega )$, by [Reference Ambrosio, Fusco and Pallara4, proposition 3.6], it follows that $E$ has finite perimeter in $\Omega$ if and only if $\chi _{E} \in BV(\Omega )$ and $P(E;\Omega )$ coincides with $|D\chi _{E}|(\Omega )$, the total variation in $\Omega$ of the distributional derivative of $\chi _{E}$. Moreover, a generalized Gauss–Green formula holds:

\[ {\int_{E}\mathrm{div}\,\varphi(x)\,{\rm d}x=\int_{\Omega}\left\langle\nu_{E}(x),\varphi(x)\right\rangle\,{\rm d}|D\chi_{E}|,\quad\forall\,\varphi\in C_{c}^{1}(\Omega;\mathbb{R}^{N})}, \]

where $D\chi _{E}=\nu _{E}|D\chi _{E}|$ is the polar decomposition of $D\chi _{E}$.

The following approximation result can be found in [Reference Baldo9].

Lemma 2.3 Let $E$ be a set of finite perimeter in $\Omega$. Then, there exists a sequence of polyhedra $E_n$, with characteristic functions $\chi _n$, such that $\chi _n\to \chi$ in $L^1(\Omega ;\{0,1\})$ and $P (E_n;\Omega )\to P(E;\Omega )$.

2.2 BD and LD functions

We now recall some facts about functions of bounded deformation. More details can be found in [Reference Ambrosio, Coscia and Dal Maso3, Reference Barroso, Fonseca and Toader12, Reference Bellettini, Coscia and Dal Maso16, Reference Temam51, Reference Temam and Strang52].

A function $u\in L^{1}(\Omega ;{\mathbb {R}}^{N})$ is said to be of bounded deformation, and we write $u\in BD(\Omega )$, if the symmetric part of its distributional derivative $Du$, $Eu: = {(Du + Du^T)}/{2},$ is a matrix-valued bounded Radon measure. The space $BD(\Omega )$ is a Banach space when endowed with the norm

\[ \|u\|_{BD(\Omega)} = \|u\|_{L^1(\Omega; {\mathbb{R}}^N)} + |Eu|(\Omega). \]

We denote by $LD(\Omega )$ the subspace of $BD(\Omega )$ comprised of functions $u$ such that $Eu \in L^1(\Omega ;{\mathbb {R}}^{N\times N}_s)$, a counterexample due to [Reference Ornstein44] shows that $W^{1,1}(\Omega ;\mathbb {R}^N) \subsetneq LD(\Omega )$.

The intermediate topology in the space $BD(\Omega )$ is the one determined by the distance

\[ d(u,v) := \|u-v\|_{L^1(\Omega;{\mathbb{R}}^N)} + \left| |Eu|(\Omega) - |Ev |(\Omega)\right|,\;u, v \in BD(\Omega). \]

Hence, a sequence $\{u_n\} \subset BD(\Omega )$ converges to a function $u \in BD(\Omega )$ with respect to this topology, written $u_n \stackrel {i}{\to }u$, if and only if, $u_n \to u$ in $L^1(\Omega ;{\mathbb {R}}^N)$, $Eu_n \stackrel {*}{\rightharpoonup } Eu$ in the sense of measures and $|Eu_n|(\Omega ) \to |Eu|(\Omega )$.

Recall that if $u_n \to u$ in $L^1(\Omega ;{\mathbb {R}}^N)$ and there exists $C > 0$ such that $|Eu_n|(\Omega ) \leq C, \forall n \in {\mathbb {N}}$, then $u \in BD(\Omega )$ and

(2.3)\begin{equation} |Eu|(\Omega) \leq \liminf_{n\to +\infty}|Eu_n|(\Omega). \end{equation}

By the Lebesgue decomposition theorem, $Eu$ can be split into the sum of two mutually singular measures $E^{a}u$ and $E^{s}u$, the absolutely continuous part and the singular part, respectively, of $Eu$ with respect to the Lebesgue measure $\mathcal {L}^N$. The Radon–Nikodým derivative of $E^{a}u$ with respect to $\mathcal {L}^N$, is denoted by ${\mathcal {E}} u$ so we have

\[ Eu= {\mathcal{E}} u\,\mathcal{L}^N \lfloor \Omega + E^{s}u. \]

With these notations we may write

\[ LD(\Omega):=\{u \in BD(\Omega): E^s u=0\} \]

and (cf. [Reference Temam51]) $LD(\Omega )$ is a Banach space when endowed with the norm

\[ \|u\|_{LD(\Omega)}: = \|u\|_{L^1(\Omega;\mathbb{R}^N)} +\|{\mathcal{E}} u\|_{L^1(\Omega;\mathbb{R}^N)}. \]

If $\Omega$ is a bounded, open subset of ${\mathbb {R}}^N$ with Lipschitz boundary $\Gamma$, then there exists a linear, surjective and continuous, both with respect to the norm and to the intermediate topologies, trace operator

\[ \text{tr }: BD(\Omega) \to L^1(\Omega;{\mathbb{R}}^N) \]

such that tr $u = u$ if $u \in BD(\Omega ) \cap C(\overline {\Omega };{\mathbb {R}}^N)$. Furthermore, the following Gauss–Green formula holds

(2.4)\begin{equation} \int_{\Omega}(u \odot D \varphi)(x)\,{\rm d}x + \int_{\Omega}\varphi(x)\,{\rm d}Eu(x) =\int_{\Gamma}\varphi (x)(\text{tr}\,u \odot \nu)(x)\,{\rm d}\mathcal{H} ^{N-1}(x), \end{equation}

for every $\varphi \in C^1(\overline {\Omega })$ (cf. [Reference Ambrosio, Coscia and Dal Maso3, Reference Temam51]).

The following lemma is proved in [Reference Barroso, Fonseca and Toader12].

Lemma 2.4 Let $u \in BD(\Omega )$ and let $\rho \in C_0^{\infty }({\mathbb {R}}^N)$ be a non-negative function such that ${\rm supp}(\rho ) \subset \subset B(0,1)$, $\rho (-x) = \rho (x)$ for every $x \in {\mathbb {R}}^N$ and $\int _{{\mathbb {R}}^N}\rho (x)\,{\rm d}x = 1$. For any $n \in {\mathbb {N}}$ set $\rho _n(x) : = n^N\rho (nx)$ and

\[ u_n(x) := (u *\rho_n)(x) = \int_{\Omega}u(y)\rho_n(x-y)\,{\rm d}y,\quad\text{for}\ x \in \left\{y \in \Omega : {\rm dist}(y,\partial \Omega) > \frac{1}{n}\right\}. \]

Then $u_n \in C^{\infty }\left (\left \{y \in \Omega : {\rm dist}(y,\partial \Omega ) > \frac {1}{n}\right \};{\mathbb {R}}^N\right )$ and

  1. (i) for any non-negative Borel function $h : \Omega \to {\mathbb {R}}$

    \[ \int_{B(x_0,\varepsilon)}h(x) |{\mathcal{E}} u_n(x)|\,{\rm d}x \leq\int_{B(x_0,\varepsilon+ {1}/{n})}(h*\rho_n)(x)\,{\rm d}|E u|(x), \]
    whenever ${\varepsilon } + \frac {1}{n} < {\rm dist}(x_0,\partial \Omega );$
  2. (ii) for any positively homogeneous of degree one, convex function $\theta : {\mathbb {R}}^{N\times N}_{\rm sym} \to [0,+\infty [$ and any ${\varepsilon } \in \, ]0,{\rm dist}(x_0,\partial \Omega )[$ such that $|E u|(\partial B(x_0,{\varepsilon })) = 0$,

    \[ \lim_{n\rightarrow +\infty}\int_{B(x_0,\varepsilon)} \theta({\mathcal{E}} u_n(x))\,{\rm d}x =\int_{B(x_0,\varepsilon)}\theta\left(\frac{{\rm d}Eu}{{\rm d}|Eu|}\right)\,{\rm d}|Eu|, \]
  3. (iii) $\lim _{n \to + \infty }u_n(x) = \widetilde u(x)$ and $\lim _{n \to + \infty }(|u_n -u| * \rho _n)(x) = 0$ for every $x \in \Omega {\setminus} S_u$, whenever $u \in L^{\infty }(\Omega ;{\mathbb {R}}^N)$.

