On some non-local approximation of nonisotropic Griffith-type functionals

1.   Introduction

The scope of this paper is to provide a generalization of recent results, obtained in ^[23], concerning the approximation of brittle fracture energies for linearly elastic materials, by means of nonlocal functionals defined on Sobolev spaces, which are easier to handle also from a computational point of view.

In ^[23] an approach originally devised by Braides and Dal Maso ^[6] for the approximation of the Mumford-Shah functional has been generalized to the linearly elastic context. Namely, it was shown that, for a given bounded increasing function $f\colon \mathbb{R}^+\to \mathbb{R}^+$ the energies

$\Gamma$-converge to the functional

with $\alpha = f^\prime(0)$ and $\beta = \lim_{t\to +\infty}f(t)$, in the $L^1(\Omega)$-topology. Above, $W(\mathcal Eu(y))$ is a convex elastic energy depending on the linearized strain $\mathcal Eu$, given by the symmetrized gradient of a vector-valued displacement $u$, whose jump set $J_u$ represents the cracked part of a material. The energy space of the limit functional is the one of generalized functions with bounded deformation, introduced in ^[17].

It is noteworthy that the above result allowed one for a general (convex) bulk energy $W$ having $p$-growth for $p > 1$. The proof strategy must then avoid, at least when estimating the bulk part, any slicing procedure. This latter is instead successful in the special case^* $W(\xi) = |\xi|^p$, considered for instance in ^[22]. There, non-local convolution-type energies of the form

^* We remark that this particular case is however not the most relevant one from a mechanical point of view, as even for an isotropic material additional terms in the bulk energy are expected to appear.

are considered, where $\rho$ is a convolution kernel whose support is a convex bounded domain and $\rho_ \varepsilon(z)$ is the usual sequence of convolution kernels $\rho(z/ \varepsilon)/ \varepsilon^n$. The $\Gamma$-limit of (1.1) with respect to the $L^1$ convergence is given by

where the anisotropic surface density $\phi_\rho$ depends on the geometry and on the size of ${\rm supp }\rho$. A similar effort of generalizing the results of ^[6] to Mumford-Shah type energies with non-isotropic surface part has been previously performed in ^[13].

In this paper, we extend the focus of ^[22,23] by showing that general Griffith-type functionals of the form

where $\phi$ is any norm on $\mathbb{R}^n$, can be obtained as variational limit of non-local convolution-type functionals

Above, $f$ is again a bounded nondecreasing function with $\alpha = f^\prime(0)$ and $\beta = \lim_{t\to +\infty}f(t)$, and the unscaled kernel $\rho$ has the bounded convex symmetric domain $\overline{S}: = \; \{\xi\in \mathbb{R}^n \colon \phi(\xi)\le 1\}$ as its support. This is the analogue, in the linear elastic setting, of the results in ^[13].

The proof strategy we devise is based on a localization method and involves nontrivial adaptions to the method used in ^[23], in particular when estimating the bulk term in the $\Gamma$-liminf inequality (Proposition 4.2). There, we have to impose (and this is the only point in the paper) an additional restriction on the convolution kernel $\rho$, namely of being nonincreasing with respect to the given norm $\phi$ (see Assumption (N2) below). This is namely needed in order to be able to estimate from below the size of the nonlocal approximations of the bulk term in an anisotropic tubular neighborhood of the set where they exceed the threshold $\frac{\beta}\alpha$, which heuristically corresponds to the breaking of the elastic bonds. With this, a set $K^\prime_\varepsilon$ with small area and bounded perimeter, where the fracture energy concentrates can be explicitly constructed. This yields an estimate of the $\Gamma$-liminf which has an optimal constant in front of the bulk term, although being non-optimal for the surface energy.

Another non-optimal estimate for the $\Gamma$-liminf, but with an optimal constant for the surface energy can be instead obtained by a slicing procedure, involving a comparison argument and the convexity of the open set $S$ (Proposition 4.3). As bulk and surface energy in (1.2) are mutually singular as measures, a localization procedure entails then the $\Gamma$-liminf inequality (Proposition 4.6). Finally, the $\Gamma$-limsup inequality (Proposition 5.1) can be obtained by a direct construction for a regular class of competitors having a "nice" jump set, and which are dense in energy. Notice indeed that such an approximation (see Theorem 2.3 for a precise statement) is possibile also with respect to an anisotropic norm $\phi$, combining the recent results in ^[8] with the ones in ^[12].

As a final remark, it would be desirable to get rid of the structural assumption (N2) on the convolution kernels, which is used only in Proposition 4.2. It is our opinion that this is going to require quite a delicate abstract analysis of the $\Gamma$-limit of nonlocal functionals which approximate free-discontinuity problems in $GSBD$, possibly including also finite-difference models which are well suited to numerical approximations (see ^[15] for a recent discrete finite-difference approximation of some Griffith-type functionals in $GSBD$). A similar analysis for the $SBV$ setting has been performed in ^[11], where integral representation formulas for the limit energy have been provided. Furthermore, nontrivial sufficient conditions have been given under which the bulk part of the energy can be recovered by only considering weakly compact sequences in Sobolev spaces. We plan to defer this abstract analysis to a forthcoming contribution. For the asymptotic analysis via $\Gamma$-convergence of local free-discontinuity functionals in linear elasticity and the related issues, we refer the reader to the very recent papers ^[7,14,20].

Outline of the paper: The paper is structured as follows. In Section 2 we fix the basic notation and results on the function spaces we will deal with (Section 2.2), together with some technical lemmas (Section 2.3) which will be useful throughout the paper. In Section 3 we list the main assumptions, introduce our model (Eq (3.4)), and state the main results of the paper, provided {in} Theorem 3.1 and Theorem 3.2. Section 4 is devoted to the proof of the compactness statements in the main Theorems (Proposition 4.1), and to the $\Gamma$-liminf inequality, which is proved in Section 4.3 combining the estimates in Sections 4.1 and 4.2. The proof of the upper bound is given in Section 5.

