# Existence of minimizers for the SDRI model in $\mathbb{R}^{n}$: Wetting and
dewetting regimes with mismatch strain

Shokhrukh Yu. Kholmatov  Fakultät für Mathematik
Universität Wien
Oskar-Morgenstern Platz 1
1090 Wien (Austria<EMAIL_ADDRESS>and  Paolo Piovano
Politecnico di Milano, Dipartimento di Matematica, P.zza Leonardo da Vinci 32,
20133 Milano, Italy, & WPI c/o Research Platform MMM “Mathematics-Magnetism-
Materials”, Fak. Mathematik Univ. Wien, A1090 Vienna<EMAIL_ADDRESS>
###### Abstract.

The existence and the regularity results obtained in [37] for the variational
model introduced in [36] to study the optimal shape of crystalline materials
in the setting of stress-driven rearrangement instabilities (SDRI) are
extended from two dimensions to any dimensions $n\geq 2$. The energy is the
sum of the elastic and the surface energy contributions, which cannot be
decoupled, and depend on configurational pairs consisting of a set and a
function that model the region occupied by the crystal and the bulk
displacement field, respectively. By following the physical literature, the
“driving stress” due to the mismatch between the ideal free-standing
equilibrium lattice of the crystal with respect to adjacent materials is
included in the model by considering a discontinuous mismatch strain in the
elastic energy. Since two-dimensional methods and the methods used in the
previous literature where Dirichlet boundary conditions instead of the
mismatch strain and only the wetting regime were considered, cannot be
employed in this setting, we proceed differently, by including in the analysis
the dewetting regime and carefully analyzing the fine properties of energy-
equibounded sequences. This analysis allows to establish both a compactness
property in the family of admissible configurations and the lower-
semicontinuity of the energy with respect to the topology induced by the
$L^{1}$-convergence of sets and a.e. convergence of displacement fields, so
that the direct method can be applied. We also prove that our arguments work
as well in the setting with Dirichlet boundary conditions.

###### Key words and phrases:

Minimal configurations, elastic energy, surface energy, mismatch strain,
Dirichlet Boundary conditions, existence, regularity, compactness, lower-
semicontinuity, density estimates, SDRI, interface instabilities, thin films,
crystal cavities, fractures.

###### 2010 Mathematics Subject Classification:

49J45, 35R35, 74G65

##  1\. Introduction

Elastic effects can strongly affect the structure of crystalline materials by
inducing morphological destabilizations from the optimal free-standing
crystalline equilibrium, that are often referred to as the family of _stress-
driven rearrangement instabilities_ (SDRI) [4, 19, 30, 34, 48]. In order to
relieve the strain, atoms move from their crystalline order possibly inducing
both bulk deformations and interface irregularities. The latter can be
originated in various forms, such as the roughness of the exposed crystalline
boundaries, the formation of internal cracks in the bulk, the nucleation of
dislocations in the crystalline lattice, and the delamination at contact edges
with adjacent materials. However, such corrugations and extra boundary
interfaces are not favorable with respect to the surface energy, which would
instead prescribe regular specific Wulff/Winterbottom-type shapes [44, 45, 50,
51]. Therefore, the surface energy competes against the destabilizing effect
of the elastic energy with a regularizing effect: a delicate microscopical
compromise between such opposite mechanisms must then be reached strongly
affecting in a variety of ways the original crystalline-material macroscopical
properties.

In the strive of capturing such interplay between elastic and (anisotropic)
surface energy described by the physical literature [25, 35, 39, 46, 47, 49,
52], various mathematical models with a variational nature have been
introduced in relation to the different settings relevant for the
applications. A non-exhaustive list includes [6, 9, 20, 21, 27, 33, 38] for
epitaxially-strained thin films deposited on supporting materials, [10, 11,
29] for fractures, [5, 40] for delamination, and, e.g., [28] for crystalline
cavities. Establishing the existence of minimizers for such models even in
dimension $n=2$ is a challenging task especially due to compactness issues.
Such issues were first solved in simplified settings, by working under the
antiplane-shear assumption [8, 16], or by distinguishing the applications with
_adhoc_ geometric assumptions on the morphology of the crystalline materials,
such as adopting graph-type and star-shapedness constraints on film profiles
and crystal cavities, respectively. More recently, the development of several
techniques related to GSBD-functions, a specific subclass of functions of
bounded deformation [18], have been sucessfully applied to models related to
the Griffith energy [11, 12, 13, 14, 18, 29]. Following this progress, there
has been a growing effort [15, 17, 36, 37] to develop mathematical frameworks
enabling the simultaneous treatment of the various mechanisms of mass
rearrangement and boundary instabilities, which is of crucial importance, as
often such phenomena concomitantly occur in applications.

The aim of this paper is to extend to dimension $n\geq 2$, and hence including
the physical relevant case of $n=3$, the existence and the regularity results
obtained in [37] for $n=2$ for the SDRI model introduced in [36]. In regard of
the existence, such an extension was previously achieved in [17] for the
_wetting regime_ , i.e., the case for which it is more convenient for the
crystal material to always cover the surface of a (supporting) adjacent
material rather than letting it exposed, and the setting in which the stress
driving effect characterizing SDRI is mathematically prescribed by introducing
_boundary Dirichlet conditions_. Here we address also the dewetting regime
and, as previously done by the authors in [36, 37] for $n=2$, by following the
physical literature [4, 19, 30, 34, 47, 48, 52] we avoid the use of any
Dirichlet boundary conditions and we directly introduce a _mismatch strain_ in
the elastic energy. As suggested by its name, such strain is induced in the
_free crystal_ , i.e., the crystal of which we are studying the morphology, by
the mismatch between its ideal free-standing equilibrium lattice and the
lattice of adjacent materials. Since the approach used in [17] cannot be
applied to this setting without boundary conditions as it is described below
(see also [37]), we have developed an alternative strategy that allows us to
tackle both the case with mismatch strain and the one with Dirichlet
conditions (see Remark 2.10 for more details). Finally, the method of this
paper extends (also to both the settings with and without Dirichlet
conditions) the regularity results for the bulk displacements and the
morphologies of the energy minimizing configurations obtained by the authors
in [37] for $n=2$ (besides extending the existence results of [37] to the
presence of different adjacent materials and to Griffith-type models with
mismatch strain and delamination).

To facilitate this generalization, we adopt the terminology introduced in [36,
37], by referring to the bounded region $\Omega$ in the space $\mathbb{R}^{n}$
where the free crystal is located as the _container_ in analogy to capillarity
problems, and to the region $S$ occupied by adjacent materials outside the
container, i.e., $S\subset\mathbb{R}^{n}\setminus\Omega$, as the _substrate_
in analogy to the thin-film setting where $S$ is the supporting material on
which the film is being deposited. We notice that the _contact region_ between
the container and the substrate $\Sigma:=\partial\Omega\cap\partial S$ is
assumed to be a Lipschitz $(n-1)$-manifold and that $S$ can be given by a
finite number of different connected components possibly modeling different
adjacent materials. The free crystals are then represented by configurational
pairs of set-function type $(A,u)$, where $A\subset\Omega$ is a set of finite
perimeter denoting the region occupied by the free crystal and subject to the
volume constraint $|A|=\mathtt{v}$ with $\mathtt{v}\in(0,|\Omega|]$, and $u$
is a vector valued faction in $GSBD^{2}(\rm{Int}(A\cup\Sigma\cup
S))\color[rgb]{0,0,0}\cap H_{\mathrm{loc}}^{1}(S)\color[rgb]{0,0,0}$ denoting
the displacement field of the free-crystal and substrate bulk materials with
respect to their optimal equilibrium arrangements. The family of all such
admissible configurational pairs $(A,u)$ is denoted by $\mathcal{C}$.

The configurational energy of any free-crystal pair $(A,u)\in\mathcal{C}$ is
defined by

$\mathcal{F}(A,u)=\mathcal{W}(A,u)+\mathcal{S}(A,u),$ (1.1)

where $\mathcal{S}$ and $\mathcal{W}$ represent the elastic and the surface
energy, respectively. The elastic energy $\mathcal{W}$ in (1.1) is defined as
in [27] by

$\mathcal{W}(A,u)=\int_{A\cup
S}\mathbb{C}(x)[\mathcal{E}u-\mathbf{M}_{0}]:[\mathcal{E}u-\mathbf{M}_{0}]\,\mathrm{d}x,$

where $\mathbb{C}$ is a bounded measurable tensor-valued map $\mathbb{C}$ in
$\Omega\cup S$ satisfying the coercivity assumption $\mathbb{C}\geq
c\,\mathbb{I}>0$ (in the sense of linear operators), where $\mathbb{I}$ is the
identity tensor, $\mathcal{E}u$ is the approximate symmetric gradient of $u$
(see (2.2)) and $\mathbf{M}_{0}$ is the (discontinuous) _mismatch strain_
defined as

$\mathbf{M}_{0}=\begin{cases}\mathcal{E}u_{0}&\text{in $\Omega,$}\\\
0&\text{in $S$}\end{cases}$ (1.2)

for some fixed $u_{0}\in H^{1}(\mathbb{R}^{n})$. In the special case in which
the equilibrium lattice of the free crystal and of the substrate matches at
$\Sigma$, we take $u_{0}\equiv 0$. The surface energy $\mathcal{S}$ in (1.1)
is defined as

$\mathcal{S}(A,u):=\int_{\partial^{*}A\cup
J_{u}}\psi(x,\nu(x))\mathrm{d}\mathcal{H}^{n-1},$

where $\partial^{*}A$ is the reduced boundary of $A$, $J_{u}$ is the jump set
of $u$, and the surface energy density $\psi(\cdot,\nu(\cdot))$ is given by