The following result, proved in [Reference Temam51], see also [Reference Barroso, Fonseca and Toader12, theorem 2.6], shows that it is possible to approximate any $BD(\Omega )$ function $u$ by a sequence of smooth functions which preserve the trace of $u$.

Theorem 2.5 Let $\Omega$ be a bounded, connected, open set with Lipschitz boundary. For every $u \in BD(\Omega )$, there exists a sequence of smooth functions $\{u_n\} \subset C^{\infty }(\Omega ;{\mathbb {R}}^N) \cap W^{1,1}(\Omega ;{\mathbb {R}}^N)$ such that $u_n \stackrel {i}{\to }u$ and tr $u_n =$ tr $u$. If, in addition, $u \in LD(\Omega )$, then ${\mathcal {E}} u_n \to {\mathcal {E}} u$ in $L^1(\Omega ;\mathbb{R}_s^{N\times N}).$

It is also shown in [Reference Temam51] that if $\Omega$ is an open, bounded subset of ${\mathbb {R}}^N$, with Lipschitz boundary, then $BD(\Omega )$ is compactly embedded in $L^q(\Omega ;{\mathbb {R}}^N)$, for every $1 \leq q < {N}/{(N-1)}.$ In particular, the following result holds.

Theorem 2.6 Let $\Omega$ be an open, bounded subset of $\mathbb {R}^N$, with Lipschitz boundary and let $1\leq q <{N}/{(N-1)}$. If $\{u_n\}$ is bounded in $BD(\Omega )$, then there exist $u \in BD(\Omega )$ and a subsequence $\{ u_{n_k}\}$ of $\{u_n\}$ such that $u_{n_k}\to u$ in $L^q (\Omega ;\mathbb {R}^N)$.

If $u \in BD(\Omega )$ then $J_u$ is countably $(N-1)$-rectifiable, see [Reference Ambrosio, Coscia and Dal Maso3], and the following decomposition holds

\[ Eu= {\mathcal{E}} u \mathcal{L}^N \lfloor \Omega + [u] \odot \nu_u {\mathcal{H}}^{N-1}\lfloor J_u + E^cu, \]

where $[u] = u^+ - u^-$, $u^{\pm }$ are the traces of $u$ on the sides of $J_u$ determined by the unit normal $\nu _u$ to $J_u$ and $E^cu$ is the Cantor part of the measure $Eu$ which vanishes on Borel sets $B$ with $\mathcal {H}^{N-1}(B) < + \infty.$

We end this subsection by pointing out that the equivalent of (2.1), with ${\mathcal {E}} u(x)$ replacing $\nabla u(x)$, is false (see [Reference Ambrosio, Coscia and Dal Maso3]). However the following result holds (cf. [Reference Ambrosio, Coscia and Dal Maso3, theorem 4.3] and [Reference Ebobisse28, theorem 2.5]).

Theorem 2.7 (Approximate symmetric differentiability)

If $u \in BD(\Omega ),$ then, for $\mathcal {L}^N$-a.e. $x \in \Omega$, there exists an $N\times N$ matrix $\nabla u(x)$ such that

(2.5)\begin{equation} \lim_{\varepsilon \rightarrow 0^+} \frac{1}{\varepsilon^{N+1}}\int_{B_\varepsilon(x)} |u(y) - u(x) - \nabla u(x)(y-x)|\,{\rm d}y = 0, \end{equation}
(2.6)\begin{equation} \lim_{\varepsilon \rightarrow 0^+} \frac{1}{\varepsilon^{N}}\int_{B_\varepsilon(x)} \frac{|\langle u(y) - u(x) - {\mathcal{E}} u(x)(y-x), y-x\rangle|}{|y-x|^2}\,{\rm d}y = 0, \end{equation}

for $\mathcal {L}^N$-a.e. $x \in \Omega$. Furthermore

\[ {\mathcal{L}}^N(\{x\in \Omega:|\nabla u(x)|>t \})\leq\frac{C(N,\Omega)}{t}\|u\|_{BD(\Omega)},\quad\forall\ t>0, \]

with $C(N,\Omega )>0$ depending only on $N$ and $\Omega$.

From (2.5) and (2.6) it follows that ${\mathcal {E}} u={(\nabla u+\nabla u^T)}/{2}$.

We denote by $\mathcal {R}$ the kernel of the linear operator $E$ consisting of the class of rigid motions in $\mathbb {R}^N$, i.e. affine maps of the form $Mx + b$ where $M$ is a skew-symmetric $N \times N$ matrix and $b\in \mathbb {R}^N$. $\mathcal {R}$ is therefore closed and finite-dimensional so it is possible to define the orthogonal projection $P : BD(\Omega )\to \mathcal {R}$. This operator belongs to the class considered in the following Poincaré–Friedrichs type inequality for $BD$ functions (see [Reference Ambrosio, Coscia and Dal Maso3, Reference Kohn38, Reference Temam51]).

Theorem 2.8 Let $\Omega$ be a bounded, connected, open subset of $\mathbb {R}^N$, with Lipschitz boundary, and let $R : BD(\Omega )\to \mathcal {R}$ be a continuous linear map which leaves the elements of $\mathcal {R}$ fixed. Then there exists a constant $C(\Omega, R)$ such that

\[ \int_\Omega |u(x) - R(u)(x)|\,{\rm d}x \leq C(\Omega,R)\,|E u|(\Omega),\ \text{ for every}\ u\in BD(\Omega). \]

2.3 Notions of quasiconvexity

Definition 2.9 ([Reference Barroso, Fonseca and Toader12], definition 3.1)

A Borel measurable function $f:\mathbb {R}^{N\times N}_s\to \mathbb {R}$ is said to be symmetric quasiconvex if

(2.7)\begin{equation} f(\xi)\leq\int_Q f(\xi+{\mathcal{E}} \varphi(x))\,{\rm d}x, \end{equation}

for every $\xi \in \mathbb {R}^{N\times N}_s$ and for every $\varphi \in C^{\infty }_{\rm per}(Q;\mathbb {R}^N)$.

Remark 2.10 The above property (2.7) is independent of the size, orientation and centre of the cube over which the integration is performed. Also, if $f$ is upper semicontinuous and locally bounded from above, using Fatou's lemma and the density of smooth functions in $LD(Q)$, it follows that in (2.7) $C^{\infty }_{\rm per}(Q;\mathbb {R}^N)$ may be replaced by $LD_{\rm per}(Q).$

Given $f:\mathbb {R}^{N\times N}_s\to \mathbb {R}$, the symmetric quasiconvex envelope of $f$, $SQf$, is defined by

(2.8)\begin{equation} SQf(\xi): = \inf \left\{\int_Q f(\xi+{\mathcal{E}} \varphi(x))\,{\rm d}x : \varphi \in C^{\infty}_{\rm per}(Q;\mathbb{R}^N)\right\}. \end{equation}

It is possible to show that $SQf$ is the greatest symmetric quasiconvex function that is less than or equal to $f$. Moreover, definition (2.8) is independent of the domain, i.e.