2.   Notation and preliminary results

2.1.   Notation

The symbol $|\cdot|$ denotes the Euclidean norm in any dimension, while $\langle\cdot, \cdot\rangle$ stands for the scalar product in $\mathbb{R}^n$. We will always denote by $\Omega$ an open, bounded subset of $\mathbb{R}^n$ {with Lipschitz boundary}, and by $\mathbb{S}^{n-1}$ the $(n-1)$-dimensional unit sphere. The Lebesgue measure in $\mathbb{R}^n$ and the $s$-dimensional Hausdorff measure are written as $\mathcal{L}^n$ and $\mathcal{H}^s$, respectively. $\mathcal{A}(\Omega)$ stands for the family of the open subsets of $\Omega$.

Let $S$ be a bounded, open, convex and symmetrical set, i.e., $S = -S$. For $\eta > 0$, we denote by $\eta S$ the $\eta$-dilation of $S$ and we will often use the shorthand {$S(x, \eta)$} in place of $x+\eta S$. We consider $|\cdot|_S$ the norm induced by $S$, defined as

whose unit ball $\{|x|_S < 1\}$ coincides with $S$, and, correspondingly, we introduce the distance to a closed bounded set $K\subset \mathbb{R}^n$; namely,

2.2.   $GBD$ and $GSBD$ functions

In this section we recall some basic definitions and results on generalized functions with bounded deformation, as introduced in ^[17]. Throughout the paper we will use standard notations for the spaces $(G)SBV$ and $(G)SBD$, referring the reader to ^[2] and ^[1,3,24], respectively, for a detailed treatment on the topics.

Let $\xi\in \mathbb{R}^n\backslash\{0\}$ and $\Pi^\xi = \{y\in \mathbb{R}^n:\, \langle\xi, y\rangle = 0\}$. If $\Omega\subset \mathbb{R}^n$ and $y\in\Pi^\xi$ we set $\Omega_{\xi, y}: = \{t\in \mathbb{R}:\, y+t\xi\in \Omega\}$ and $\Omega_\xi: = \{y\in \Pi^\xi:\, \Omega_{\xi, y}\neq\emptyset\}$. Given $u:\Omega\to \mathbb{R}^n$, $n\geq2$, we define $u^{\xi, y}: \Omega_{\xi, y}\to \mathbb{R}$ by

while if $v: \Omega\to \mathbb{R}$, the symbol $v^{\xi, y}$ will denote the restriction of $v$ to the set $\Omega_{\xi, y}$; namely,

Let $\xi\in \mathbb{S}^{n-1}$. For any $x\in \mathbb{R}^n$ we denote by $x_\xi$ and $y_\xi$ the projections onto the subspaces $\Xi: = \{t\xi:\, \, t\in \mathbb{R}\}$ and $\Pi^\xi$, respectively. For $\sigma, r > 0$ and $x\in \mathbb{R}^n$ we define the cylinders

Note that $C_{\sigma, r}^\xi(x) = (x_\xi-\sigma, x_\xi+\sigma)\times B^{n-1}_{r}(y_\xi)$, where $B^{n-1}$ denotes a ball in the $(n-1)$-dimensional space $\Pi^\xi$.

Definition 2.1. An $\mathcal L^{n}$-measurable function $u:\Omega\to \mathbb{R}^{n}$ belongs to $GBD(\Omega)$ if there exists a positive bounded Radon measure $\lambda_u$ such that, for all $\tau \in C^{1}(\mathbb{R}^{n})$ with $-\frac12 \le \tau \le \frac12$ and $0\le \tau'\le 1$, and all $\xi \in \mathbb{S}^{n-1}$, the distributional derivative $D_\xi (\tau(\langle u, \xi\rangle))$ is a bounded Radon measure on $\Omega$ whose total variation satisfies

for every Borel subset $B$ of $\Omega$.

If $u\in GBD(\Omega)$ and $\xi\in \mathbb{R}^n\backslash\{0\}$ then, in view of [17,Theorem 9.1,Theorem 8.1], the following properties hold:

${\rm(a)}$ $\dot{u}^{\xi, y}(t) = \langle\mathcal{E}u(y+t\xi)\xi, \xi\rangle$ for a.e. $t\in \Omega_{\xi, y}$;

${\rm(b)}$ $J_{u^{\xi, y}} = (J_u^\xi)_{\xi, y}$ for $\mathcal{H}^{n-1}$-a.e. $y\in\Pi^\xi$, where

Definition 2.2. A function $u \in GBD(\Omega)$ belongs to the subset $GSBD(\Omega)$ of special functions of bounded deformation if, in addition, for every $\xi \in \mathbb{S}^{n-1}$ and $\mathcal H^{n-1}$-a.e. $y \in \Pi^\xi$, it holds that $u^{\xi, y}\in SBV_{\mathrm{loc}}(\Omega_{\xi, y})$.

The inclusions $BD(\Omega)\subset GBD(\Omega)$ and $SBD(\Omega)\subset GSBD(\Omega)$ hold (see [17,Remark 4.5]). Although they are, in general, strict, relevant properties of $BD$ functions are retained also in this weak setting. In particular, $GBD$-functions have an approximate symmetric differential $\mathcal{E}u(x)$ at $\mathcal L^{n}$-a.e. $x\in \Omega$. Furthermore the jump set $J_u$ of a $GBD$-function is $\mathcal H^{n-1}$-rectifiable (this is proven in [17,Theorem 6.2 and Theorem 9.1], but it has been recently shown that this property is actually a general one for measurable functions ^[18]).

Let $p > 1$. The space $GSBD^p(\Omega)$ is defined as

Every function in $GSBD^p(\Omega)$ can be approximated with the so-called "piecewise smooth" $SBV$-functions, denoted $\mathcal{W}(\Omega; \mathbb{R}^n)$, characterized by the three properties

This is stated by the following result, which combines [8,Theorem 1.1] with [12,Theorem 3.9].