$\psi(x,\nu(x)):=\begin{cases}\varphi(x,\nu_{A}(x))&\text{if
$x\in\Omega\cap\partial^{*}A,$}\\\ 2\varphi(x,\nu_{J_{u}}(x))&\text{if $x\in
A^{(1)}\cap J_{u},$}\\\ \beta(x)&\text{if
$x\in[\Sigma\cap\partial^{*}A]\setminus J_{u},$}\\\
\varphi(x,\nu_{\Sigma}(x))&\text{if $x\in\Sigma\cap\partial^{*}A\cap
J_{u}$,}\end{cases}$ (1.3)

where $\nu_{U}(x)$ denotes the outward-pointing normal vector to $U$ at
$x\in\partial^{*}U$ for any set of finte perimeter $U\subset\mathbb{R}^{n}$,
$\nu_{\Sigma}:=\nu_{S}$, $\nu_{J_{u}}$ is the normal on $J_{u}$, $A^{(1)}$ is
the set of points of density $1$ for $A$, $\varphi\in
C(\overline{\Omega}\times\mathbb{R}^{n})$ is a a Finsler norm denoting the
_anisotropic surface tension_ of the free-crystal material, and $\beta\in
L^{\infty}(\Sigma)$ represents the _relative adhesion coefficient_ of $\Sigma$
for which we assumed, as in capillarity theory (see, e.g., [24]), that

$|\beta(x)|\leq\varphi(x,\nu_{\Sigma})\quad\text{for \color[rgb]{0,0,0}a.e.\
\color[rgb]{0,0,0}$x\in\Sigma$.}$ (1.4)

We notice that the weights in (1.3), which forbid to decouple the surface
energy from the elastic energy making the energy $\mathcal{F}$ highly
nonlocal, are consistent with the ones chosen in [17, 27, 28, 36, 37], where
they were crucial to prove energy lower-semicontinuity-type properties. In
particular, the anisotropy on _internal cracks_ $A^{(1)}\cap J_{u}$ is
weighted twice as much as the _free boundary_ $\Omega\cap\partial^{*}A$ of the
exposed boundary of the free crystal, because cracks can be approximated by
“closing voids” as in [17, 27, 36]. The presence of the surface energy over
$\Sigma\cap\partial^{*}A\cap J_{u}$ allows to consider a more general
framework for thin films depositing on a substrate, in which cracks are
allowed to appear not only inside the film material, but also along the
surface of the substrate characterizing the _delamination region_ , where
debonding between the atoms of the two materials occurs, and as such, the
corresponding surface tension in (1.3) is regarded as the same of the one on
the free-crystal exposed boundary. Finally, on the complementary region to the
delamination in $\Sigma\cap\partial^{*}A$ where the bulk displacement is
continuous, the relative adhesion coefficient $\beta$ is considered.

We observe that in the case of total wetting case, i.e., if
$\beta(x)=-\varphi(x,\nu_{\Sigma}(x))$ for a.e. $x\in\Sigma,$ we reduce to the
setting of material voids considered in [17] (with the mismatch strain
$\mathbf{M}_{0}$ replaced by a Dirichlet boundary condition). On the contrary,
in the total dewetting case, i.e., if $\beta(x)=\varphi(x,\nu_{\Sigma}(x))$
for a.e. $x\in\Sigma,$ then one can readily check that the energy
$\mathcal{F}$ is minimized by configurational pairs with displacement $u\equiv
u_{0}$ in $\Omega$ and null otherwise, and so characterized by having a zero
elastic energy: the model reduces to the dewetted capillarity setting, or in
other words, to the anisotropic isoperimetric problem in a container. Finally,
in the case with $\mathtt{v}=|\Omega|$, we reduces to the Griffith model with
the inclusion of possible delamination at the substrate boundary, which
generalize also for $n=2$ the setting considered by the authors in [36, 37]
together with $S\neq\emptyset$.

We now present the two main results of the paper (see Section 2.2 for more
detailed statements) and comment their proofs. We begin by observing that,
since the values of the admissible displacement fields $u$ in the void regions
$\Omega\setminus A$ do not play any role in the energy of $(A,u)$, as only a
formal difference with respect to the previous presentation of the SDRI models
introduced in [36, 37], for every $(A,u)\in\mathcal{C}$ we can redefine $u$ in
$\Omega\setminus A$ with a properly chosen constant such that
$\Omega\cap\partial^{*}A\subset J_{u}$ (see Remark 2.1), and so without
changing the value of $\mathcal{F}(A,u)$. We make use of this observation in
the following.

######  Theorem 1.1 (Existence of minimizing configurations).

The minimum problem

$\min\limits_{(A,u)\in\mathcal{C},\,|A|=\mathtt{v}}\,\,\mathcal{F}(A,u)$ (1.5)

admits a solution.

We refer the Reader to Theorem 2.4 for a more detailed and comprehensive
statement of the existence result of Theorem 1.1.

Theorem 1.1 is established by means of the _direct method of the calculus of
variations_ with respect to a properly chosen topology $\tau_{\mathcal{C}}$
with which we equip $\mathcal{C}$, and that is characterized by the
convergence:

$(A_{k},u_{k})\overset{\tau_{\mathcal{C}}}{\longrightarrow}(A,u)\qquad\Longleftrightarrow\qquad\begin{cases}A_{k}\to
A\quad\text{in $L^{1}(\mathbb{R}^{n}),$}\\\ u_{k}\to
u\quad\text{\color[rgb]{0,0,0}a.e.\ \color[rgb]{0,0,0}in $\Omega\cup
S.$}\end{cases}$

In order to establish the $\tau_{\mathcal{C}}$-lower semicontinuity of
$\mathcal{F}$ in Theorem 2.5 we consider the positive Radon measures $\mu_{k}$
and $\mu$ in $\mathbb{R}^{n}$ associated to the localized energy versions of
$\mathcal{F}(A_{k},u_{k})$ and $\mathcal{F}(A,u)$, respectively, for which it
holds that

$\liminf\limits_{k\to+\infty}\mathcal{F}(A_{k},u_{k})\geq\mathcal{F}(A,u)\qquad\Longleftrightarrow\qquad\liminf\limits_{k\to+\infty}\mu_{k}(\mathbb{R}^{n})\geq\mu(\mathbb{R}^{n}).$
(1.6)

Then, we observe that, up to a subsequence, $\mu_{k}$ weakly* converges to
some positive Radon measure $\mu_{0}$, and that $\mu$ is absolutely continuous
with respect to $\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits(\partial^{*}A\cup
J_{u}\cup\Sigma)+\mathcal{L}^{n}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits(\Omega\cup S),$ and we
establish the following estimates for the Radon-Nikodym derivatives:

$\displaystyle\frac{\mathrm{d}\mu_{0}}{\mathrm{d}\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits(\partial^{*}A\cup
J_{u}\cup\Sigma)}\geq\frac{\mathrm{d}\mu}{\mathrm{d}\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits(\partial^{*}A\cup
J_{u}\cup\Sigma)}\quad\text{$\mathcal{H}^{n-1}$ -a.e. on $\partial^{*}A\cup
J_{u}\cup\Sigma,$}$ (1.7)
$\displaystyle\frac{\mathrm{d}\mu_{0}}{\mathrm{d}\mathcal{L}^{n}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits(\Omega\cup
S)}\geq\frac{\mathrm{d}\mu}{\mathrm{d}\mathcal{L}^{n}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits(\Omega\cup
S)}\quad\text{$\mathcal{L}^{n}$-a.e. in $\Omega\cup S,$}$ (1.8)

which imply that
$\lim\mu_{k}(\mathbb{R}^{n})=\mu_{0}(\mathbb{R}^{n})\geq\mu(\mathbb{R}^{n})$
and, in view of (1.6), conclude the proof of the lower-semicontinuity. For the
estimate (1.7) we need to distinguish between the estimate at the reduced
boundary of $A$ and at $\Sigma\setminus J_{u}$, where we can implement
techniques developed in capillarity theory [1, 24], from the estimate at the
(approximate) jump points of $u$, where we employ arguments based on the
slicing properties of GSBD-functions as in the Griffith model [13, 14, 15],
for which though extra care is needed: unless $\mathtt{v}=|\Omega|,$ we cannot
directly apply those arguments because at jump points we need to obtain
different weights with respect to the ones at the reduced boundary of $A.$
Rather, we replace $J_{u_{k}}$ in small “holes” up to some error by means of
Corollaries 3.3 and 3.5 in such a way that each slice intersects the boundary
of those holes at least in two points (see the proof of Proposition 4.1),
which in turns yields the desired estimate with weight $2$ at such jump points
(see Corollary 4.2). Finally, we prove (1.8) by using the convexity of
$\mathcal{W}(A,\cdot)$ and by observing that the condition $u_{k}\to u$ a.e.
in $\Omega\cup S,$ together with the compactness result [14, Theorem 1.1],
allows us to conclude that $\mathcal{E}u_{k}\rightharpoonup\mathcal{E}u$ in
$L^{2}(\Omega\cup S)$. We recall that in [17] the authors prove the lower
semicontinuity of an energy for crystalline voids via relaxation arguments.
Namely, the authors start in the regular family of pair configurations given
by voids with a Lipschitz boundary and Sobolev displacement fields, and then
in the relaxation, the jump set appears as the void boundaries collapse,
resulting in a coefficient $2$ in front of the jump energy of $\mathcal{S}$.
We are here actually arguing in the reverse direction: first we start in
$\mathcal{C}$ with admissible pairs allowing displacements with jump sets, and
then we carefully create an at most countable family of voids around them.