(2.9)\begin{equation} SQf(\xi): = \inf \left\{\frac{1}{{\mathcal{L}} ^N(D)}\int_{D} f(\xi+{\mathcal{E}} \varphi(x))\,{\rm d}x : \varphi \in C^{\infty}_{0}(D;\mathbb{R}^N)\right\} \end{equation}

whenever $D \subset {\mathbb {R}}^N$ is an open, bounded set with ${\mathcal {L}} ^N(\partial D) = 0$.

In [Reference Ebobisse27], a Borel measurable function $f:\mathbb {R}^{N\times N}_s\to \mathbb {R}$ is said to be symmetric quasiconvex if and only if

(2.10)\begin{equation} f(\xi)\leq \frac{1}{\mathcal{L}^N(D)}\int_D f(\xi +{\mathcal{E}}\varphi(x))\,{\rm d}x\text{ for all }\varphi \in W^{1,\infty}_0(D;\mathbb{R}^N), \end{equation}

and it is stated that $f$ is symmetric quasiconvex if and only if $f\circ \pi$ is quasiconvex in the sense of Morrey, where $\pi$ is the projection of $\mathbb {R}^{N \times N}$ onto $\mathbb {R}^{N\times N}_{s}$.

Let us show that these two notions coincide. Observe first that, for any $\varphi \in C^\infty _0(D;\mathbb {R}^N)$,

(2.11)\begin{align} SQf(\xi)\leq \frac{1}{\mathcal{L}^N(D)}\int_D SQf(\xi+ {\mathcal{E}} \varphi(x))\,{\rm d}x=\frac{1}{\mathcal{L}^N(D)}\int_D (SQf \circ \pi)(\xi+ \nabla\varphi(x))\,{\rm d}x. \end{align}

If $f$ is upper semicontinuous and satisfies a growth condition from above as in (1.5), then $SQf$ in (2.9) is symmetric quasiconvex also in the sense of [Reference Ebobisse27]. Indeed, $SQf$ satisfies the same growth condition (1.5) and a density argument as in [Reference Ball and Murat10] shows that $SQf \circ \pi$ is $W^{1,1}$-quasiconvex, hence $W^{1,\infty }$-quasiconvex, i.e. $\varphi$ can be chosen in $W^{1,\infty }_0(D;\mathbb {R}^N)$. Thus,

(2.12)\begin{equation} SQf(\xi)\leq \frac{1}{\mathcal{L}^N(D)}\int_D SQf(\xi+ {\mathcal{E}} \varphi(x))\,{\rm d}x\leq \frac{1}{\mathcal{L}^N(D)}\int_D f(\xi+ {\mathcal{E}} \varphi(x))\,{\rm d}x, \end{equation}

for every $\varphi \in W^{1,\infty }_0(D;\mathbb {R}^N)$. Therefore, denoting by $SQf_E$ the symmetric quasiconvexification

(2.13)\begin{equation} SQf_E(\xi) : = \inf \left\{\frac{1}{{\mathcal{L}} ^N(D)}\int_{D} f(\xi+{\mathcal{E}} \varphi(x))\,{\rm d}x : \varphi \in W^{1,\infty}_{0}(D;\mathbb{R}^N)\right\}, \end{equation}

and by $SQf$ the symmetric quasiconvexification defined through (2.9), trivially $SQf_E\leq SQf$ and by (2.12) we have equality.

Actually, under linear growth conditions and upper semicontinuity of $f$, we may also conclude that

\[ SQf_E(\xi) : = \inf \left\{\frac{1}{{\mathcal{L}}^N(D)}\int_{D} f(\xi+{\mathcal{E}} \varphi(x))\,{\rm d}x : \varphi \in W^{1,1}_{0}(D;\mathbb{R}^N)\right\}. \]

3. Auxiliary results

We recall that for $u \in BD(\Omega )$ and $\chi \in BV(\Omega ;\{0,1\})$ the energy under consideration is

(3.1)\begin{equation} F(\chi,u;\Omega) : =\int_{\Omega}\chi(x) W_1({\mathcal{E}} u(x))+ (1 - \chi(x))W_0({\mathcal{E}} u(x))\,{\rm d}x + |D\chi|(\Omega), \end{equation}

and our aim is to obtain an integral representation for the localized relaxed functionals, defined for $A \in \mathcal {O}(\Omega )$, by

(3.2)\begin{align} \mathcal{F}\left(\chi,u;A\right)& :=\inf\left\{ \liminf_{n\rightarrow +\infty} F(\chi_n,u_n;A): u_{n} \in W^{1,1}(A;\mathbb{R}^{N}),\chi_{n} \in BV(A;\{0,1\}),\right.\nonumber\\ & \quad \left.\vphantom{\liminf_{n\rightarrow +\infty}} u_{n}\to u\text{ in }L^{1}(A;\mathbb{R}^{N}),\chi_{n}\to\chi\text{ in }L^1(A;\{0,1\})\right\}, \end{align}
(3.3)\begin{align} \mathcal{F}_{LD}\left(\chi,u;A\right)& :=\inf\left\{ \liminf_{n\rightarrow +\infty} F(\chi_n,u_n;A): u_{n} \in LD(A), \chi_{n} \in BV(A;\{0,1\}),\right.\nonumber\\ & \quad \left.\vphantom{\liminf_{n\rightarrow +\infty}} u_{n}\to u\text{ in }L^{1}(A;\mathbb{R}^{N}),\chi_{n}\to\chi\text{ in }L^1(A;\{0,1\})\right\}, \end{align}

where the densities $W_i$, $i = 0,1$, are continuous functions such that

(3.4)\begin{equation} \exists\,\alpha, \beta > 0 \text{ such that }\alpha |\xi| \leq W_i(\xi) \leq \beta (1 + |\xi|),\quad \forall \xi \in {\mathbb{R}}^{N \times N}_s, \end{equation}

and where, for purposes of notation, we let $f: \{0,1\} \times \mathbb {R}^{N \times N}_s \to [0,+\infty )$ be defined as

(3.5)\begin{equation} f\left(q,\xi\right):=q W_1(\xi)+ (1-q)W_0(\xi). \end{equation}

It follows from the definition of the recession function (1.7) and from the growth conditions (3.4) that for every $q \in \{0,1\}$ and every $\xi \in {\mathbb {R}}^{N \times N}_s$

(3.6)\begin{equation} \alpha |\xi| \leq f^{\infty}(q,\xi) \leq \beta |\xi|. \end{equation}

It is an immediate consequence of (3.4) that

(3.7)\begin{equation} |f(q_1,\xi) - f(q_2,\xi)| \leq \beta \, |q_1 - q_2|(1 + |\xi|),\quad\forall q_1, q_2 \in \{0,1\},\ \forall \xi \in {\mathbb{R}}^{N \times N}_s, \end{equation}

from which it follows that

(3.8)\begin{equation} |f^{\infty}(q_1,\xi) - f^{\infty}(q_2,\xi)| \leq \beta\,|q_1 - q_2| \,|\xi|,\quad \forall q_1, q_2 \in \{0,1\},\ \forall \xi \in {\mathbb{R}}^{N \times N}_s. \end{equation}

The following additional hypothesis will be used to write the density of the jump term in the form given in (1.11)

(3.9)\begin{equation} \exists \,0 < \gamma \leq 1, \exists \, C, L > 0 : \; t\,|\xi| > L \Rightarrow \left| f^{\infty}(q,\xi) - \frac{f(q,t\xi)}{t}\right| \leq C \frac{|\xi|^{1-\gamma}}{t^\gamma}, \end{equation}

for every $q \in \{0,1\}$ and every $\xi \in {\mathbb {R}}^{N \times N}_s$. As pointed out in [Reference Fonseca and Müller32], this can be stated equivalently as

(3.10)\begin{equation} \exists \, 0 < \gamma \leq 1, \exists \, C > 0 \text{ such that }\left| f^{\infty}(q,\xi) - f(q,\xi)\right| \leq C \left(1 + |\xi|^{1-\gamma}\right), \end{equation}

for every $q \in \{0,1\}$ and every $\xi \in {\mathbb {R}}^{N \times N}_s$.