Theorem 2.3. Let $\phi$ be a norm on $\mathbb{R}^n$. Let $\Omega\subset\mathbb{R}^n$ be a bounded open Lipschitz set, and let $u\in GSBD^p(\Omega; \mathbb{R}^n)$. Then there exists a sequence $(u_j)$ such that $u_j\in \mathcal{W}(\Omega; \mathbb{R}^n)$ and

Moreover, if $\int_\Omega \psi(|u|)\, \mathrm{d}x$ is finite for $\psi:[0, +\infty)\to[0, +\infty)$ continuous, increasing, with

then

As observed in [9,Remark 4.3], we may even approximate through functions $u$ such that, besides (2.5), have a closed jump set strictly contained in $\Omega$ made of pairwise disjoint $(n{-}1)$-dimensional simplexes, with $J_u \cap \Pi_i \cap \Pi_l = \emptyset$ for any two different hyperplanes $\Pi_i$, $\Pi_l$.

We recall the following general $GSBD^p$ compactness result from ^[10], which generalizes [17,Theorem 11.3].

Theorem 2.4 ($GSBD^p$ compactness). Let $\Omega \subset \mathbb{R}^n$ be an open, bounded set, and let $(u_j)_j \subset GSBD^p(\Omega)$ be a sequence satisfying

Then there exists a subsequence, still denoted by $(u_j)$, such that the set ${A^\infty} : = \lbrace x\in \Omega: \, |u_j(x)| \to +\infty \rbrace$ has finite perimeter, and there exists $u \in GSBD^p(\Omega)$ such that

where $\partial^*$ denotes the essential boundary of a set with finite perimeter.

Remark 2.5. If in the statement above one additionally assumes that

for a positive, continuous and increasing function $\psi$ with $\lim_{s\to +\infty}\psi(s) = +\infty$, then $A^\infty = \emptyset$, so that $|u|$ is finite a.e., and (i) holds on $\Omega$. Moreover, if $\psi$ is superlinear at infinity, that is

by the Vitali dominated convergence theorem one gets that $u \in L^1(\Omega)$ and ${\rm (i)}$ holds with respect to the $L^1$-convergence in $\Omega$.

2.3.   Some lemmas

We recall here (without adding the standard proofs) some properties of integral convolutions in the setting of Sobolev spaces.

Proposition 2.6. Let $w\in W^{1, p}(\Omega; \mathbb{R}^n)$ and $\rho\in L^\infty(\mathbb{R}^n)$ be a convolution kernel, with ${\rm supp}\, \rho\subset\overline{S}$ for some bounded, open and convex set $S\subset\Omega$. Set $\rho_\theta(x): = \frac1{\theta^n}\rho\left(\frac x\theta\right)$. Then the following holds:

(i) let $\Omega' \subset\subset \Omega$ and $0\le \theta \le \mathrm{dist}_S(\Omega', \partial \Omega)$. The convolution

belongs to $W^{1, p}(\Omega'; \mathbb{R}^n)$. Moreover, it holds that

(ii) assume that $w_{ \varepsilon}\rightarrow w$ in $L^1(\Omega; \mathbb{R}^n)$ and let $\theta_\varepsilon$ be any sequence with $\theta_\varepsilon \to 0$ when $\varepsilon \to 0$. Then the sequence

satisfies $\hat{w}_ \varepsilon \to c w$ in $L^1(\Omega; \mathbb{R}^n)$, where $c = \int_{ \mathbb{R}^n} \rho(x)\, \mathrm{d}x$.

We also recall the following convergence property of one-dimensional sections of averaged functions (see, e.g., [23,Lemma 2.7(ii)]).

Lemma 2.7. Assume that $w_{ \varepsilon}\rightarrow w$ in $L^1(\Omega; \mathbb{R}^n)$ and let $\eta_\varepsilon$ be any sequence with $\eta_\varepsilon \to 0$ when $\varepsilon \to 0$. Then for all $\xi \in \mathbb{S}^{n-1}$ and a.e. $y \in \Pi^{\xi}$, the sequence

satisfies $\hat{w}^{\xi, y}_ \varepsilon \to w^{\xi, y}$ in $L^1(\Omega_{\xi, y}; \mathbb{R}^n)$, where $w^{\xi, y}(t): = w(y+t\xi)$.

We will also make use of the following localization result, dealing with the supremum of a family of measures (see, e.g., [4,Proposition 1.16]).

Lemma 2.8. Let $\mu:\mathcal{A}(\Omega)\longrightarrow[0, +\infty)$ be a superadditive function on disjoint open sets, let $\lambda$ be a positive measure on $\Omega$ and let $\varphi_h: \Omega\longrightarrow[0, +\infty]$ be a countable family of Borel functions such that $\mu(A)\geq\int_A\varphi_h\, \mathrm{d}\lambda$ for every $A\in\mathcal{A}(\Omega)$. Then, setting $\varphi: = \sup_{h\in \mathbb{N}}\varphi_h$, it holds that

for every $A\in\mathcal{A}(\Omega)$.

Lower semicontinuous increasing functions can be approximated from below with truncated affine functions. We refer the reader to [23,Lemma 2.10] for a proof of the following result.

Lemma 2.9. Consider a lower semicontinuous increasing function $f:[0, +\infty)\to[0, +\infty)$ such that there exist $\alpha, \beta > 0$ with

Then there exist two positive sequences $(a_i)_{i\in \mathbb{N}}$, $(b_i)_{i\in \mathbb{N}}$ with

and $\min\{a_i t, b_i\}\le f(t)$ for all $i \in \mathbb{N}$ and $t \in \mathbb{R}$.

2.4.   $\Gamma$-convergence

Let $(X, d)$ be a metric space. We recall here the definition of $\Gamma$-convergence for families of functionals $F_\varepsilon:X\to[-\infty, +\infty]$ depending on a real parameter $\varepsilon$ (see, e.g., ^[5,16]).