The $\tau_{\mathcal{C}}$-compactness of an energy-equibounded sequence
$\\{(A_{k},u_{k})\\}\subset\mathcal{C}$ is established in Theorem 2.6. We
easily get the uniform bounds on the perimeters of $A_{k}$, the
$\mathcal{H}^{n-1}$\- measure of the jumps $J_{u_{k}}$, and the $L^{2}$-norm
of $\mathcal{E}u_{k}$ by the assumptions on the anisotropic surface tensions
and the elasticity tensor (see Remark 2.3). Thus, we can directly deduce the
convergence in $L^{1}(\mathbb{R}^{n})$ up to a non-relabelled subsequence of
$A_{k}$ to some set $A\subset\Omega$ of finite perimeter. However,
establishing the $\mathcal{L}^{n}$ a.e. convergence of the displacements
$u_{k}$ is delicate: by [14, Theorem 1.1] there could be an exceptional set
$E$ with $\mathcal{L}^{n}$ positive measure, in which $|u_{k}|\to+\infty$. The
presence of such an exceptional set has been previously treated by prescribing
Dirichlet boundary conditions [13, 14, 17]. For instance, in [17] the
compactness issue is solved by considering in the proof an auxiliary more
general class $GSBD_{\infty}^{p}$, $p>1$, of displacements (which are allowed
to attain the infinite value on a subset of their domain of also
$\mathcal{L}^{n}$ positive measure) and then, by using the Dirichlet condition
imposed on the displacements at the boundary, the authors are able to prove
that the minimizing displacements belong to the original space $GSBD^{p}.$
However, as in the setting with the mismatch strain (1.2), we cannot rely on
any fixed boundary condition, one cannot even exclude the situation with
$E=\Omega\cup S$ and hence, this issue unfortunately forbids the
implementation of the strategy of [17] to our SDRI setting. The other option
of excluding the presence of the exceptional set is based on the employment of
Poincaré-Korn inequality for GSBD-functions citeCCF:2016 with small jump: the
set $\Omega$ is partitioned into a Caccioppoli family $\\{P_{j}\\}$ of sets
$P_{j}$ in which a sequence $\\{a_{k}^{j}\\}$ of rigid displacements are
defined in such a way that $u_{k}-a_{k}^{j}$ is convergent pointwise a.e. in
$P_{j}$, so that one can conclude that the sequence

$v_{k}:=u_{k}-\sum\limits_{j}a_{k}^{j}\chi_{P_{j}}$ (1.9)

converges to some $u\in GSBD^{p}(\Omega)$ a.e. in $\Omega$,
$\mathcal{E}v_{k}\rightharpoonup\mathcal{E}u$ in $L^{p}(\Omega)$, and

$\lim\limits_{k\to+\infty}\mathcal{H}^{n-1}(J_{u_{k}})\geq\mathcal{H}^{n-1}\Big{(}J_{u}\cup(\Omega\cap\bigcup_{j}\partial^{*}P_{j})\Big{)}$

(see [15, Theorem 1.1]). However, also this approach seems not implementable
in our SDRI setting, since the functions $v_{k}$ defined in (1.9) may admit
extra jumps along the boundary of the partition phases $P_{j}$ that should be
counted with different weights in our setting with different surface tensions.

In view of these issues, in order to prove compactness we use a different
strategy in this paper by directly partitioning the sets $A$ and $A_{k}$ (not
only $A$!) into Caccioppoli families (that need to be created by starting from
the connected components of the substrate) up to a controllable error (see
Figures 3 and 5). Such strategy is a reminiscence of the ideas already used by
the authors in [36, Theorem 2.7], of partitioning $A_{k}$ by means of
introducing extra circles closing the shrinking “necks”, which though works
only for $n=2$ and under the constraint assumed in [36] on the number of
boundary components for the admissible free-crystal regions. More precisely,
we proceed here arguing as follows: First, by the classical Poincaré-Korn
inequality we partition $S$ in a family $\\{S^{i}\\}_{i\geq 1}$ of sets
$S^{i}$ such that for each $i\geq 1$ the set $S^{i}$ is a union of connected
components of $S$ and there exists a sequence of rigid displacements
$\\{a_{k}^{i}\\}$ such that, up to a subsequence, $u_{k}-a_{k}^{i}$ converges
a.e. in $S^{i}$ and $|a_{k}^{i}-a_{k}^{j}|\to+\infty$ a.e. in $\mathbb{R}^{n}$
for every $j\neq i$. Second, by applying [14, Theorem 1.1] with
$u_{k}-a_{k}^{i}$ we construct a family $\\{F^{i}\\}_{i\geq 0}$ of pairwise
disjoint Caccioppoli subsets of $A,$ such that for $i\geq 1$ the sequence
$u_{k}-a_{k}^{i}$ converges a.e. in $F^{i}\cup S^{i}$ and diverges to infinity
otherwise, and $F^{0}:=A\setminus\bigcup_{i\geq 1}F^{i}$. Furthermore, since
$F^{0}$ is the portion of the free crystal, so-called in the following
“hanging phase” (see Figure 1), that does not “interact” with any substrate
component, we can redefine the displacements in $F^{0}$ as $u_{0}$ (see
(1.2)), which corresponds to providing a zero contribution to the overall
elastic energy. Third, by using the $\mathcal{H}^{n-1}$-rectifiability of
$\partial^{*}F^{i}$ and Propositions 4.1 and 5.2, we construct for any
$\delta>0$ a union $G_{k}^{\delta}\subset\Omega$ of open sets covering
$\bigcup\partial^{*}F^{i}$ up to some error of order $O(\sqrt{\delta})$ and
whose perimeter and volume are controlled, and we set

$B_{k}^{\delta}:=A_{k}\setminus G_{k}^{\delta}\quad\text{and}\quad
v_{k}^{\delta}:=u_{0}\chi_{F^{0}}+\sum\limits_{i\geq
1}(u_{k}-a_{k}^{i})\chi_{S^{i}\cup(F^{i}\setminus
G_{k}^{\delta})}+u_{0}\chi_{F^{0}}.$ (1.10)

We notice that actually the definition of the $v_{k}^{\delta}$ in (1.10) is
more involved (see (5.3)), as we need also to control the possible large jumps
created along $\Sigma$, that though in the limit disappear (becoming wetting
layer), by creating artificial small jumps in $A_{k}^{\delta}\setminus A$ and
redefining $v_{k}^{\delta}$ in that set near $\Sigma$. The obtained
configurations satisfy

$\mathcal{F}(A_{k},u_{k})\geq\mathcal{F}(B_{k}^{\delta},v_{k}^{\delta})-c\sqrt{\delta}\Big{(}\mathcal{H}^{n-1}(\partial^{*}A_{k})+\mathcal{H}^{n-1}(J_{u_{k}})+\sum\limits_{h=0}^{m}P(F^{h})\Big{)}$
(1.11)

for some constant $c>0$ (see Proposition 5.1), from which Theorem 2.6 follows
by a diagonal argument.

Figure 1. The partitions of the substrate and the free crystal into,
respectively, the families $\\{S^{i}\\}_{i\geq 1}$ and $\\{F^{i}\\}_{i\geq 0}$
of Caccioppoli sets, which are used to prove the
$\tau_{\mathcal{C}}$-compactness result, are depicted by representing the
various phases of the free crystal that are interacting with the substrate
with different line patterns and the remaining “hanging phase” $F^{0}$ with a
point pattern.

We also notice that in the case with Dirichlet boundary conditions, one see at
most 2 elements in the partition, the hanging phase $F^{0}$ and a phase
$F^{1}$ interacting with the substrate, since in this case we do not need to
add any rigid displacements. Apart from this simplification, the methods used
in the proof of Theorems 2.5 and 2.6 still work, even by relaxing the
assumptions on the convex elastic energy densities, i.e., by allowing for a
$p$-growth with respect to the strains (see Section 2.3). This allows us in
particular to recover in Remark 2.10 the existence results for the model
representing material voids in the framework with Dirichlet boundary
conditions of [17] and the existence and regularity results for the Griffith
fracture model with Dirichlet boundary conditions of [13].

The second main result of the paper relates to properties of partial
regularity satisfied by the minimizers $(A,u)$ of $\mathcal{F}$, such as the
essential closedness of $J_{u}$ and $\partial^{*}A.$

######  Theorem 1.2 (Regularity results for minimizing configurations).

Let $(\widetilde{A},\widetilde{u})$ be a solution of (1.5). Then the pair
$(A,u)$ defined by

$A\color[rgb]{0,0,0}:=\color[rgb]{0,0,0}\mathrm{Int}(A^{(1)})\qquad\text{and}\qquad
u\color[rgb]{0,0,0}:=\color[rgb]{0,0,0}\widetilde{u}\chi_{A\cup
S}+\xi\chi_{\Omega\setminus A},$

where $\xi\in\mathbb{R}^{n}$ is chosen such that
$\Omega\cap\partial^{*}A\subset J_{u}$ (see Remark 2.1), is also a solution of
(1.5). Furthermore, we have that

$\mathcal{H}^{n-1}(\widetilde{A}^{(1)}\setminus
A)<+\infty,\quad\mathcal{H}^{n-1}(J_{u}\setminus
J_{u}^{*})=0,\qquad\text{and}\qquad\mathcal{H}^{n-1}(\overline{J_{u}^{*}}\setminus
J_{u}^{*})=0,$

where

$J_{u}^{*}:=\\{x\in J_{u}:\,\,\theta(J_{u},x)=1\\}$

with $\theta(J_{u},x)$ denoting the $(n-1)$-dimensional density of $J_{u}$ at
$x.$ Finally, there exists a constant $c>0$ such that if $E\subset A$ is a
“hanging” component of $A,$ i.e., if
$\mathcal{H}^{n-1}([\partial^{*}E\cap\Sigma]\setminus J_{u})=0,$ then $|E|\geq
c.$

We refer the Reader to Theorem 2.7 for a more detailed statement of Theorem
1.2.