Under our assumed growth conditions (3.4), we observe that if $f$ satisfies (3.9), or equivalently (3.10), then the same holds for its symmetric quasiconvex envelope $SQf$. To this end, we recall that, under the hypothesis (3.4), the recession function of a symmetric quasiconvex function is still symmetric quasiconvex (see [Reference Rindler46, remarks 8 and 9]) and we begin by stating the following results (cf. [Reference Carita and Zappale21, (iv) and (v) in remark 3.2] and [Reference Ribeiro and Zappale45, propositions 2.6 and 2.7] for the quasiconvex counterpart).

Proposition 3.1 Let $f:\{0,1\} \times \mathbb {R}^{N \times N}_s\to [0, + \infty )$ be a continuous function as in (3.5) and satisfying (3.4) and (3.9). Let $f^\infty$ and $SQf$ be its recession function and its symmetric quasiconvex envelope, defined by (1.7) and (2.8), respectively. Then

(3.11)\begin{equation} SQ(f^\infty)(q,\xi)= (SQf)^\infty(q,\xi)\quad\text{for every}\ (q,\xi) \in \{0,1\} \times \mathbb{R}^{N \times N}_s. \end{equation}

Proposition 3.2 Let $f:\{0,1\} \times \mathbb {R}^{N \times N}_s\to [0, + \infty )$ be a continuous function as in (3.5), satisfying (3.4) and (3.9). Then, there exist $\gamma \in [0,1)$ and $C>0$ such that

\[ \displaystyle\left| (SQf)^\infty(q,\xi)- SQf(q, \xi)\right|\leq C\left( 1+|\xi|^{1-\gamma}\right),\quad \forall\ (q,\xi)\in \{0,1\} \times \mathbb{R}^{N \times N}_s. \]

The growth conditions (3.4), as well as standard diagonalization arguments, allow us to prove the following properties of the functional $\mathcal {F} (\chi,u;A)$ defined in (3.2).

Proposition 3.3 Let $A \in \mathcal {O}(\Omega )$, $u \in BD(A)$, $\chi \in BV(A;\{0,1\})$ and $F(\chi,u;A)$ be given by (3.1). If $W_i$, $i = 0,1$, satisfy (3.4), then

  1. (i) there exists $C > 0$ such that

    \[ C\left(|Eu|(A) + |D\chi|(A)\right) \leq \mathcal{F} (\chi,u;A) \leq C\left({\mathcal{L}}^N(A) + |Eu|(A) + |D\chi|(A)\right); \]
  2. (ii) $\mathcal {F} (\chi,u;A)$ is always attained, that is, there exist sequences $\{u_n\} \subset W^{1,1}(A;\mathbb {R}^{N})$ and $\{\chi _{n}\} \subset BV(A;\{0,1\})$ such that $u_{n}\to u$ in $L^{1}(A;\mathbb {R}^{N})$, $\chi _{n}\to \chi$ in $L^1(A;\{0,1\})$ and

    \[ \mathcal{F}(\chi,u;A) = \lim_{n\rightarrow\infty}F(\chi_n,u_n;A); \]
  3. (iii) if $\{u_n\} \subset W^{1,1}(A;\mathbb {R}^{N})$ and $\{\chi _{n}\} \subset BV(A;\{0,1\})$ are such that $u_{n}\to u$ in $L^{1}(A;\mathbb {R}^{N})$ and $\chi _{n}\to \chi$ in $L^1(A;\{0,1\})$, then

    \[ \mathcal{F}(\chi,u;A) \leq \liminf_{n\to +\infty}\mathcal{F}(\chi_n, u_n;A). \]

Proof. $(i)$ The upper bound follows from the growth condition from above of $W_i$, $i=0,1$ and by fixing $\chi _n = \chi$ as a test sequence for $\mathcal {F} (\chi,u;A)$, whereas the lower bound is a consequence of the inequality from below in (3.4) and (2.3) and the lower semicontinuity of the total variation of Radon measures.

The conclusions in $(ii)$ and $(iii)$ follow by standard diagonalization arguments.

Remark 3.4 Analogous conclusions also hold for the functional $\mathcal {F}_{LD} (\chi,u;A)$.

Remark 3.5 Assuming that the continuous functions $W_0$ and $W_1$ satisfy the growth hypothesis (3.4), it follows from the density of smooth functions in $LD(\Omega )$ and a diagonalization argument that

\[ \mathcal{F}\left(\chi,u;A\right) = \mathcal{F}_{LD}\left(\chi,u;A\right),\ \text{for every}\ \chi \in BV(A;\left\{0,1\right\}), u \in BD(\Omega), A \in {\mathcal{O}}(\Omega). \]

Proof. As $W^{1,1}(A;\mathbb {R}^N) \subset LD(A)$, one inequality is trivial. In order to show the reverse one, let $\{u_n\} \subset LD(A)$, $\{\chi _n\} \subset BV(A;\left \{0,1\right \})$ be such that $u_{n}\to u$ in $L^{1}(A;\mathbb {R}^{N})$, $\chi _{n}\to \chi$ in $L^1(A;\{0,1\})$ and

\begin{align*} & \mathcal{F}_{LD}\left(\chi,u;A\right)\\ & \quad = \lim_n \left[\int_{A}\chi_n(x) W_1({\mathcal{E}} u_n(x)) + (1 - \chi_n(x))W_0({\mathcal{E}} u_n(x))\,{\rm d}x + |D\chi_n|(A) \right]. \end{align*}

By theorem 2.5, for each $n \in {\mathbb {N}}$, let $v_{n,k} \in W^{1,1}(A;\mathbb {R}^N)$ be such that $v_{n,k} \to u_n$ in $L^{1}(A;\mathbb {R}^{N})$, as $k \to + \infty$, and ${\mathcal {E}} v_{n,k} \to {\mathcal {E}} u_n$ in $L^{1}(A;\mathbb {R}^{N\times N}_s)$, as $k \to + \infty$. By passing to a subsequence, if necessary, assume also that $\lim _{k\rightarrow +\infty }{\mathcal {E}} v_{n,k}(x) = {\mathcal {E}} u_n(x)$, for a.e. $x \in A$. By (3.4) and Fatou's lemma we obtain

\begin{align*} & \int_A\chi_n(x)\left[C(1 + |{\mathcal{E}} u_n(x)|) - W_1({\mathcal{E}} u_n(x))\right]\,{\rm d}x \\ & \quad \leq\liminf_{k\to +\infty}\int_A\chi_n(x) \left[C(1 + |{\mathcal{E}} v_{n,k}(x)|) - W_1({\mathcal{E}} v_{n,k}(x))\right]\,{\rm d}x \end{align*}

so that

\[ \int_A\chi_n(x)W_1({\mathcal{E}} u_n(x))\,{\rm d}x \geq\limsup_{k\to +\infty}\int_A\chi_n(x)W_1({\mathcal{E}} v_{n,k}(x))\,{\rm d}x, \]

and likewise for the term involving $(1 - \chi _n)W_0$. From the previous inequalities we conclude that

\[ F(\chi_n,u_n;A) \geq \limsup_{k\to +\infty}F(\chi_n,v_{n,k};A). \]