For all $u\in X$, we define the lower $\Gamma$-limit of $(F_\varepsilon)$ as $\varepsilon\to0^+$ by

and the upper $\Gamma$-limit of $(F_\varepsilon)$ as $\varepsilon\to0^+$ by

We then say that $(F_\varepsilon)$ $\Gamma$-converges to $F:X\to[-\infty, +\infty]$ as $\varepsilon\to0^+$ iff

2.5.   A one-dimensional $\Gamma$-convergence result

The following one-dimensional $\Gamma$-convergence result will be useful in the proof of the lower bound for the surface term. In the statement below, functions in $L^1(I)$ with $I \subset \mathbb{R}$ are extended by $0$ outside $I$, so that the functionals $H_\varepsilon$ are well-defined (actually, the result is not affected by the considered extension).

Theorem 2.10. Let $p > 1$, let $I\subset \mathbb{R}$ be a bounded interval and consider a lower semicontinuous, increasing function $f:[0, +\infty)\to[0, +\infty)$ complying with

for some $\alpha, \beta > 0$. Let $H_\varepsilon:L^1(I)\to[0, +\infty]$ be defined by

where it is understood that

if $u\not\in W^{1, p}(x-\varepsilon, x+\varepsilon)$. Then the functionals $(H_\varepsilon)$ $\Gamma$-converge as $\varepsilon\to0^+$ to the functional

in $L^1(I)$.

Proof. The proof can be found, e.g., in [4,Theorem 3.30].

3.   The non-local model and main results

In this section we list our assumptions and introduce the main results of the paper. Let $\Omega\subset \mathbb{R}^n$ be an open set with Lipschitz boundary, let $1 < p < +\infty$ and $f:[0, +\infty)\to[0, +\infty)$ a lower semicontinuous, increasing function satisfying

Let $\rho\in L^\infty(\mathbb{R}^n; [0, +\infty))$ be a convolution kernel. The minimal assumption is that

(N1) $\rho$ is Riemann integrable with $\|\rho\|_1 = 1$ and $S = S_\rho: = \{x\in \mathbb{R}^n:\, \, \rho(x)\neq0\}$ is a bounded, open, convex and symmetrical set.

As every Riemann integrable function is continuous at almost every point, we may also suppose, up to a modification on a null set, that $\rho$ is lower semicontinuous. Also notice that, by a simple scaling argument, one can always consider the case of kernels with unit mass, up to modifying the constant $\alpha$ in (3.1).

A sequence $(\rho_\varepsilon)_{\varepsilon > 0}$ of convolution nuclei is then obtained by setting, for every $x\in \mathbb{R}^n$ and $\varepsilon > 0$,

For every $v\in \mathbb{R}^n$ we define

Under the previous assumptions on $S$, the function $\phi_\rho$ turns out to be a norm on $\mathbb{R}^n$.

To obtain our main result, we will have to couple (N1) with the additional assumption that the convolution kernel is a non-increasing function with respect to the norm $|\cdot|_S$, that is

(N2) $|x|_S\ge |y|_S\Longrightarrow\rho(x)\le \rho(y)$ for all $x$, $y \in \mathbb{R}^n$.

Equivalently, we require that it exists a non-increasing function $\varrho\colon \mathbb{R}^+\to \mathbb{R}^+$ such that $\rho(x) = \varrho(|x|_S)$. Notice that, in the case $S = B_1$, every non-increasing radial function $\rho$ complies with (N2).

Let $W: \mathbb{R}^{n\times n}\to \mathbb{R}$ be a convex positive function on the subspace $\mathbb{M}^{n\times n}_{sym}$ of symmetric matrices, such that

For every $\varepsilon > 0$ we consider the functional $F_\varepsilon: L^1(\Omega; \mathbb{R}^n)\to [0, +\infty]$ defined as

We will deal with a localized version of the energies (3.4). Namely, for every $A\in\mathcal{A}(\Omega)$, we will denote by $F_\varepsilon(u, A)$ the same functional as in (3.4) with the set $A$ in place of $\Omega$. When $A = \Omega$, we simply write $F_\varepsilon(u)$ in place of $F_\varepsilon(u, \Omega)$.

The following theorem is the first main result of this paper. We notice that the additional assumption (N2) on the structure of the convolution kernel is required in (ii) below only to obtain the optimal lower bound for the bulk term of the energy.

Theorem 3.1. Let $\rho\in L^\infty(\mathbb{R}^n; [0, +\infty))$ be a convolution kernel as in $({\rm{N1}})$, and let $F_\varepsilon$ be defined as in $(3.4)$. Under assumptions $(3.1)$ and $(3.3)$, it holds that

$\rm(i)$ there exists a constant $c_0$ independent of $\varepsilon$ such that, for all $(u_\varepsilon)\subset L^p(\Omega; \mathbb{R}^n)$ satisfying $F_\varepsilon(u_\varepsilon) \leq C$ for every $\varepsilon>0$, one can find a sequence $\overline{u}_\varepsilon \in {GSBV^p(\Omega; \mathbb{R}^n)}$ with

$\rm(ii)$ If, in addition, $\rho$ complies with $({\rm{N2}})$, then the functionals $(F_\varepsilon)$ $\Gamma$-converge, as $\varepsilon\to0$, to the functional

with respect to the $L^1$ convergence in $\Omega$.

The $L^1$-convergence on the whole $\Omega$ can be enforced with the addition of a lower order fidelity term, as we have discussed in Remark 2.5. This motivates the statement below, {where} we consider a continuous increasing function $\psi:[0, +\infty)\to[0, +\infty)$ such that

and we set for every $A\in\mathcal{A}(\Omega)$

As before, we simply write $G_\varepsilon(u)$ in place of $G_\varepsilon(u, \Omega)$.

Then we have the following result.

Theorem 3.2. Under assumptions $(3.1)$, $({\rm{N1}})$, $(3.3)$, and $(3.6)$ it holds that

$\rm(i)$ If $(u_\varepsilon)\subset L^p(\Omega; \mathbb{R}^n)$ is such that $G_\varepsilon(u_\varepsilon) \leq C$ for every $\varepsilon>0$, then $(u_\varepsilon)$ is compact in $L^1(\Omega; \mathbb{R}^n)$.

$\rm(ii)$ If, in addition, $({\rm{N2}})$ holds, the {functionals} $(G_\varepsilon)$ $\Gamma$-converge, as $\varepsilon\to0$, to the functional

with respect to the $L^1$ convergence in $\Omega$.