The proof of Theorem 1.2 is carried out by implementing in the SDRI setting
the methods for the partial regularity of the minimizers of the Griffith model
by means of the ideas already employed by the authors in [37] for $n=2$: we
introduce a localized version of $\mathcal{F}$ and establish uniform lower and
upper $\mathcal{H}^{n-1}$ density estimates for the jump sets (see Section 6).
by paying extra care to treat the presence of voids and of the different
weights for the surface tension in the surface energy, which is a crucial
difference from the Griffith model. We overcome such difficulties by means of
the strategy employed in [43] and based on the _relative isoperimetric
inequality_ [3] to distinguish in the _Decay Lemma_ the blows up “inside the
free crystal” from the ones “in the voids”, and by applying the approximation
result of [12, Theorem 3].

The paper is organized as follows: In Section 2 we introduce the SDRI model,
some preliminary results related to sets of finite perimeter and GSBD-
functions, and state the main results. In Section 3 we provide some technical
results which allows to replace a part of jump set with an open set without
modifying too much the corresponding SDRI energy. Section 4 is devoted to the
proof of the lower semicontinuity of $\mathcal{F}.$ Section 5 contains the
proof of the compactness for energy-equibounded sequences. In Section 6 we
prove the decay estimates for $\mathcal{F}$ and the regularity results of
Theorem 2.7. Finally, we conclude the paper with the Appendix containing the
results related to the equivalence of the volume-constrained minimum problem
with the volume-uncontrained penalized minimum problem, and to some properties
of GSBD-functions.

##  2\. Mathematical setting and formulation of the main results

### Notation

Unless otherwise stated, all sets we consider are subsets of $\mathbb{R}^{n},$
in which the coordinates $(x_{1},\ldots,x_{n})$ of $x\in\mathbb{R}^{n}$ are
given with respect to the standard basis $\\{{\bf e}_{1},\ldots,{\bf
e}_{n}\\}.$ The symbol $B_{r}(x)$ stands for the open ball in $\mathbb{R}^{n}$
centered at $x$ and of radius $r>0.$ The symbol
$Q_{r}(x):=x+[-\frac{r}{2},\frac{r}{2}]^{n}$ stands for the standard
$n$-dimensional (hyper) cube in $\mathbb{R}^{n}$ of sidelength $r$ centered at
$x.$ We write $Q_{r}:=[-\frac{r}{2},\frac{r}{2}]^{n}.$ Given $r>0,$
$\nu\in\mathbb{S}^{n-1}$ and $x\in\mathbb{R}^{n}$ we denote by $Q_{r,\nu}(x)$
the cube of sidelength $r$ centered at $x$ whose sides are either parallel or
perpendicular to $\nu$. The characteristic function of a Lebesgue measurable
set $F$ is denoted by $\chi_{F}$ and its Lebesgue measure by $|F|;$ we set
also $\omega_{n}:=|B_{1}(0)|.$ We denote by $E^{c}$ the complement of $E$ in
$\mathbb{R}^{n}.$ By $\mathcal{H}^{n-1}$ we denote by $(n-1)$-dimensional
Hausdorff measure in $\mathbb{R}^{n}$ and we write $K=_{\mathcal{H}^{n-1}}L$
and $K\subset_{\mathcal{H}^{n-1}}L$ to mean $\mathcal{H}^{n-1}(K\Delta L)=0$
and $\mathcal{H}^{n-1}(K\setminus L)=0.$

Given an open set $U\subset\mathbb{R}^{n},$ the set of $L^{1}(U)$-functions
having bounded total variation in $U$ is denoted by $BV(U)$ and the elements
of

$BV(U;\\{0,1\\}):=\\{E\subseteq U:\,\,\chi_{E}\in BV(U)\\}$

are called sets of finite perimeter in $U$. The standard references for
$BV$-functions and sets of finite perimeter are for instance [3, 32, 41].

Given $E\in BV(U,\\{0,1\\})$ we denote

  * –

by $P(E,U):=\int_{U}|D\chi_{E}|$ the perimeter of $E$ in $U;$

  * –

by $\partial E$ the measure-theoretic boundary of $E,$ i.e.,

$\partial E:=\\{x\in\mathbb{R}^{n}:\,\,0<|B_{\rho}\cap
E|<|B_{\rho}|\quad\forall\rho>0\\};$

  * –

by $\partial^{*}E$ the reduced boundary of $E,$ i.e.,

$\partial^{*}E:=\Big{\\{}x\in\mathbb{R}^{n}:\,\,\exists\nu_{E}(x):=-\lim\limits_{r\to
0}\frac{D\chi_{E}(B_{r}(x))}{|D\chi_{E}|(B_{r}(x))}\quad\text{and}\quad|\nu_{E}(x)|=1\Big{\\}}.$

  * –

by $\nu_{E}$ the outer measure-theoretic unit normal to $\partial^{*}E.$

Given a Lebesgue measurable set $E\subseteq\mathbb{R}^{n}$ and
$\alpha\in[0,1]$ we define

$E^{(\alpha)}:=\left\\{x\in\mathbb{R}^{n}:\,\,\lim\limits_{\rho\to
0^{+}}\frac{|B_{\rho}(x)\cap E|}{|B_{\rho}(x)|}=\alpha\right\\}.$

Given a set $K\subset\mathbb{R}^{n}$ and a point $x_{0}\in\mathbb{R}^{n},$ we
denote by

$\theta_{*}(K,x_{0}):=\liminf\limits_{r\to
0}\frac{\mathcal{H}^{n-1}(B_{r}(x_{0})\cap K)}{\omega_{n-1}r^{n-1}}$

and

$\theta^{*}(K,x_{0}):=\limsup\limits_{r\to
0}\frac{\mathcal{H}^{n-1}(B_{r}(x_{0})\cap K)}{\omega_{n-1}r^{n-1}}$

the $(n-1)$-dimensional lower and upper density of $K$ at $x_{0}$,
respectively (see e.g., [3, page 78]). When these densities coincide, we
denote their common value by $\theta(K,x_{0})$. Recall that by [3, Theorem
2.63], $K$ is $\mathcal{H}^{n-1}$-rectifiable if and only if $\theta(K,x)=1$
for $\mathcal{H}^{n-1}$-a.e. $x\in K.$

Given $x\in\mathbb{R}^{n}$ and $r>0,$ the blow-up map $\sigma_{x,r}$ is
defined as

$\sigma_{x,r}(y)=\frac{y-x}{r}.$ (2.1)

Given an open set $U\subset\mathbb{R}^{n}$ and a metric space $X$ we denote by
$\mathrm{Lip}(U;X)$ the family of all Lipschitz functions $\psi:U\to X.$ We
denote by $\mathrm{Lip}(\psi)$ the Lipschitz constant of
$\psi\in\mathrm{Lip}(U;X).$

By $GSBD(U;\mathbb{R}^{n})$ we denote the collection of all _generalized
special functions of bounded deformation_ (see [14, 18] for their definition
and properties). Given $u\in GSBD(U;\mathbb{R}^{n})$ we denote by
$\mathcal{E}u\in\mathbb{M}^{n\times n}_{\rm sym}$ the _approximate symmetric
gradient_ and by $J_{u}$ the jump set of $u;$ we recall that by [18, Theorem
9.1]

$\mathop{\mathrm{ap\,\,lim}}\limits_{y\to
x}\frac{[u(y)-u(x)-\mathcal{E}u(x)(y-x)]\cdot(y-x)}{|y-x|^{2}}=0\qquad\text{for
a.e.\ $x\in U$}$ (2.2)

and by [18, Theorem 6.2] $J_{u}$ is $\mathcal{H}^{n-1}$-rectifiable. Let us
also define

$GSBD^{2}(U):=\\{u\in GSBD(U;\mathbb{R}^{n}):\,\,\mathcal{E}u\in
L^{2}(U;\mathbb{M}^{n\times n}_{\rm sym})\\}.$

Given a $\mathcal{H}^{n-1}$-rectifiable set $K\subset\overline{U},$ we
consider a normal vector $\nu_{K}$ to its approximate tangent space and we
denote by $u_{K}^{+}$ and $u_{K}^{-}$ the approximate limits of $u\in
GSBD(U;\mathbb{R}^{n})$ with respect to $\nu_{K},$ i.e.,

$u_{K}^{+}(x):=\mathop{\mathrm{ap\,\,lim}}\limits_{\begin{subarray}{c}(y-x)\cdot\nu_{K}>0,\\\
y\in U\end{subarray}}\,\,u(y)\quad\text{and}\quad
u_{K}^{-}(x):=\mathop{\mathrm{ap\,\,lim}}\limits_{\begin{subarray}{c}(y-x)\cdot\nu_{K}<0\\\
y\in U\end{subarray}}\,\,u(y)$

for every $x\in K$ whenever they exist [18, Definition 2.4]. We refer to
$u_{K}^{+}$ and $u_{K}^{-}$ as the _two-sided traces_ of $u$ at $K$ and we
notice that they are uniquely determined up to a permutation when changing the
sign of $\nu_{K}.$

Let us recall some notation from [14] related to GSBD-functions. For
$\xi\in\mathbb{S}^{n-1},$ $y\in\mathbb{R}^{n},$ $B\subset\mathbb{R}^{n}$ and
$v:B\to\mathbb{R}^{n}$ let

$\Pi_{\xi}:=\\{x\in\mathbb{R}^{n}:\,x\cdot\xi=0\\},\qquad
B_{y}^{\xi}:=\\{t\in\mathbb{R}:\,\,y+t\xi\in B\\},$

and

$v_{y}^{\xi}(t):=v(y+t\xi),\qquad\widehat{v}_{y}^{\xi}(t):=v_{y}^{\xi}(t)\cdot\xi.$