Since $v_{n,k} \to u_n$ in $L^{1}(A;\mathbb {R}^{N})$, as $k \to + \infty$, and $u_{n}\to u$ in $L^{1}(A;\mathbb {R}^{N})$, by a diagonalization argument there exists a sequence $k_n \to + \infty$ such that $v_{n,k_n} \to u$ in $L^{1}(A;\mathbb {R}^{N})$ and

\[ F(\chi_n,v_{n,k_n};A) \leq F(\chi_n,u_n;A) + \frac{1}{k_n}. \]

As $\{\chi _n\}$, $\{v_{n,k_n}\}$ are admissible for $\mathcal {F}\left (\chi,u;A\right )$ it follows that

\begin{align*} \mathcal{F}\left(\chi,u;A\right) & \leq \liminf_{n\to +\infty}F(\chi_n,v_{n,k_n};A)\\ & \leq \limsup_{n\to +\infty} \left(F(\chi_n,u_n;A) + \frac{1}{k_n}\right) = \mathcal{F}_{LD}\left(\chi,u;A\right). \end{align*}

A straightforward adaptation of the proof of [Reference Barroso, Fonseca and Toader12, proposition 3.7] yields the following result which enables us to prove the nested subadditivity property of the functional $\mathcal {F}\left (\chi,u;\cdot \right )$.

Proposition 3.6 Let $A \in \mathcal {O} (\Omega )$ and assume that $W_0, W_1$ satisfy the growth condition (3.4). Let $\{\chi _n\} \subset BV(A;\{0,1\})$ and $\{u_n\}, \{v_n\} \subset BD(A;\mathbb {R}^N)$ be sequences satisfying $u_{n} - v_n \to 0$ in $L^{1}(A;\mathbb {R}^{N})$, $\sup _n |Eu_n|(A) < + \infty$, $|Ev_n| \mathop {\rightharpoonup }\limits ^{\ast } \mu$ and $|Ev_n| \to \mu (A)$. Then there exist subsequences $\{v_{n_k}\}$ of $\{v_n\}$, $\{\chi _{n_k}\}$ of $\{\chi _n\}$ and there exists a sequence $\{w_k\} \subset BD(A)$ such that $w_k = v_{n_k}$ near $\partial A$, $w_k - v_{n_k} \to 0$ in $L^{1}(A;\mathbb {R}^{N})$ and

\[ \limsup_{k\to +\infty}F(\chi_{n_k},w_k;A) \leq\liminf_{n\to +\infty}F(\chi_n,u_n;A). \]

It is clear from the proof that if the original sequences $\{u_n\}, \{v_n\}$ belong to $W^{1,1}(A;\mathbb {R}^{N})$ then the sequence $\{w_k\}$ will also be in this space.

Proposition 3.7 Assume that $W_0$ and $W_1$ are continuous functions satisfying (3.4). Let $u \in BD(\Omega )$, $\chi \in BV(\Omega ;\{0,1\})$ and $S, U, V \in \mathcal {O}(\Omega )$ be such that $S \subset \subset V \subset U.$ Then

\[\mathcal{F}\left(\chi,u;U\right) \leq \mathcal{F}\left(\chi,u;V\right)+ \mathcal{F}\left(\chi,u;U{\setminus} \overline S\right). \]

Proof. By proposition 3.3, $(ii)$, let $\{v_n\} \subset W^{1,1}(V;\mathbb {R}^{N})$, $\{w_n\} \subset W^{1,1}(U{\setminus} \overline S;\mathbb {R}^{N})$, $\{\chi _n\} \subset BV(V;\{0,1\})$ and $\{\theta _n\} \subset BV(U{\setminus} \overline S;\{0,1\})$ be such that $v_n \to u$ in $L^{1}(V;\mathbb {R}^{N})$, $w_n \to u$ in $L^{1}(U{\setminus} \overline S;\mathbb {R}^{N})$, $\chi _{n}\to \chi$ in $L^1(V;\{0,1\})$ $\theta _{n}\to \chi$ in $L^1(U{\setminus} \overline S;\{0,1\})$ and

(3.12)\begin{align} \mathcal{F}\left(\chi,u;V\right) & = \lim_{n\rightarrow+\infty}F(\chi_n,v_n;V) \end{align}
(3.13)\begin{align} \mathcal{F}\left(\chi,u;U{\setminus} \overline S\right) & = \lim_{n\rightarrow+\infty}F(\theta_n,w_n;U{\setminus} \overline S). \end{align}

Let $V_0 \in \mathcal {O}_{\infty }(\Omega )$ satisfy $S \subset \subset V_0 \subset \subset V$ and $|E u| (\partial V_0) = 0$, $|D\chi |(\partial V_0) = 0$. Applying proposition 3.6 to $\{v_n\}$ and $u$ in $V_0$, we obtain a subsequence $\{\overline \chi _n\}$ of $\{\chi _n\}$ and a sequence $\{\overline v_n\} \subset W^{1,1}(V_0;\mathbb {R}^{N})$ such that $\overline v_n = u$ near $\partial V_0$, $\overline v_n \to u$ in $L^{1}(V_0;\mathbb {R}^{N})$ and

(3.14)\begin{equation} \limsup_{n\to +\infty}F(\overline \chi_n,\overline v_n;V_0) \leq \liminf_{n\to +\infty}F(\chi_n,v_n;V_0). \end{equation}

A further application of proposition 3.6, this time to $\{w_n\}$ and $u$ in $U {\setminus} \overline V_0$, yields a subsequence $\{\overline \theta _n\}$ of $\{\theta _n\}$ and a sequence $\{\overline w_n\} \subset W^{1,1}(U {\setminus} \overline V_0;\mathbb {R}^{N})$ such that $\overline w_n = u$ near $\partial V_0$, $\overline w_n \to u$ in $L^{1}(U {\setminus} \overline V_0;\mathbb {R}^N)$ and

(3.15)\begin{equation} \limsup_{n\to +\infty}F(\overline \theta_n,\overline w_n;U {\setminus} \overline V_0) \leq \liminf_{n\to +\infty}F(\theta_n,w_n;U {\setminus} \overline V_0). \end{equation}

Define

\[ z_n: = \begin{cases} \overline v_n, & {\rm in}\ V_0\\ \overline w_n, & {\rm in}\ U {\setminus} V_0, \end{cases} \]

note that, by the properties of $\{\overline v_n\}$ and $\{\overline w_n\}$, $\{z_n\} \subset W^{1,1}(U;\mathbb {R}^{N})$ and $z_n \to u$ in $L^{1}(U;\mathbb {R}^{N})$.