4.   Compactness and estimate from below of the $\Gamma$-limit

With the following proposition, we prove the compactness statements in Theorem 3.1(i), and Theorem 3.2(i), respectively. These results can be easily inferred by a comparison with non-local integral energies whose densities are averages of the gradient on balls with small radii, for which a compactness result has been provided in [23,Proposition 4.1]. In order to do that, we will only require assumption (N1) on the convolution kernel $\rho$.

Proposition 4.1. Let $A\in\mathcal{A}(\Omega)$, and let $F_\varepsilon$, $G_\varepsilon$ be defined as in $(3.4)$, and $(3.7)$, respectively, where $\rho\in L^\infty(\mathbb{R}^n; [0, +\infty))$ satisfies $({\rm{N1}})$. Then:

(i) Assume $(3.1)$, $(3.3)$. If $(u_\varepsilon)\subset L^p(\Omega; \mathbb{R}^n)$ is such that $F_\varepsilon(u_\varepsilon, A) \leq C$ for every $\varepsilon > 0$, one can find a sequence $\overline{u}_\varepsilon \in {GSBV^p(A; \mathbb{R}^n)}$ with

for some $c_0 > 0$.

(ii) Assume $(3.1)$, $(3.3)$, and $(3.6)$. If $(u_\varepsilon)\subset L^p(\Omega; \mathbb{R}^n)$ is such that $G_\varepsilon(u_\varepsilon, A) \leq C$ for every $\varepsilon > 0$, then $(u_\varepsilon)$ is compact in $L^1(A; \mathbb{R}^n)$.

Proof. Let $\eta\in(0, 1)$ be fixed such that $B_\eta(0)\subset\subset S$, and denote by $m_\eta$ the minimum of $\rho$ on $\overline{B_\eta}$, which is strictly positive as we are assuming that $\rho > 0$ on $S$. Setting $\tilde{f}(t): = f(m_\eta\omega_n\eta^nt)$ and for any $\varepsilon > 0$, we consider the energies

Since $B_\eta(0)\subseteq S$ and $\rho\geq m_\eta$ on $B_\eta(0)$, a simple computation shows that

for every $u\in W^{1, p}(\Omega; \mathbb{R}^n)$ and $A\subseteq\Omega$ open set. By virtue of (4.2), to obtain (i) it will suffice to apply the argument of [23,Proposition 4.1] to the sequence $\widetilde{F}_\varepsilon$ in (4.1). We then omit the details.

We now come to (ii). If additionally $G_\varepsilon(u_\varepsilon, A) \leq C$, following the argument for [23,Proposition 4.1(ii)], it can be shown that the sequence $(\bar{u}_ \varepsilon)$ constructed in (i) complies with

for all $\varepsilon$. Therefore, in view of the growth assumption (3.6) on $\psi$, Theorem 2.4 and Remark 2.5 apply, and this provides the compactness of the sequence $(\overline{u}_\varepsilon)$ in $L^1(A; \mathbb{R}^n)$. Then, since $\overline{u}_\varepsilon-u_\varepsilon \to 0$ in measure on $A$, with the Vitali dominated convergence Theorem we infer that $(u_\varepsilon)$ is compact in $L^1(A; \mathbb{R}^n)$ as well. This concludes the proof of (ii).

Now, we turn to provide a first estimate of the $\Gamma$-liminf of the functionals $F_ \varepsilon$. This estimate is optimal, up to a small error, only for the bulk part of the energy, and this is the only very point where we need to require the additional assumptions (N2) on the convolution kernels (see Section 4.1). The proof of an optimal estimate for the surface term, instead, will be derived separately by means of a slicing argument (see Proposition 4.3 below) for more general kernels complying only with (N1) providing the comparison estimate (4.2). As the two parts of the energy are mutually singular, the localization method of Lemma 2.8 will eventually allow us to get the $\Gamma$-liminf inequality.

4.1.   Estimate from below of the bulk term

We begin by giving the announced estimate for the bulk term.

Proposition 4.2. Let $A\in\mathcal{A}(\Omega)$ with $A\subset\subset \Omega$, and consider a sequence $u_ \varepsilon\in W^{1, p}(\Omega; \mathbb{R}^n)$ converging to $u$ in $L^1(\Omega; \mathbb{R}^n)$. Assume $(3.1)$ and $(3.3)$, let $\eta\in(0, 1)$ be fixed and let $\rho$ comply with (N1)–(N2). Suppose that

Then, for every fixed $0 < \delta < 1$, there exist a constant $M_{\delta, \eta}$ only depending on $f$, $\delta$ and $\eta$, a constant $\sigma_\eta$ depending on $\rho, \eta$ such that $\sigma_\eta\to0$ as $\eta\to0$, and a sequence of functions $(v_\varepsilon^{\delta, \eta})\subset {GSBV^p(A; \mathbb{R}^n)}$ such that

${\rm (i)}$ $\alpha(1-\sigma_\eta)^2(1-\delta)^{2n+1}\int_{A} W(\mathcal{E}v_\varepsilon^{\delta, \eta}(x))\, \mathrm{d}x\leq F_\varepsilon(u_ \varepsilon, A)$;

${\rm (ii)}$ $\mathcal{H}^{n-1}(J_{v_\varepsilon^{\delta, \eta}})\leq M_{\delta, \eta}\, F_\varepsilon(u_ \varepsilon, A)$;

$(iii)$ $v_\varepsilon^{\delta, \eta} \to u$ in $L^1(A; \mathbb{R}^n)$ as $\varepsilon \to 0$.

Proof. We first consider the case $f(t) = \min\{at, b\}$, with $a, b > 0$. For given $\eta$, we introduce the truncated kernel $\rho^\eta(x): = \frac{1}{1-\sigma_\eta}\rho(x)(1-\chi_{\eta S}(x))$, where the constant $\sigma_\eta$ is given by

and $\sigma_\eta\to0$ as $\eta\to0$. Notice that with this choice of $\sigma_\eta$ one has $\int_{ \mathbb{R}^n}\rho^\eta(x)\, \mathrm{d}x = 1$.