We denote by $\pi_{\xi}$ the projection of $\mathbb{R}^{n}$ onto $\Pi_{\xi},$
i.e.,

$\pi_{\xi}:=x-(x\cdot\xi)\xi.$

Recall that if $v\in GSBD^{2}(U)$ for an open set $U\subset\mathbb{R}^{n},$
then $\widehat{v}_{y}^{\xi}\in SBV^{2}_{\mathrm{loc}}(U_{y}^{\xi})$ for every
$\xi\in\mathbb{S}^{n-1}$ and $\mathcal{H}^{n-1}$-a.e. $y\in\Pi_{\xi}.$ We
denote by $\dot{u}_{y}^{\xi}$ the the absolutely continuous part of
$Du_{y}^{\xi}$ w.r.t. $\mathcal{L}^{1}.$ Let us introduce

$I_{y,\xi}^{U}(v):=\int_{U_{y}^{\xi}}|\dot{v}_{y}^{\xi}|^{2}\mathrm{d}t$

and

$II_{y,\xi}^{U}(v):=|D[\tau(v\cdot\xi)]_{y}^{\xi}|(U_{y}^{\xi}),$

where $\tau\in C^{1}(\mathbb{R},(-\frac{1}{2},\frac{1}{2}))$ and satisfies
$0\leq\tau^{\prime}\leq 1.$ By [14, Eq. 3.8]

$\int_{\Pi_{\xi}}I_{y,\xi}^{U}(v)\mathrm{d}\mathcal{H}^{n-1}(y)=\int_{U}|\mathcal{E}v(x)\xi\cdot\xi|^{2}\mathrm{d}x\leq\int_{U}|\mathcal{E}v|^{2}\mathrm{d}x$
(2.3)

and by [14, Eq. 3.9] and obvious estimate $a\leq 1+a^{2}$

$\displaystyle\int_{\Pi_{\xi}}II_{y,\xi}^{U}(v)\mathrm{d}\mathcal{H}^{n-1}(y)=|D_{\xi}[\tau(v\cdot\xi)]|(U)\leq$
$\displaystyle\int_{U}|\mathcal{E}v|\mathrm{d}x+\mathcal{H}^{n-1}(U\cap
J_{v})$ $\displaystyle\leq$
$\displaystyle|U|+\int_{U}|\mathcal{E}v|^{2}\mathrm{d}x+\mathcal{H}^{n-1}(U\cap
J_{v}).$ (2.4)

By the Fubini Theorem and the equality

$\int_{\mathbb{S}^{n-1}}|\nu\cdot\xi|\,\mathrm{d}\mathcal{H}^{n-1}(\xi)=2\omega_{n-1},\qquad\nu\in\mathbb{S}^{n-1},$

for any $\mathcal{H}^{n-1}$-rectifiable Borel set $L\subset\mathbb{R}^{n}$ and
an open set $U\subset\mathbb{R}^{n}$ we have

$\displaystyle\mathcal{H}^{n-1}(U\cap L)=$
$\displaystyle\frac{1}{2\omega_{n-1}}\,\int_{\mathbb{S}^{n-1}}\mathrm{d}\mathcal{H}^{n-1}(\xi)\int_{U\cap
L}|\nu_{L}\cdot\xi|\,\mathrm{d}\mathcal{H}^{n-1}(y)$ $\displaystyle=$
$\displaystyle\frac{1}{2\omega_{n-1}}\,\int_{\mathbb{S}^{n-1}}\mathrm{d}\mathcal{H}^{n-1}(\xi)\int_{\Pi_{\xi}}\mathcal{H}^{0}(U_{y}^{\xi}\cap
L_{y}^{\xi})\,\mathrm{d}\mathcal{H}^{n-1}(y),$ (2.5)

where we applied the area formula with $\pi_{\xi}$ in the second equality.

A linear function $a:\mathbb{R}^{n}\to\mathbb{R}^{n}$ satisfying $\nabla
a=-(\nabla a)^{T}$ is called an (infinitesimal) _rigid displacement_.

###  2.1. The SDRI model

Given nonempty open sets $\Omega\subset\mathbb{R}^{n}$ and
$S\subset\mathbb{R}^{n}\setminus\Omega,$ we define the space of admissible
configurations by

$\displaystyle\mathcal{C}:=\Big{\\{}(A,u):$ $\displaystyle A\in
BV(\Omega;\\{0,1\\}),\,\,u\in GSBD^{2}({\rm Int}{(\Omega\cup
S\cup\Sigma)})\cap H_{\mathrm{loc}}^{1}(S)\Big{\\}}$

where $\Sigma:=\partial S\cap\partial\Omega.$

The energy of admissible configurations is given by

$\mathcal{F}:\mathcal{C}\to[-\infty,+\infty],\qquad\mathcal{F}:=\mathcal{S}+\mathcal{W},$

where $\mathcal{S}$ and $\mathcal{W}$ are the surface and elastic energies of
the configuration, respectively. The surface energy of $(A,u)\in\mathcal{C}$
is defined as

$\displaystyle\mathcal{S}(A,u):=$
$\displaystyle\int_{\Omega\cap\partial^{*}A}\varphi(x,\nu_{A}(x))\,\mathrm{d}\mathcal{H}^{n-1}(x)$
$\displaystyle+\int_{A^{(1)}\cap
J_{u}}\big{[}\varphi(x,\nu_{J_{u}}(x))+\varphi(x,-\nu_{J_{u}}(x))\big{]}\,\mathrm{d}\mathcal{H}^{n-1}(x)$
$\displaystyle+\int_{\Sigma\cap\partial^{*}A\setminus
J_{u}}\beta(x)\,\mathrm{d}\mathcal{H}^{n-1}(x)+\int_{\Sigma\cap\partial^{*}A\cap
J_{u}}\varphi(x,-\nu_{\Sigma}(x))\,\mathrm{d}\mathcal{H}^{n-1}(x),$

where $\varphi:\overline{\Omega}\times\mathbb{S}^{n-1}\to(0,+\infty)$ and
$\beta:\Sigma\to\mathbb{R}$ are Borel functions denoting the anisotropy of
crystal and the relative adhesion coefficient of the substrate boundary,
respectively, and $\nu_{\Sigma}:=\nu_{S}$. In applications instead of
$\varphi(x,\cdot)$ it is more convenient to use its positively one-homogeneous
extension $|\xi|\varphi(x,\xi/|\xi|).$ With an abuse of notation we denote
this extension also by $\varphi.$

The elastic energy of $(A,u)\in\mathcal{C}$ is defined as

${\mathcal{W}}(A,u):=\int_{A\cup
S}W(x,\mathcal{E}u-\mathbf{M}_{0})\mathrm{d}x,$

where the elastic energy density $W$ is a quadratic form

$W(x,\mathbf{M}):=\mathbb{C}(x)\mathbf{M}:\mathbf{M},$

determined by a tensor-valued measurable map $x\in\Omega\cup
S\to\mathbb{C}(x),$ the so-called stress-tensor, in the Hilbert space
$\mathbb{M}^{n\times n}_{\rm sym}$ of all $n\times n$-symmetric matrices with
the natural inner product

$\mathbf{M}:\mathbf{N}=\sum\limits_{i,j=1}^{n}M_{ij}N_{ij}.$

The mismatch strain $x\in\Omega\cup
S\mapsto\mathbf{M}_{0}(x)\in\mathbb{M}^{n\times n}_{\rm sym}$ is given by

$\mathbf{M}_{0}:=\begin{cases}\mathcal{E}u_{0}&\text{in $\Omega,$}\\\
0&\text{in $S,$}\end{cases}$

for a fixed $u_{0}\in H^{1}(\mathbb{R}^{n})$.

######  Remark 2.1 (Values of displacements outside a set).

  * (i)

The functional $\mathcal{F}(A,u)$ does not “see” the values of $u$ in
$\Omega\setminus A,$ i.e.,

$\mathcal{F}(A,u)=\mathcal{F}(A,u\chi_{A\cup S}+v\chi_{\Omega\setminus
A})\quad\text{for any $v\in GSBD^{2}(\Omega).$}$

Thus, we can redefine $u$ in $\Omega\setminus A$ arbitrarily without changing
the energy of the configuration $(A,u)$.

  * (ii)

For any $(A,u)\in\mathcal{C}$ there exists an at most countable set
$\Xi^{(A,u)}\subset\mathbb{R}^{n}$ such that for any
$\xi\in\mathbb{R}^{n}\setminus\Xi^{(A,u)}$ the function

$u^{\xi}:=u\chi_{A\cup S}+\xi\chi_{\Omega\setminus A}$ (2.6)

satisfies

$J_{u^{\xi}}=_{\mathcal{H}^{n-1}}(\Omega\cap\partial^{*}A)\cup(\Sigma\cap
J_{u})\cup(A^{(1)}\cap J_{u})\cup(\Sigma\setminus\partial^{*}A).$ (2.7)

Indeed, for $\xi\in\mathbb{R}^{n}$ let
$E_{\xi}^{(A,u)}:=\\{x\in\partial^{*}A\cup\Sigma:\,\,\mathrm{tr}_{A\cup
S}u(x)=\xi\\}\subset\Sigma\cup\partial^{*}A$ and let

$\Xi^{(A,u)}:=\\{\xi\in\mathbb{R}^{n}:\,\,\mathcal{H}^{n-1}(E_{\xi}^{(A,u)})>0\\}.$

Since $\mathcal{H}^{n-1}(\partial^{*}A\cup\Sigma)<+\infty$ and
$E_{\xi}^{(A,u)}\cap E_{\eta}^{(A,u)}=\emptyset$ for $\xi\neq\eta,$ by slicing
arguments (see e.g. [37, Proposition A.2]) the set $\Xi^{(A,u)}$ is at most
countable. By the definition of jump, for any
$\xi\in\mathbb{R}^{n}\setminus\Xi^{(A,u)}$ the function $u^{\xi}$ satisfies
(2.7).