We must now build a transition sequence $\{\eta _n\}$ between $\{\overline \chi _n\}$ and $\{\overline \theta _n\}$, in such a way that an upper bound for the total variation of $\eta _n$ is obtained. In order to connect these functions without adding more interfaces, we argue as in [Reference Barroso, Matias, Morandotti and Owen13] (see also [Reference Barroso and Zappale14]). For $\delta > 0$ consider

\[ V_\delta:=\{x \in V: {\rm dist}(x,V_0) < \delta\}, \]

where $\delta$ is small enough so that $\overline w_n = u$ in $V_{\delta } {\setminus} \overline V_0$ and

(3.16)\begin{equation} \int_{V_{\delta} {\setminus} \overline V_0}C(1 + |u(x)|)\,{\rm d}x = O(\delta). \end{equation}

Given $x \in V$, let ${\rm d}(x) := {\rm dist}(x;V_0)$. Since the distance function to a fixed set is Lipschitz continuous, applying the change of variables formula (see theorem 2, § 3.4.3 in [Reference Evans and Gariepy30]) yields

\[ \int_{V_\delta {\setminus} \overline{V_0}}|\overline\chi_n(x)-\overline\theta_n(x)||{\rm det\,}\nabla\,{\rm d}(x)|\,{\rm d}x = \int_0^\delta \left[\int_{{\rm d}^{{-}1}(y)} |\overline\chi_n(x)-\overline\theta_n(x)| \,{\rm d}{\mathcal{H}}^{N-1}(x) \right]{\rm d}y \]

and, as $|{\rm det}\,\nabla \,{\rm d}(x)|$ is bounded and $\overline \chi _n - \overline \theta _n \to 0$ in $L^1(V \cap (U {\setminus} \overline S);\{0,1\})$, it follows that, for almost every $\rho \in [0; \delta ]$, we have

(3.17)\begin{align} & \lim_{n\to +\infty}\int_{{\rm d}^{{-}1}(\rho)}|\overline\chi_n(x) - \overline\theta_n(x)| \,{\rm d}{\mathcal{H}}^{N-1}(x)\notag\\ & \quad = \lim_{n\to +\infty}\int_{\partial V_\rho}|\overline\chi_n(x)-\overline\theta_n(x)| \,{\rm d}{\mathcal{H}}^{N-1}(x) = 0. \end{align}

Fix $\rho _0\in [0; \delta ]$ such that $|D\chi |(\partial V_{\rho _0})= 0$ and (3.17) holds. We observe that $V_{\rho _0}$ is a set with locally Lipschitz boundary since it is a level set of a Lipschitz function (see, e.g. [Reference Evans and Gariepy30]). Hence, for every $n$, we can consider $\overline \chi _n, \overline \theta _n$ on $\partial V_{\rho _0}$ in the sense of traces and define

\[ \eta_n :=\begin{cases} \overline\chi_n, & \text{in } V_{\rho_0}\\ \overline\theta_n, & \text{ in } U{\setminus} V_{\rho_0}. \end{cases} \]

Then $\{\eta _n\} \subset BV(U;\{0,1\})$, $\eta _{n}\to \chi$ in $L^1(U;\{0,1\})$ and so $\{\eta _n\}$ and $\{z_n\}$ are admissible for $\mathcal {F}\left (\chi,u;U\right )$. Therefore, by (3.17), (3.4), (3.14), (3.15), (3.16), (3.12) and (3.13),

\begin{align*} \mathcal{F}\left(\chi,u;U\right) & \leq \liminf_{n\to +\infty}F(\eta_n,z_n;U) \\ & = \liminf_{n\to +\infty}\left[F(\overline\chi_n,\overline v_n; V_0) +\int_{V_{\rho_0}{\setminus} \overline V_0}\overline\chi_n(x)W_1({\mathcal{E}} u(x))\right.\\ & \quad + (1 - \overline \chi_n(x))W_0({\mathcal{E}} u(x))\,{\rm d}x + |D\overline \chi_n|(V_{\rho_0}{\setminus} V_0) + F(\overline\theta_n,\overline w_n; U {\setminus} V_{\rho_0})\\ & \quad + \left.\int_{\partial V_{\rho_0} }|\overline \chi_n(x) - \overline \theta_n(x)| \,{\rm d}{\mathcal{H}}^{N-1}(x)\right] \\ & \leq \limsup_{n\to +\infty}F(\overline\chi_n,\overline v_n; V_0) +\limsup_{n\to +\infty}F(\overline\theta_n,\overline w_n; U {\setminus} \overline V_{0})\\ & \quad + \int_{V_{\rho_0}{\setminus} \overline V_0}C \left(1 + |{\mathcal{E}} u(x)|\right)\,{\rm d}x + \limsup_{n\to +\infty} |D\chi_n|(V_{\rho_0}{\setminus} V_0) \\ & \leq \liminf_{n\to +\infty}F(\chi_n,v_n; V_0) +\liminf_{n\to +\infty}F(\theta_n,w_n; U {\setminus} \overline V_{0})\\ & \quad + O(\delta) + \limsup_{n\to +\infty} |D\chi_n|(V_{\rho_0}{\setminus} V_0) \\ & \leq \limsup_{n\to +\infty}F(\chi_n,v_n; V) +\limsup_{n\to +\infty}F(\theta_n,w_n; U {\setminus} \overline S) + O(\delta) \\ & = \mathcal{F}\left(\chi,u;V\right) +\mathcal{F}\left(\chi,u;U{\setminus} \overline S\right) + O(\delta) \end{align*}

so the result follows by letting $\delta \to 0^+$.

Proposition 3.8 Let $W_0$ and $W_1$ be continuous functions satisfying (3.4). For every $u \in BD(\Omega )$, $\chi \in BV(\Omega ;\{0,1\})$, $\mathcal {F}\left (\chi,u;\cdot \right )$ is the restriction to $\mathcal {O}(\Omega )$ of a Radon measure.

Proof. By proposition 3.3 $(ii)$, let $\{u_n\} \subset W^{1,1}(\Omega ;\mathbb {R}^{N})$, $\{\chi _n\} \subset BV(\Omega ;\{0,1\})$, be such that $u_n \to u$ in $L^{1}(\Omega ;\mathbb {R}^{N})$, $\chi _{n}\to \chi$ in $L^1(\Omega ;\{0,1\})$ and

\[ \mathcal{F}\left(\chi,u;\Omega\right) = \lim_{n\rightarrow+\infty}F(\chi_n,u_n;\Omega). \]

Let $\mu _n = f(\chi _n(\cdot ), {\mathcal {E}} u_n (\cdot )) {\mathcal {L}}^N\lfloor {\Omega }+ |D \chi _n|$ and extend this sequence of measures outside of $\Omega$ by setting, for any Borel set $E \subset \mathbb {R}^N$,

\[ \lambda_n(E) = \mu_n(E \cap \Omega). \]

Passing, if necessary, to a subsequence, we can assume that there exists a non-negative Radon measure $\mu$ (depending on $\chi$ and $u$) on $\overline {\Omega }$ such that $\lambda _n \stackrel {*}{\rightharpoonup } \mu$ in the sense of measures in $\overline {\Omega }$. Let $\varphi _k \in C_0(\overline {\Omega })$ be an increasing sequence of functions such that $0 \leq \varphi _k \leq 1$ and $\varphi _k(x) \to 1$ a.e. in $\overline {\Omega }$. Then, by Fatou's lemma and by the choice of $\{u_n\}$, $\{\chi _n\}$, we have

\begin{align*} \mu(\overline{\Omega}) & = \int_{\overline{\Omega}}\liminf_{k \to + \infty} \varphi_k(x)\,{\rm d}\mu \leq \liminf_{k \to +\infty}\int_{\overline{\Omega}}\varphi_k(x)\,{\rm d}\mu \\ & = \liminf_{k \to +\infty}\lim_{n \to + \infty}\left(\int_{\Omega}\varphi_k(x) f(\chi_n(x),{\mathcal{E}} u_n (x))\,{\rm d}x + \int_{\Omega}\varphi_k(x)\,{\rm d} |D\chi_n| \right)\\ & \leq \lim_{n \to + \infty} \left(\int_{\Omega}f(\chi_n(x),{\mathcal{E}} u_n (x))\,{\rm d}x + |D\chi_n|(\Omega) \right) = {\mathcal{F}}(\chi,u;\Omega), \end{align*}

so that

(3.18)\begin{equation} \mu(\overline{\Omega})\leq {\mathcal{F}}(\chi, u; \Omega). \end{equation}

On the other hand, by the upper semicontinuity of weak $\ast$ convergence of measures on compact sets, for every open set $V \subset \Omega$, it follows that