For fixed $\delta\in(0, 1)$, we then define

and the functions

Observe that, since $W\ge 0$, by the definition of $\rho^\eta$ and assumption (N2), we get

for all $x\in A$. Define now the following sets, depending on $\delta, \eta$ and $S$:

We prove the inclusion

For this, if $x\in K^{\prime}_\varepsilon$ then there exists $z\in K_\varepsilon$ such that $|x-z|_S\leq \delta\eta\varepsilon$. Now, by the triangle inequality, for every $y\in\Omega$ it holds that

whence

if and only if ${|z-y|_S}\geq (1-\delta)\eta\varepsilon$. In this case, since $\rho$ is non-increasing with respect to $|\cdot|_S$, we have

Notice that this inequality holds true also if $|z-y|_S < (1-\delta)\eta\varepsilon$. In this case, indeed, one has $\frac{y-z}{(1-\delta)\varepsilon}\in \eta S$ and hence $\rho^\eta\left(\frac{y-z}{(1-\delta)\varepsilon}\right) = 0$ by definition of $\rho^\eta$. Rescaling the kernels and using (4.4) we get

so that

and the proof of (4.8) is concluded.

Now, from the inclusion (4.8) and the fact that ${f}(t) = b$ for $t\geq \frac{b}{a}$, we deduce that

Applying the coarea formula (see for instance [19,Theorem 3.14]) to the $1$-Lipschitz function $g(x): = \mathrm{dist}_S(x, K_ \varepsilon)$ in the open set $\{0 < g(x) < \eta\delta \varepsilon\}\subset K^{\prime}_\varepsilon$ we get

It follows that we can choose $0 < \delta^\prime_ \varepsilon < \eta\delta \varepsilon$ such that, for

it holds

We define a sequence $(v_\varepsilon^{\delta, \eta})$ of functions in ${GSBV^p(A; \mathbb{R}^n)}$ as

Since $\|\rho^\eta\|_1 = 1$, by Proposition 2.6(ii) (applied for $\theta_\varepsilon = \varepsilon(1-\delta)$) and the fact that, by construction and (4.9), it holds $\mathcal L^n(K^{\prime\prime}_\varepsilon)\to 0$ when $\varepsilon \to 0$, we have that $v_\varepsilon^{\delta, \eta} \to u$ in $L^1(A; \mathbb{R}^n)$ as $\varepsilon \to 0$. We also have $\mathcal{H}^{n-1}(J_{v_\varepsilon^{\delta, \eta}})\le \mathcal{H}^{n-1}(\partial K^{\prime\prime}_\varepsilon)$, so that with (4.11) we deduce (ii) for $M_{\delta, \eta} = \frac1{\eta\delta b}$.

Now, since $K_ \varepsilon \subset K^{\prime\prime}_ \varepsilon$ and $A\subset\subset \Omega$, it holds $\psi_\varepsilon^{\eta, \delta}(x) < {C_{\delta, \eta}}\frac ba$ for all $x\in K^{\prime\prime}_ \varepsilon$. As $f(t) = \min\{at, b\}$, this gives

for all $x \in A \setminus K^{\prime\prime}_ \varepsilon$. Now, since the function $f$ is concave and $f(0) = 0$, it holds $f(\lambda t) \geq \lambda f(t)$ for all $\lambda\in[0, 1]$. Combining with the monotonicity of $f$ and (4.5), we have

for all $x \in A$. With this, using (4.13), the Jensen's inequality, (2.11), (4.12), and since $W(\bf 0) = 0$, we get

For a general $f$ complying with (3.1), use Lemma 2.9 to find $a_\delta, b_\delta > 0$ with $a_\delta\ge \alpha(1-\delta)$ and $f(t) \ge \min\{a_\delta t, b_\delta\}$ for all $t \in \mathbb{R}$, and perform the same construction as in the previous step. This gives (iii), (ii) (with $M_\delta: = \frac1{\delta\eta b_\delta}$) and

that is (i).

4.2.   Estimate from below of the surface term

In this section we derive by slicing a lower bound for the surface term in the energy. It is worth mentioning that, by virtue of (4.17), the desired estimate could be probably also obtained by adapting to the $GSBD$-setting the semi-discrete approach of [22,Proposition 6.4]. Nonetheless, that argument is quite delicate for our purposes, and more complicated than we need. It indeed aimed to provide an optimal lower bound for both the bulk and the surface terms in a unique proof by means of a slicing procedure. In our case, the general form of the bulk energy we are considering does not comply with slicing arguments. Therefore, on the one hand, the two terms have to be estimated separately. On the other hand, an independent and simpler strategy can be followed to provide a lower bound with optimal constant in front of the surface energy.

We set

for every $\xi\in\mathbb{S}^{n-1}$.

Proposition 4.3. Let $\rho\in L^\infty(\mathbb{R}^n; [0, +\infty))$ be a convolution kernel complying with $({\rm{N1}})$, and let $F_\varepsilon$ be defined as in $(3.4)$. Assume $(3.1)$ and $(3.3)$. Let $\delta\in(0, 1)$ be fixed, and consider a sequence $\varepsilon_j \to 0$. Let $A\in\mathcal{A}(\Omega)$ and $u_j\in W^{1, p}(A; \mathbb{R}^n)$ converging to $u$ in $L^1(A; \mathbb{R}^n)$. Assume that

Then $u \in GSBD^p(A)$ and

for every $\xi\in \mathbb{S}^{n-1}$.