  * (iii)

For any countable set $\mathcal{U}\subset\mathcal{C}$ there exists an at most
countable set $\Xi_{\mathcal{U}}\subset(0,1)^{n}$ such that for any
$\xi\in(0,1)^{n}\setminus\Xi_{\mathcal{U}}$ and $(A,u)\in\mathcal{U}$ the
function $\widetilde{u}^{\xi}$, defined as in (2.6), satisfies (2.7). Indeed,
it is enough to define

$\Xi_{\mathcal{U}}:=\bigcup_{(A,u)\in\mathcal{U}}\Xi^{(A,u)}.$

We introduce a topology in $\mathcal{C}$ as follows.

######  Definition 2.2.

We say that a sequence $\\{(A_{k},u_{k})\\}$ converges to
$(A,u)\in\mathcal{C}$ in the $\tau_{\mathcal{C}}$-topology (or shortly
$\tau_{\mathcal{C}}$-converges) and denote as
$(A_{k},u_{k})\overset{\tau_{\mathcal{C}}}{\to}(A,u)$ if

  * •

$A_{k}\to A$ in $L^{1}(\mathbb{R}^{n}),$

  * •

$u_{k}\to u$ a.e. in $\Omega\cup S.$

###  2.2. Main results

Unless otherwise stated, throughout the paper the parameters $\Omega,$ $S,$
$\varphi,$ $\beta,$ $\mathbb{C}$ of SDRI energy and volume constant
$\mathtt{v}$ are assumed to satisfy the following:

  * (H0)

$\Omega$ and $S$ are bounded Lipschitz open sets, $S$ has finitely many
connected components, $\Sigma:=\partial\Omega\cap\partial S$ is a Lipschitz
$(n-1)$-manifold;

  * (H1)

$\varphi\in C^{0}(\overline{\Omega}\times\mathbb{R}^{n})$ and is a Finsler
norm, i.e., there exist $b_{2}\geq b_{1}>0$ such that for every
$x\in\overline{\Omega},$ $\varphi(x,\cdot)$ is a norm in $\mathbb{R}^{n}$
satisfying

$b_{1}|\xi|\leq\varphi(x,\xi)\leq b_{2}|\xi|,\qquad
x\in\overline{\Omega},\quad\xi\in\mathbb{R}^{n};$ (2.8)

  * (H2)

$\beta\in L^{\infty}(\Sigma)$ and satisfies

$-\varphi(x,\nu_{\Sigma}(x))\leq\beta(x)\leq\varphi(x,\nu_{\Sigma}(x))\qquad\text{$\mathcal{H}^{n-1}$-a.e.\
$x\in\Sigma$};$ (2.9)

  * (H3)

$\mathbb{C}\in L^{\infty}(\Omega\cup S)\cap C^{0}(\overline{\Omega})$ and
there exists $b_{4}\geq b_{3}>0$ such that

$2b_{3}\,\mathbf{M}:\mathbf{M}\leq\mathbb{C}(x)\mathbf{M}:\mathbf{M}\leq
2b_{4}\,\mathbf{M}:\mathbf{M},\qquad x\in\Omega\cup
S,\quad\mathbf{M}\in\mathbb{M}^{n\times n}_{\rm sym};$ (2.10)

  * (H4)

$\mathtt{v}\in(0,|\Omega|].$

######  Remark 2.3 (A priori bounds).

Hypotheses (H1)-(H3) are important to get a priori estimates for energy-
equibounded countable families. Indeed, let $\mathcal{U}\subset\mathcal{C}$ be
any at most countable family of $\mathcal{C}$ such that

$M:=\sup\limits_{(A,u)\in\mathcal{U}}\,\mathcal{F}(A,u)<+\infty.$

Then by (2.8) and (2.9)

$\mathcal{S}(A,u)\leq M\quad\text{and}\quad\mathcal{W}(A,u)\leq
M+\int_{\Sigma}|\beta|\mathrm{d}\mathcal{H}^{n-1}\leq
M+b_{2}\mathcal{H}^{n-1}(\Sigma).$

Moreover:

  * (i)

for any $(A,u)\in\mathcal{U}$

$P(A)+\mathcal{H}^{n-1}(A^{(1)}\cap
J_{u})\leq\frac{M+b_{2}\mathcal{H}^{n-1}(\Sigma)}{b_{1}}+P(\Omega)$

and

$\int_{A\cup
S}|\mathcal{E}u|^{2}\mathrm{d}x\leq\frac{2M+2b_{2}\mathcal{H}^{n-1}(\Sigma)}{b_{3}}+3\int_{\Omega}|\mathcal{E}u_{0}|^{2}\mathrm{d}x;$

  * (ii)

if $\mathcal{U}\ni(A_{k},u_{k})\overset{\tau_{\mathcal{C}}}{\to}(A,u)$ for
some $(A,u)\in\mathcal{C},$ then111Indeed, let
$\Xi_{\mathcal{U}}\subset\mathbb{R}^{n}$ be the countable set, given by Remark
2.1 (c), and let $\xi\in(0,1)^{n}\setminus\Xi_{\mathcal{U}}.$ Since the values
of $u_{k}$ are not important in $\Omega\setminus A_{k},$ we may assume
$u=u^{\xi},$ where $u^{\xi}$ is given as (2.6). Then $u_{k}\to u\chi_{A\cup
S}+\xi\chi_{\Omega\setminus A}$ a.e. and hence, (2.11) follows from [14,
Theorem 1.1].

$\chi_{A_{k}\cup S}\mathcal{E}u_{k}\rightharpoonup\chi_{A\cup
S}\mathcal{E}u\quad\text{in $L^{2}({\rm Int}{(\Omega\cup S\cup\Sigma)})$.}$
(2.11)

Now we formulate main results of the paper. First we deal with the existence
of admissible configurations with minimal energy.

######  Theorem 2.4 (Existence of minimizing configurations).

The minimum problem

$\inf\limits_{(A,u)\in\mathcal{C},\,\,|A|=\mathtt{v}}\mathcal{F}(A,u)$ (2.12)

has a solution. Moreover, there exists $\lambda_{0}>0$ such that
$(A,u)\in\mathcal{C}$ is a solution of (2.12) if and only if it solves

$\inf\limits_{(A,u)\in\mathcal{C}}\mathcal{F}^{\lambda}(A,u)$ (2.13)

for any $\lambda\geq\lambda_{0},$ where

$\mathcal{F}^{\lambda}(A,u):=\mathcal{F}(A,u)+\lambda\big{|}|A|-\mathtt{v}\big{|}.$

To prove Theorem 2.4 we will apply direct methods of Calculus of Variations.
To this aim we establish the $\tau_{\mathcal{C}}$-lower semicontinuity of
$\mathcal{F}$ and the $\tau$-compactness of energy-equibounded sequences in
$\mathcal{C}.$

######  Theorem 2.5 (Lower semicontinuity).

Assume that the sequence $\\{(A_{k},u_{k})\\}\subset\mathcal{C}$
$\tau_{\mathcal{C}}$-converges to $(A,u)\in\mathcal{C}.$ Then

$\liminf\limits_{k\to+\infty}\,\mathcal{F}(A_{k},u_{k})\geq\mathcal{F}(A,u).$
(2.14)

######  Theorem 2.6 (Compactness).

Let $\\{(A_{k},u_{k})\\}\in\mathcal{C}$ be such that

$M:=\sup_{k}\,\mathcal{F}(A_{k},u_{k})<+\infty.$

Then there exists a subsequence $\\{(A_{k_{l}},u_{k_{l}})\\},$ a sequence
$\\{(B_{l},v_{l})\\}\subset\mathcal{C}$ and $(A,u)\in\mathcal{C}$ such that
$(B_{l},v_{l})\overset{\tau_{\mathcal{C}}}{\to}(A,u),$ $|A_{k_{l}}\Delta
B_{l}|\to 0$ and

$\liminf\limits_{l\to+\infty}\,\mathcal{F}(A_{k_{l}},u_{k_{l}})\geq\liminf\limits_{l\to+\infty}\,\mathcal{F}(B_{l},v_{l})\geq\mathcal{F}(A,u).$

Notice that our compactness result is analogous to those in [27, 36].
According to the proof, in general we have $|B_{l}|\leq|A_{k_{l}}|,$ i.e., the
volume constraint may not be preserved. Rather, Theorems 2.5 and 2.6 allow to
solve the unconstrained minimum problem (2.13), and then, as in [26, Theorem
1], using the equivalence of the minimum problems (2.12) and (2.13) (see
Proposition A.1), we establish the existence of a volume-constraint minimizer.

It is worth to remark that in both Theorems 2.5 and 2.6 (and hence, in the
existence) the assumption $\mathbb{C}\in C(\overline{\Omega})$ can be relaxed
to $\mathbb{C}\in L^{\infty}(\Omega)$. The continuity of $\mathbb{C}$ is
important in the (partial) regularity of minimizers of $\mathcal{F}$.