(3.19)\begin{equation} {\mathcal{F}}(\chi, u; V) \leq \liminf_{n \to +\infty} F(\chi_n, u_n; V)= \liminf_{n \to +\infty} \mu_n(V) \leq \limsup_{n \to +\infty} \mu_n(\overline{V})\leq \mu(\overline{V}). \end{equation}

Now let $V \in \mathcal {O}(\Omega )$ and ${\varepsilon }>0$ be fixed and consider an open set $S \subset \subset V$ such that $\mu (V {\setminus} S) < {\varepsilon }$. Then

(3.20)\begin{equation} \mu(V) \leq \mu(S) + {\varepsilon} = \mu(\overline \Omega) - \mu(\overline \Omega {\setminus} S) + {\varepsilon}, \end{equation}

and so, by (3.20), (3.18), (3.19) and proposition 3.7 we have

\[ \mu(V)\leq \mu({\overline \Omega})- \mu( {\overline\Omega} {\setminus} S)+ \varepsilon\leq {\mathcal{F}}(\chi, u; \Omega)- {\mathcal{F}}(\chi, u; \Omega {\setminus} {\overline S}) + \varepsilon\leq {\mathcal{F}}(\chi, u; V)+\varepsilon. \]

Letting ${\varepsilon } \to 0^+$, we obtain

\[ \mu(V)\leq {\mathcal{F}}(\chi, u; V), \]

whenever $V$ is an open set such that $V \subset \subset \Omega$. For a general open subset $V \subset \Omega$ we have

\[ \mu(V) = \sup \{\mu(O) : O \subset{\subset} V \}\leq \sup \{{\mathcal{F}}(\chi, u; O) : O \subset{\subset} V \} \leq {\mathcal{F}}(\chi, u; V). \]

It remains to show that ${\mathcal {F}}(\chi, u; U) \leq \mu (U)$, $\forall \, U \in \mathcal {O}(\Omega )$. Fix ${\varepsilon } > 0$ and choose $V, S \in \mathcal {O}(\Omega )$ such that $S \subset \subset V \subset \subset U$ and $\mathcal {L}^N(U {\setminus} \overline S) + |Eu|(U {\setminus} \overline S) + |D\chi |(U {\setminus} \overline S) < {\varepsilon }$. By proposition 3.3 $(i)$, (3.19) and the nested subadditivity result, it follows that

\begin{align*} {\mathcal{F}}(\chi, u; U) & \leq {\mathcal{F}}(\chi, u; V) +{\mathcal{F}}(\chi, u; U {\setminus} {\overline S}) \\ & \leq \mu(\overline V) +C\left(\mathcal{L}^N(U {\setminus} \overline S) + |Eu|(U {\setminus} \overline S) + |D\chi|(U {\setminus} \overline S)\right) \leq \mu(U) + C\varepsilon, \end{align*}

so it suffices to let ${\varepsilon } \to 0^+$ to conclude the proof.

Combining the arguments given in the proofs of propositions 3.6 and 3.7 it is possible to obtain the following refined version of proposition 3.6.

Proposition 3.9 Let $A \in \mathcal {O} (\Omega )$ and assume that $W_0, W_1$ satisfy the growth condition (3.4). Let $\{u_n\}, \{v_n\} \subset BD(A;\mathbb {R}^N)$ and $\{\chi _n\}, \{\theta _n\} \subset BV(A;\{0,1\})$ be sequences satisfying $u_{n} - v_n \to 0$ in $L^{1}(A;\mathbb {R}^{N})$, $\chi _{n} - \theta _n \to 0$ in $L^1(A;\{0,1\})$, $\sup _n |E^su_n|(A) < + \infty$, $|Ev_n| \mathop {\rightharpoonup }\limits ^{\ast } \mu$, $|Ev_n| \to \mu (A)$, $\sup _n |D\chi _n|(A) < + \infty$ and $\sup _n |D\theta _n|(A) < + \infty$. Then there exist subsequences $\{v_{n_k}\}$ of $\{v_n\}$, $\{\theta _{n_k}\}$ of $\{\theta _n\}$ and there exist sequences $\{w_k\} \subset BD(A)$, $\{\eta _k\} \subset BV(A;\{0,1\})$ such that $w_k = v_{n_k}$ near $\partial A$, $\eta _k = \theta _{n_k}$ near $\partial A$, $w_k - v_{n_k} \to 0$ in $L^{1}(A;\mathbb {R}^{N})$, $\eta _{k} - \theta _{n_k} \to 0$ in $L^1(A;\{0,1\})$ and

\[ \limsup_{k\to +\infty}F(\eta_{k},w_k;A) \leq \liminf_{n\to +\infty}F(\chi_n,u_n;A). \]

As in proposition 3.6, the new sequence $\{w_k\}$ has the same regularity as the original sequences $\{u_n\}, \{v_n\}$ as it is obtained through a convex combination of these ones using smooth cut-off functions.

The following proposition, whose proof is standard (cf. for instance [Reference Ribeiro and Zappale45, lemma 3.1] or [Reference Carita and Zappale22, proposition 2.14]), allows us to assume without loss of generality that $f$ is symmetric quasiconvex.

Proposition 3.10 Let $W_0$ and $W_1$ be continuous functions satisfying (3.4) and consider the functional $F:BV(\Omega ;\{0,1\})\times BD(\Omega )\times \mathcal {O}(\Omega )$ defined in (3.1). Consider furthermore the relaxed functionals given in (3.2) and

(3.21)\begin{align} \mathcal{F}_{SQf}(\chi,u;A) & :=\inf\left\{\liminf_{n\to +\infty}\int_A SQf(\chi_n(x),{\mathcal{E}} u_n(x))\,{\rm d}x + |D \chi_n|(A) :\right.\nonumber\\ & \quad (\chi_n,u_n) \in BV(A;\{0,1\})\times LD(A),u_{n}\to u\text{ in }L^{1}(A;\mathbb{R}^{N}),\notag\\ & \quad \left.\chi_{n} \to \chi \text{ in } L^1(A;\{0,1\})\vphantom{\liminf_{n\to +\infty}}\right\}. \end{align}

Then, ${\mathcal {F}}(\cdot,\cdot ;\cdot )$ coincides with ${\mathcal {F}}_{SQf}(\cdot, \cdot ; \cdot )$ in $BV(\Omega ;\{0,1\})\times BD(\Omega )\times \mathcal {O}(\Omega )$.

In the sequel we rely on the result of proposition 3.10 and assume that $f$ is symmetric quasiconvex. Together with (3.4), this entails the Lipschitz continuity of $f$ with respect to the second variable (see [Reference De Philippis and Rindler26]). Under this quasiconvexity hypothesis, assuming in addition that (3.9) holds and taking also into account proposition 3.2, we recall (cf. (1.11)) that our relaxed surface energy density is given by

(3.22)\begin{align} & K(a,b,c,d,\nu)\notag\\ & \quad :=\inf\left\{\int_{Q_{\nu}}f^{\infty}(\chi(x),{\mathcal{E}} u(x))\,{\rm d}x+|D\chi|(Q_{\nu}):\left(\chi,u\right) \in\mathcal{A}(a,b,c,d,\nu)\right\}, \end{align}

where, for $(a,b,c,d,\nu ) \in \{0,1\} \times \{0,1\} \times \mathbb {R}^{N} \times \mathbb {R}^{N} \times S^{N-1},$ the set of admissible functions is