Proof. It follows from Proposition 4.1 and Theorem 2.4 that $u \in GSBD^p(A)$. To prove (4.16), we first note that, by virtue of the growth assumption (3.3), we have

for every $\xi\in \mathbb{S}^{n-1}$. Thus, for every fixed $\xi$, since $f$ is non-decreasing, it will be sufficient to provide a lower estimate for the energies

We proceed by a slicing argument, and for each $x\in A$ we denote by $x_\xi$ and $y_\xi$ the projections of $x$ onto $\Xi$ and $\Pi^\xi$, respectively. Since $S$ is open and convex, for every fixed $\xi\in \mathbb{S}^{n-1}$ we can find a radius $r = r(\delta, S) > 0$ such that the cylinder

where $\lambda_{\xi, \delta}: = \frac{\tau_\xi}{2}(1-\delta)$ and $\tau_\xi$ is the length of the section ${S_\xi}$. Indeed, since $S$ is open, some $\eta > 0$ can be found such that $\overline{B_{\eta}(0)}$ is contained in $S$. Now, if $t = (1-\delta)s$ for some $s\in {S_\xi}$ and $y\in\xi^\perp$ with $|y|\leq\eta$, then $t\xi+\delta y\in S$ from the convexity of $S$. Thus, it will suffice to choose $r: = \delta\eta$.

If we denote by $m_C$ the minimum of $\rho$ on $\overline{C_{(1-\delta), r}^\xi}$, we then have

where $\tilde{f}(t): = f(c\, m_C t)$. Note that $\tilde{f}(t)\to\beta$ as $t\to+\infty$.

We now set

We denote (with a slight abuse of notation) still with $z$ the $(n-1)$-dimensional variable in $B^{n-1}_{r \varepsilon_j}(y_\xi)$. Set ${w}_j^{\xi, y_\xi}(t): = {\rlap{-} \smallint }_{B^{n-1}_{r \varepsilon_j}(y_\xi)}\langle u_j(z+t\xi)), \xi\rangle\, \mathrm{d}z$.

By virtue of Lemma 2.7(ii), applied with $\theta_{\varepsilon_j} = r\varepsilon_j$, we have that ${w}_j^{\xi, y_\xi}$ converges to $u^{\xi, y_\xi}$ in $L^1(A_{\xi, y_\xi})$ for a.e. $y_\xi$. Furthermore, setting $g(t): = \tilde{f}(\omega_{n-1}r^{n-1} t)$, Fubini's Theorem, Jensen's inequality and the monotonicity of $\tilde{f}$ entail that

Now, for the function $t\mapsto g(t)$ it still holds $g(t)\to \beta$ when $t\to +\infty$. Hence, applying Theorem 2.10 to the one-dimensional energies

we obtain the lower bound

Therefore, using (4.20) and (4.21) we deduce

With (4.19) and Fatou's Lemma we finally have

where in the last equality we used the Area Formula. This concludes the proof of (4.16).

4.3.   Proof of the $\Gamma$-liminf inequality

For any $A\in\mathcal{A}(\Omega)$, we denote by $F'(u, A)$ and $G'(u, A)$ the lower $\Gamma$-limits of $F_\varepsilon(u, A)$ and $G_\varepsilon(u, A)$, respectively, as defined in (2.12). It holds that $G'(u, A)\geq F'(u, A)$ for each $A\in\mathcal{A}(\Omega)$ and $u \in L^1(A; \mathbb{R}^n)$ (see, e.g., [16,Proposition 6.7]). The results of the previous subsection lead to the following estimate.

Proposition 4.4. Assume $(3.1)$, $(3.3)$, and $(3.6)$. Let $F_ \varepsilon$ and $G_ \varepsilon$ be defined as in $(3.4)$ and $(3.7)$, respectively, and let $\rho$ comply with (N1) -(N2). Let $u\in L^1(\Omega; \mathbb{R}^n)$, $A\in\mathcal{A}(\Omega)$, and define $F'(u, A)$ and $G'(u, A)$ by $(2.12)$ in correspondence of $F_\varepsilon$ and $G_\varepsilon$, respectively. If $F'(u, A) < +\infty$, then $u\in GSBD^p(A)$ and

${\rm (i)}$ $F'(u, A)\geq \alpha \int_A W(\mathcal{E}u)\, \mathrm{d}x$

${\rm (ii)}$ $G'(u, A)\geq F'(u, A)\geq \beta \int_{J_u^\xi\cap A}\tau_\xi|\langle \nu_u, \xi\rangle|\, \mathrm{d}\mathcal{H}^{n-1}$

for every $\xi\in \mathbb{S}^{n-1}$. If in addition $G'(u, A) < +\infty$ holds, then one also has

${\rm (iii)}$ $G'(u, A)\geq \alpha \int_A W(\mathcal{E}u)\, \mathrm{d}x+ \int_A \psi(|u|)\, \mathrm{d}x$.

Proof. With (2.12) and a diagonal argument, one may find (not relabeled) subsequences $(u_j)$ and $(\tilde{u}_j)$} converging to $u$ in $L^1(A; \mathbb{R}^n)$ such that

With the first equality and Proposition 4.3 we have that, if $F'(u, A) < +\infty$, then $u\in GSBD^p(A)$. By the second one, the superadditivity of the liminf, Fatou's lemma and (2.12), we have

Hence, (iii) will follow once we have proved (i).

We therefore only have to check (i) and (ii). To this aim, let $\eta, \delta\in(0, 1)$ be fixed. Then, by applying Proposition 4.2 to the sequence $(u_j)$, we can find a sequence $(v_j^{\delta, \eta})\subset {GSBV^p(A; \mathbb{R}^n)}$ such that $v_j^{\delta, \eta}\to u$ in $L^1(A)$ as $\varepsilon_j \to 0$ and

(a) $\alpha (1-\sigma_\eta)^2(1-\delta)^{2n+1}\int_{A} W(\mathcal{E}v_j^{\delta, \eta}(x))\, \mathrm{d}x\leq F_{\varepsilon_j}(u_j, A)$;

(b) $\mathcal{H}^{n-1}(J_{v_j^{\delta, \eta}}\cap A)\leq M_{\delta, \eta} F_{\varepsilon_j}(u_j, A)$.

Now, the equiboundedness of $F_{\varepsilon_j}(u_j, A)$ combined with the bounds (a) and (b) allows to apply the lower semicontinuity part of Theorem 2.4 to the sequence $(v_j^{\delta, \eta})$. Taking into account that ${A^\infty} = \emptyset$ because $u\in L^1(A; \mathbb{R}^n)$, by the convexity of $W$ and (2.10), (ii), we have

We then obtain (i) by letting $\delta\to0$ and $\eta\to0$ above.