######  Theorem 2.7 (Properties of minimizing configurations)).

Let $(\widetilde{A},\widetilde{u})\in\mathcal{C}$ be a solution of (2.12) and
let

$A=\mathrm{Int}(\widetilde{A}^{(1)})\quad\text{and}\quad
u=\widetilde{u}\chi_{A\cup S}+\xi\chi_{\Omega\setminus A},$

where $\xi\in(0,1)^{n}$ is chosen such that
$\Omega\cap\partial^{*}A\subset_{\mathcal{H}^{n-1}}J_{u}$ (see Remark 2.1),
and let

$J_{u}^{*}=\\{x\in J_{u}:\,\,\theta(J_{u},x)=1\\}.$

Then:

  * (i)

$(A,u)$ is a minimizer of $\mathcal{F}$ and

$\mathcal{H}^{n-1}(\widetilde{A}^{(1)}\setminus
A)<+\infty,\quad\mathcal{H}^{n-1}(J_{u}\setminus
J_{u}^{*})=0,\quad\mathcal{H}^{n-1}(\overline{J_{u}^{*}}\setminus
J_{u}^{*})=0;$

  * (ii)

for any $x\in\Omega$ and $r\in(0,\min\\{1,\mathrm{dist}(x,\partial\Omega)\\})$

$\frac{\mathcal{H}^{n-1}(Q_{r}(x)\cap
J_{u}}{r^{n-1})}\leq\frac{4nb_{2}+\lambda_{0}}{b_{1}},$

where $\lambda_{0}$ is given by Theorem 2.4;

  * (iii)

there exist $\varsigma_{0}=\varsigma_{0}(b_{1},b_{2},b_{3},b_{4})\in(0,1)$ and
$R_{0}=R_{0}(b_{1},b_{2},b_{3},b_{4})>0$ such that

$\frac{\mathcal{H}^{n-1}(Q_{r}(x)\cap J_{u})}{r^{n-1}}\geq\varsigma_{0}$

for all cubes $Q_{r}(x)\subset\Omega$ centered at
$x\in\Omega\cap\overline{J_{u}^{*}}$ with sidelength $r\in(0,R_{0});$

  * (iv)

if $E\subset A$ is any connected component of $A$ with
$\mathcal{H}^{n-1}([\partial^{*}E\cap\Sigma]\setminus J_{u})=0,$ then
$|E|\geq\omega_{n}\big{(}\frac{b_{1}n}{\lambda_{0}}\big{)}^{n}$ and
$u=u_{0}+a$ in $E$ for some rigid displacement $a.$

###  2.3. Generalization and extra results related to Literature models

In this section we discuss some models related to the SDRI model for which the
proofs of the main results above can be adapted, by also recovering as a
byproduct of our analysis some results already available in the Literature.

First we consider more general elastic energy densities.

######  Theorem 2.8 (Elastic density with $p$-growth).

For $p>1$ let a measurable function $W_{p}:{\rm Int}{(\Omega\cup
S\cup\Sigma)}\times\mathbb{M}^{n\times n}_{\rm sym}\to\mathbb{R}$ be such that

  * (a1)

for any $x\in{\rm Int}{(\Omega\cup S\cup\Sigma)},$ $W_{p}(x,\cdot)$ is convex
and there exist $c>0$ and $f\in L^{1}({\rm Int}{(\Omega\cup S\cup\Sigma)})$
such that

$W_{p}(x,\mathbf{M})\geq c|\mathbf{M}|^{p}+f(x)\quad\text{for a.e.\ $x\in{\rm
Int}{(\Omega\cup S\cup\Sigma)}$ and for all $\mathbf{M}\in\mathbb{M}^{n\times
n}_{\rm sym};$}$ (2.15)

  * (a2)

for any $u\in GSBD^{p}({\rm Int}{(\Omega\cup S\cup\Sigma)})$ the map $x\mapsto
W_{p}(x,\mathcal{E}u(x))$ belongs to $L^{1}({\rm Int}{(\Omega\cup
S\cup\Sigma)}).$

Let

$\mathcal{C}_{p}:=\\{(A,u):\,\,A\in BV(\Omega;\\{0,1\\}),\,\,u\in
GSBD^{p}({\rm Int}{(\Omega\cup S\cup\Sigma)})\\}$

be a class of admissible configurations and let

$\mathcal{F}_{p}=\mathcal{S}+\mathcal{W}_{p}\qquad\text{in
$\mathcal{C}_{p}$},$

where

$\mathcal{W}_{p}(A,u)=\int_{A\cup
S}W_{p}(x,\mathcal{E}u-\mathbf{M}_{0})\,\mathrm{d}x.$

Then for any $\mathtt{v}\in(0,|\Omega|]$ the minimum problem

$\min\limits_{(A,u)\in\mathcal{C}_{p},\,\,|A|=\mathtt{v}}\,\mathcal{F}_{p}(A,u)$
(2.16)

admits a solution. Moreover, there exists $\lambda_{0}>0$ such that for any
$\lambda>\lambda_{0}$ a configuration $(A,u)$ is a solution to (2.16) if and
only if it is a minimizer of

$\mathcal{F}_{p}^{\lambda}(A,u)=\mathcal{F}(A,u)+\lambda\big{|}|A|-\mathtt{v}\big{|}.$

A standard example of $W_{p}$ is

$W_{p}(x,\mathbf{M})=f(x)|\mathbf{M}|^{p}+g(x)$

for some $f\in L^{\infty}({\rm Int}{(\Omega\cup S\cup\Sigma)})$ with $f\geq
c>0$ a.e. and $g\in L^{1}({\rm Int}{(\Omega\cup S\cup\Sigma)}).$

Now we study the existence of minimizers in models related to the SDRI
setting, but with Dirichlet boundary conditions.

######  Theorem 2.9 (Dirichlet case with a $p$-growth elastic density).

For $p>1$ let

$\mathcal{C}_{\rm Dir}:=\\{(A,u):\,\,A\in BV(\Omega;\\{0,1\\}),\,\,u\in
GSBD^{p}({\rm Int}{(\Omega\cup S\cup\Sigma)}),\,\,u=u_{0}\,\,\text{in
$S$}\\},$

where $u_{0}\in H^{1}(\mathbb{R}^{n})$ is fixed, and let

$\mathcal{F}_{\rm Dir}:=\mathcal{S}+\mathcal{W}_{\rm Dir}\qquad\text{in
$\mathcal{C}_{\rm Dir}$},$

where

$\mathcal{W}_{\rm Dir}(A,u):=\int_{A}W_{p}(x,\mathcal{E}u)\,\mathrm{d}x$

and the elastic energy density $W_{p}$ satisfies all assumptions of Theorem
2.8. Then for any $\mathtt{v}\in(0,|\Omega|]$ the minimum problem

$\min\limits_{(A,u)\in\mathcal{C}_{\rm
Dir},\,\,|A|=\mathtt{v}}\,\mathcal{F}_{\rm Dir}(A,u)$ (2.17)

admits a solution. Moreover, there exists $\lambda_{0}>0$ such that for any
$\lambda>\lambda_{0}$ a configuration $(A,u)$ is a solution to (2.17) and only
if it is a minimizer of

$\mathcal{F}_{\rm Dir}^{\lambda}(A,u)=\mathcal{F}_{\rm
Dir}(A,u)+\lambda\big{|}|A|-\mathtt{v}\big{|}.$

######  Remark 2.10 (Relation to some Literature results).

As a consequence of Theorem 2.9 we have:

  * (i)

Let $\beta(x)=-\varphi(x,\nu_{\Sigma}(x))$ for $\mathcal{H}^{n-1}$-a.e.
$x\in\Sigma$ and let $W_{p}:\mathbb{M}^{n\times n}_{\rm sym}\to\mathbb{R}$
satisfy

$c^{\prime}|\mathbf{M}|^{p}-c^{\prime\prime}\leq W_{p}(\mathbf{M})\leq
c^{\prime\prime}(|\mathbf{M}|^{p}+1)$

for some $c^{\prime\prime},c^{\prime}>0.$ Then Theorem 2.9 coincides with the
existence result [17, Proposition 5.8] in the setting of material voids.

  * (ii)

Let $\beta=0$ and $W_{p}$ be as in (i). Then the minimizers of
$\mathcal{F}_{\rm Dir}$ in $\mathcal{C}_{\rm Dir}$ with volume constraint
$|\mathtt{v}|=|\Omega|$ (i.e., free-crystal regions have full
$\mathcal{L}^{n}$-measure) coincide with the (strong) Griffith minimizers in
[13] under Dirichlet boundary condition.

  * (iii)

In the proof of Theorem 2.7 we work only in $\Omega,$ i.e., we study the
regularity of $\partial^{*}A$ and $J_{u}$ only in the points of $\Omega.$
Therefore, the assertion on the essential closedness of $J_{u}$ and
$\partial^{*}A$ holds also for minimizers of $\mathcal{F}_{\rm Dir}$ with
$W_{p}(x,\mathbf{M})=\mathbb{C}(x)\mathbf{M}:\mathbf{M}.$ In particular, this
covers a partial regularity part of results in [13].

We anticipate here that we equip both $\mathcal{C}_{p}$ and $\mathcal{C}_{\rm
Dir}$ with the same type of convergence introduced in $\mathcal{C},$ i.e.

$(A_{k},u_{k})\overset{\tau}{\to}(A,u)\quad\Longleftrightarrow\quad
A_{k}\overset{L^{1}(\mathbb{R}^{n})}{\longrightarrow}A\,\,\text{and}\,\,u_{k}\to
u\,\,\text{a.e.\ in $\Omega\cup S.$}$ (2.18)

##  3\. Replacing cracks with voids

In this section we provide some technical results that allow to replace a
portion of the jump set of the displacement fields with an open set without
modifying too much the corresponding SDRI energy. These results will be used
in both the lower-semicontinuity and the compactness results. We start with
the following main ingredient of all crack-opening results.