(3.23)\begin{align} \mathcal{A}(a,b,c,d,\nu)& :=\left\{\left(\chi,u\right)\in BV_{\rm loc}\left(S_{\nu};\{0,1\}\right) \times W^{1,1}_{\rm loc}\left(S_{\nu};\mathbb{R}^{N}\right):\right.\nonumber\\ & \quad(\chi(y),u(y)) = (a,c) \text{ if } y\cdot\nu=\frac{1}{2},\ (\chi(y),u(y)) = (b,d) \text{ if } y\cdot\nu={-}\frac{1}{2},\nonumber\\ & \quad\left.(\chi, u)\text{ are 1-periodic in the directions of }\nu_{1},\ldots,\nu_{N-1}\right\}, \end{align}

$\left \{\nu _{1},\nu _{2},\ldots,\nu _{N-1},\nu \right \}$ is an orthonormal basis of $\mathbb {R}^{N}$ and $S_\nu$ is the strip given by

\[ S_\nu = \left\{x \in {\mathbb{R}}^N : |x \cdot \nu| < \frac{1}{2}\right\}. \]

The following result provides an alternative characterization of $K(a,b,c,d,\nu )$ which will be useful to obtain the surface term of the relaxed energy, under hypothesis (3.9). To this end, given $(a,b,c,d,\nu ) \in \{0,1\} \times \{0,1\}\times \mathbb {R}^{N} \times \mathbb {R}^{N} \times S^{N-1},$ we consider the functions

(3.24)\begin{equation} \chi_{a,b,\nu}(y) := \begin{cases} a, & {\rm if}\ y \cdot \nu > 0\\ b, & {\rm if}\ y \cdot \nu < 0 \end{cases}\quad{\rm and\ }\quad u_{c,d,\nu}(y) := \begin{cases} c, & {\rm if}\ y \cdot \nu > 0\\ d, & {\rm if}\ y \cdot \nu < 0. \end{cases} \end{equation}

Proposition 3.11 For every $(a,b,c,d,\nu ) \in \{0,1\} \times \{0,1\} \times \mathbb {R}^{N} \times \mathbb {R}^{N} \times S^{N-1}$ we have

\[ K(a,b,c,d,\nu) = \widetilde{K}(a,b,c,d,\nu) \]

where

(3.25)\begin{align} & \widetilde{K}(a,b,c,d,\nu)\notag\\ & \quad := \inf\left\{\liminf_{n \to + \infty}\left[\displaystyle\int_{Q_{\nu}}f^{\infty}(\chi_n(x),{\mathcal{E}} u_n(x))\,{\rm d}x + |D\chi_n|(Q_{\nu})\right]:\right.\notag\\ & \quad \chi_n \in BV\left(Q_{\nu};\{0,1\}\right), u_n \in W^{1,1}\left(Q_{\nu};\mathbb{R}^{N}\right), \chi_n \to \chi_{a,b,\nu}\ {\rm in\ }\ L^1(Q_{\nu};\{0,1\}),\notag\\ & \quad \left. u_n \to u_{c,d,\nu}\ {\rm in}\ L^1(Q_{\nu};{\mathbb{R}}^N) \vphantom{\int_{Q_{\nu}}}\right\}. \end{align}

Proof. The conclusion follows as in [Reference Barroso, Bouchitté, Buttazzo and Fonseca11, proposition 3.5], by proving a double inequality.

To show that $K(a,b,c,d,\nu ) \leq \widetilde {K}(a,b,c,d,\nu )$ we take sequences $\{\chi _n\}$, $\{u_n\}$ as in the definition of $\widetilde {K}(a,b,c,d,\nu )$ and use proposition 3.9, applied to $\{\chi _n\}$, $\{\chi _{a,b,\nu }\}$, $\{u_n\}$ and $\{v_n\}$, where $v_n$ is a regularization of $u_{c,d,\nu }$ which preserves its boundary values (cf. theorem 2.5).

The reverse inequality is based on the periodicity of the admissible functions for $K(a,b,c,d,\nu )$, together with the Riemann–Lebesgue lemma.

4. Proof of the main theorem

Given $\chi \in BV(\Omega ;\{0,1\})$ and $u \in BD(\Omega )$, by proposition 3.8 we know that $\mathcal {F}(\chi,u,;\cdot )$ is the restriction to $\mathcal {O}(\Omega )$ of a Radon measure $\mu$. By proposition 3.3 $(i)$ we may decompose $\mu$ as

\[ \mu = \mu^a \mathcal{L} ^N + \mu^j+ \mu^c,\ \text{with}\ \mu^j \ll |E^j u|+ |D \chi|. \]

Our aim in this section is to characterize the density $\mu ^a$ and the measures $\mu ^j$ and $\mu ^c$.

We point out that the measure $\mu ^j$ is given by $\sigma ^j \mathcal {H}^{N-1} \lfloor (J_\chi \cup J_u)$, for a certain density $\sigma ^j$. Indeed, due to the fact that, for $BV$ functions, $\mathcal {H}^{N-1}(S_u {\setminus} J_u) = 0$, the measure $|D\chi |$ is concentrated on $J_\chi$ apart from an $\mathcal {H}^{N-1}$-negligible set, whereas, by [Reference Ambrosio, Coscia and Dal Maso3, remark 4.2 and proposition 4.4], $|E^ju|$ is concentrated on $J_u$ and it is the only part of the measure $Eu$ that is concentrated on $(n-1)$-dimensional sets.

4.1 The bulk term

Proposition 4.1 Let $u \in BD(\Omega )$, $\chi \in BV(\Omega ;\{0,1\})$ and let $W_0$ and $W_1$ be continuous functions satisfying (3.4). Assume that $f$ given by (3.5) is symmetric quasiconvex. Then, for ${\mathcal {L}} ^N$ a.e. $x_0 \in \Omega$,

\[ \mu^a(x_0) = \dfrac{{\rm d}\mathcal{F}(\chi, u; \cdot)}{{\rm d}{\mathcal{L}}^N}(x_0) \geq f(\chi(x_0), \mathcal{E} u(x_0)). \]

Proof. Let $x_0 \in \Omega$ be a point satisfying

(4.1)\begin{equation} \mu^a(x_0) = \frac{{\rm d}\mu}{{\rm d}\mathcal{L}^N}(x_0)= \lim_{{\varepsilon} \to 0^+}\frac{\mu(Q(x_0,{\varepsilon}))}{{\varepsilon}^ N}\quad \text{exists and is finite} \end{equation}

and

(4.2)\begin{equation} \frac{{\rm d}|E^su|}{{\rm d}\mathcal{L}^N}(x_0)= 0, \quad\frac{{\rm d}|D\chi|}{{\rm d}\mathcal{L}^N}(x_0)=0. \end{equation}

Furthermore, we choose $x_0$ to be a point of approximate continuity for $u$, for ${\mathcal {E}} u$ and for $\chi$, namely we assume that

(4.3)\begin{align} & \lim_{{\varepsilon}\rightarrow 0^+}\frac{1}{{\varepsilon}^{N}}\int_{Q\left(x_0,{\varepsilon}\right)}\left\vert u(x)-u(x_0)\right\vert\,{\rm d}x = 0, \end{align}
(4.4)\begin{align} & \lim_{{\varepsilon} \to 0^+} \frac{1}{{\varepsilon}^N} \int_{Q(x_0, {\varepsilon})}|\mathcal{E} u(x) - \mathcal{E}u(x_0)|\,{\rm d}x = 0 \end{align}

and

(4.5)\begin{equation} \lim_{{\varepsilon}\rightarrow 0^+}\frac{1}{{\varepsilon}^{N}}\int_{Q\left(x_0,{\varepsilon}\right)}\left\vert \chi(x)-\chi(x_0)\right\vert\,{\rm d}x = 0. \end{equation}

We observe that the above properties hold for