For what concerns (ii), from (4.16) of Proposition 4.3 we get

for every $\xi\in \mathbb{S}^{n-1}$, so that (ii) follows by taking the limit as $\delta\to0$ again.

For the proof of the $\Gamma$-liminf inequality, we need the following lemma, which can be found in [22,Lemma 4.5].

Lemma 4.5. Let $S\subset \mathbb{R}^n$ be a bounded, convex and symmetrical set, and let $\phi_\rho$ and $\tau_\xi$ be defined as in $(3.2)$ and $(4.15)$, respectively. Then

We are now in a position to prove the $\Gamma$-liminf inequality.

Proposition 4.6. Let $\rho\in L^\infty(\mathbb{R}^n; [0, +\infty))$ be a convolution kernel satisfying $({\rm{N1}})$-$({\rm{N2}})$. Assume $(3.1)$, $(3.3)$, and $(3.6)$. Consider $F_ \varepsilon$, and $G_ \varepsilon$ given by $(3.4)$, and $(3.7)$, respectively. Let $u\in L^1(\Omega; \mathbb{R}^n)$ and let $A\in\mathcal{A}(\Omega)$, and define $F'(u, A)$ and $G'(u, A)$ by $(2.12)$ in correspondence of $F_\varepsilon$ and $G_\varepsilon$, respectively. If $F'(u, A) < +\infty$, then $u\in GSBD^p(A)$ and

If it additionally holds $G'(u, A) < +\infty$, then

Proof. The proof can be obtained by a standard localization method based on Lemma 2.8. In order to prove, e.g., the second assertion containing an additional term, we can apply Lemma 2.8 to the set function $\mu(A): = G'(u, A)$, which is superadditive on disjoint open sets since $G_\varepsilon(u, \cdot)$ is superadditive as a set function:

Then, we consider the positive measure $\lambda(A): = \mathcal{L}^n(A)+\mathcal{H}^{n-1}(J_u\cap A)$ and the sequence $(\varphi_h)_{h\geq0}$ of $\lambda$-measurable functions on $A$ defined as

where

for $(\xi_h)_{h\geq1}$ a dense sequence in $\mathbb{S}^{n-1}$.

Now, by virtue of Proposition 4.4 it holds that

for every $h = 0, 1, \dots$, so that all the assumptions of Lemma 2.8 are satisfied. The assertion then follows once we notice that, taking into account Lemma 4.5, it holds

for $\lambda$-a.e. $x\in A$.

5.   Estimate from above of the $\Gamma$-limit

We denote by $F"$ and $G"$ the upper $\Gamma$-limits of $(F_\varepsilon)$ and $(G_\varepsilon)$, respectively, as defined in (2.13).

Proposition 5.1. Let $u\in GSBD^p(\Omega)\cap L^1(\Omega; \mathbb{R}^n)$. Then

If, in addition, it holds that $\int_\Omega \psi(|u|)\, \mathrm{d}x < +\infty$, then

Proof. We only prove (5.1) by using the density result of Theorem 2.3, as (5.2) follows by an analogous construction with the additional property (2.9).

In view of Theorem 2.3 and remarks below, since we perform a local costruction and by a diagonal argument it is not restrictive to assume that $u\in \mathcal{W}(\Omega; \mathbb{R}^n)$ and that $J_u$ is a closed simplex contained in any of the coordinate hyperplanes, that we denote by $K$.

For every $h > 0$, let $K_h: = \cup_{x\in K}S(x, h)$ be the anisotropic $h$-neighborhood of $K$. As $K$ is compact and $(n-1)$-rectifiable, it holds (see for instance [21,Theorem 3.7])

(observe that a factor $2$ is already contained in our definition (3.2) of $\phi_\rho$). Let $\gamma_ \varepsilon > 0$ be a sequence such that $\gamma_ \varepsilon/ \varepsilon\to0$ as $\varepsilon\to0$. Notice that, for $\varepsilon$ small,

recalling that $K \subset \Omega$. Let $\phi_ \varepsilon$ be a smooth cut-off function between $K_{\gamma_\varepsilon}$ and $K_{\gamma_\varepsilon+\varepsilon}$, and set

Since $u\in W^{1, \infty}({\Omega}\backslash J_u; \mathbb{R}^n)$ we have $u_ \varepsilon\in W^{1, \infty}({\Omega}; \mathbb{R}^n)$. Note also that, by the Lebesgue Dominated Convergence Theorem, $u_ \varepsilon\to u$ in $L^1(\Omega; \mathbb{R}^n)$. Moreover, since $u_ \varepsilon = u$ on $S(x, \varepsilon)\cap\Omega$ if $x\not\in K_{\gamma_\varepsilon+\varepsilon}$, we have

Setting

we have that $w_ \varepsilon(x)$ converges to $w(x): = W(\mathcal{E}u(x))$ in $L^1_{\rm loc}(\Omega)$ as $\varepsilon\to0$. Since $f$ complies with (3.1) and it is increasing, there exists $\tilde{\alpha} > \alpha$ such that $f(t)\leq \tilde{\alpha} t$ for every $t\geq0$.

This gives

and, taking into account that $\lim_{t\to0^+}\frac{f(t)}{t} = \alpha$, we also infer that

Thus, by Lebesgue's Dominated Convergence Theorem,

As $\tfrac{\gamma_ \varepsilon+ \varepsilon}{ \varepsilon}\to1$ as $\varepsilon\to0$, from (5.3), (5.4), the subadditivity of the limsup and (2.13) we get (5.1).

Proof of Theorems 3.1 and 3.2. The two results follow by combining Propositions 4.1, 4.6, and 5.1

Acknowledgments

The authors are members of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of INdAM. G. Scilla and F. Solombrino have been supported by the Italian Ministry of Education, University and Research through the Project "Variational methods for stationary and evolution problems with singularities and interfaces" (PRIN 2017).

Conflict of interest

The authors declare no conflict of interest.