######  Lemma 3.1.

Let $\delta\in(0,1/4),$ $Q:=Q_{r,\nu}(x_{0})$ be a cube, $\Gamma\subset Q$ is
an $(n-1)$-dimensional Lipschitz graph and $K\subset Q$ be an
$\mathcal{H}^{n-1}$-rectifiable set. Assume that

  * (a1)

$x_{0}\in\Gamma,$ $\nu$ is the unit normal to $\Gamma$ at $x_{0}$ and
$|(x-x_{0})\cdot\nu|\leq r/2$ for all $x\in\Gamma$;

  * (a2)

$\Gamma$ separates $Q$ into two open connected components $G_{1}$ and $G_{2}$;

  * (a3)

$\theta(K,x_{0})=\theta(K\cap\Gamma,x_{0})=1,$ $\nu$ is the generalized unit
normal to $K$ at $x_{0},$ and

$(1-\delta)r^{n-1}\leq\mathcal{H}^{n-1}(K\cap\Gamma)\leq\mathcal{H}^{n-1}(\Gamma)\leq(1+\delta)r^{n-1};$

  * (a4)

$\mathcal{H}^{n-1}(K\setminus\Gamma)<\delta r^{n-1}$.

Then there exist open sets $C,D\subset\subset Q$ of finite perimeter such that

  * (i)

$C\subset G_{1},$ and $\mathcal{H}^{n-1}(\partial
C\setminus\partial^{*}C)=\mathcal{H}^{n-1}(\partial
D\setminus\partial^{*}D)=0;$

  * (ii)

$\mathcal{H}^{n-1}(K\setminus\overline{C})<2\delta r^{n-1}$ and
$\mathcal{H}^{n-1}(K\setminus D)<2\delta r^{n-1};$

  * (iii)

$|C|<\delta r^{n}$ and $|D|<\delta r^{n};$

  * (iv)

$(1-2\delta)r^{n-1}\leq\mathcal{H}^{n-1}(K\cap\partial
C\cap\Gamma)\leq\mathcal{H}^{n-1}(\partial C\cap\Gamma)<(1+\delta)r^{n-1};$

  * (v)

for any norm $\phi$ in $\mathbb{R}^{n}$ satisfying (4.1) one has

$\int_{\partial D}\phi(\nu_{D})\mathrm{d}\mathcal{H}^{n-1}\leq
2\int_{K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}+5b_{2}\delta r^{n-1}.$ (3.1)
$\int_{\partial C}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}\leq
2\int_{K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}+5b_{2}\delta r^{n-1}$ (3.2)

and

$\int_{G_{1}\cap\partial
C}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}\leq\int_{K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}+3b_{2}\delta
r^{n-1},$ (3.3)

###### Proof.

Without loss of generality we assume that $\nu={\bf e}_{n},$ $x_{0}=0$ and
$G_{1}$ lies above $\Gamma$. Since $\Gamma$ is a Lipschitz graph,
$f\in\mathrm{Lip}(V)$ such that $\Gamma={\rm graph}(f),$ where
$V=[-\frac{r}{2},\frac{r}{2}]^{n-1}\subset\mathbb{R}^{n-1}.$ By (a1),
$\|f\|_{\infty}\leq r/2,$ hence, $\Gamma$ intersects only lateral sides of
$Q.$ Let

$\epsilon:=\frac{\delta}{4(1+\mathrm{Lip}(f))}.$

Let $V^{\prime\prime}\subset\subset V^{\prime}\subset\subset V$ be any
$(n-1)$-dimensional cubes in $\mathbb{R}^{n-1}$ such that

$\mathcal{H}^{n-1}(V\setminus V^{\prime\prime})<\epsilon r^{n-1}.$ (3.4)

For $\gamma\in(0,\epsilon r)$ let $g\in\mathrm{Lip}_{c}(V;[0,\gamma])$ be such
that $g\equiv\gamma$ in $V^{\prime\prime},$
$\mathrm{supp}\,(g)=\overline{V^{\prime}}$ and $\|g\|_{\infty}\leq 1.$ Let $C$
be the open set bounded between the graphs of $f$ and $f+g$ and let $D$ be the
open set bounded between the graphs of $f+g$ and $f-g.$ Since both $\partial
C$ and $\partial D$ consists of two Lipschitz graphs, it is a set of finite
perimeter.

We claim that $C$ and $D$ satisfy the assertion of the lemma.

(i) Since $\|f\pm g\|_{\infty}<3r/4$ (by (a1) and choice of $\gamma$) and
$g=0$ on $V\setminus V^{\prime},$ $C\subset G_{1}$ and $C,D\subset\subset
Q_{r}.$ Moreover, since $V^{\prime}$ is an $(n-1)$-dimensional hypercube, by
the area formula

$\mathcal{H}^{n-1}(\partial
C\setminus\partial^{*}C)=\mathcal{H}^{n-1}(\partial
D\setminus\partial^{*}D)\leq(1+\mathrm{Lip}(f))\mathcal{H}^{n-1}(\overline{V^{\prime}}\setminus
V^{\prime})=0.$

(i) By (a4)

$\displaystyle\mathcal{H}^{n-1}(K\setminus\overline{C})\leq$
$\displaystyle\mathcal{H}^{n-1}(\Gamma\cap
K\setminus\overline{C})+\mathcal{H}^{n-1}(K\setminus\Gamma)<\mathcal{H}^{n-1}(\Gamma\setminus\overline{C})+\delta
r^{n-1}.$

Moreover, by contruction

$\Gamma\setminus\overline{C}=\Gamma\setminus\partial
C=\Gamma\setminus\overline{D}=f(V\setminus\overline{V^{\prime}}),$

and hence, by the area formula and (3.4)

$\mathcal{H}^{n-1}(\Gamma\setminus\overline{C})\leq\int_{V\setminus
V^{\prime}}\sqrt{1+|\nabla
f|^{2}}\mathrm{d}x^{\prime}\leq(1+\mathrm{Lip}(f))\mathcal{H}^{n-1}(V\setminus
V^{\prime})<\frac{\delta}{4}\,r^{n-1}.$ (3.5)

Thus, $\mathcal{H}^{n-1}(K\setminus C)<\frac{5}{4}\,\delta r^{n-1}.$
Similarly,
$\mathcal{H}^{n-1}(\Gamma\setminus\overline{D})=\mathcal{H}^{n-1}(\Gamma\setminus
D)<\frac{1}{4}\delta r^{n-1}.$

(iii) By the Fubini’s theorem, the choice of $\gamma$ and also the area
formula

$\displaystyle|C|=$
$\displaystyle\int_{V^{\prime}}(f+g-f)\mathrm{d}x\leq\gamma\mathcal{H}^{n-1}(V^{\prime})<\epsilon
r\int_{V}\sqrt{1+|\nabla f|^{2}}\mathrm{d}x^{\prime}=\epsilon
r\mathcal{H}^{n-1}(\Gamma)$

and

$\displaystyle|D|=$ $\displaystyle\int_{V^{\prime}}(f+g-(f-g))\mathrm{d}x\leq
2\gamma\mathcal{H}^{n-1}(V^{\prime})<2\epsilon r\int_{V}\sqrt{1+|\nabla
f|^{2}}\mathrm{d}x^{\prime}=2\epsilon r\mathcal{H}^{n-1}(\Gamma)$

Hence, by (a3) $|C|<\frac{\delta(1+\delta)}{4}r^{n}$ and
$|C|<\frac{\delta(1+\delta)}{2}r^{n}.$

(iv) By (a3)

$\mathcal{H}^{n-1}(\partial
C\cap\Gamma)<\mathcal{H}^{n-1}(\Gamma)\leq(1+\delta)r^{n-1}.$

Moreover, by (3.5)

$\mathcal{H}^{n-1}(K\cap\Gamma)-\mathcal{H}^{n-1}(K\cap\partial
C\cap\Gamma)=\mathcal{H}^{n-1}(K\cap\Gamma\setminus\partial
C)\leq\mathcal{H}^{n-1}(\Gamma\setminus\partial
C)=\mathcal{H}^{n-1}(\Gamma\setminus C)<\frac{\delta}{4}r^{n-1}.$

Hence, by (a3)

$\mathcal{H}^{n-1}(K\cap\partial
C\cap\Gamma)\geq\mathcal{H}^{n-1}(K\cap\Gamma)-\frac{\delta}{4}r^{n-1}>(1-\tfrac{5}{4}\delta)r^{n-1}.$

(v) By the definition of $C,$ the area formula, the convexity of $\phi$, the
definition of $g$, (4.1) and (3.4)

$\displaystyle\int_{G_{1}\cap\partial
C}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}=$ $\displaystyle\int_{G_{1}\cap{\rm
graph}(f+g)}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}=\int_{V^{\prime}}\phi(-\nabla(f+g),1)\mathrm{d}\mathcal{H}^{n-1}$
$\displaystyle\leq$ $\displaystyle\int_{V^{\prime}}\phi(-\nabla
f,1)\mathrm{d}\mathcal{H}^{n-1}+\int_{V^{\prime}}\phi(-\nabla
g,0)\mathrm{d}\mathcal{H}^{n-1}$ $\displaystyle\leq$
$\displaystyle\int_{V}\phi(-\nabla
f,1)\mathrm{d}\mathcal{H}^{n-1}+\int_{V^{\prime}\setminus
V^{\prime\prime}}\phi(-\nabla g,0)\mathrm{d}\mathcal{H}^{n-1}$
$\displaystyle\leq$
$\displaystyle\int_{\Gamma}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}+b_{2}\|g\|_{\infty}\mathcal{H}^{n-1}(V^{\prime}\setminus
V^{\prime\prime})$ $\displaystyle\leq$
$\displaystyle\int_{\Gamma}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}+\frac{b_{2}\delta}{4}\,r^{n-1}.$

Moreover, by (a3)

$\mathcal{H}^{n-1}(\Gamma\setminus
K)=\mathcal{H}^{n-1}(\Gamma)-\mathcal{H}^{n-1}(\Gamma\cap K)\leq 2\delta
r^{n-1},$

and hence, by (4.1)

$\int_{\Gamma}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}\leq\int_{K\cap\Gamma}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}+b_{2}\mathcal{H}^{n-1}(\Gamma\setminus
K)\leq\int_{K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}+\frac{9b_{2}}{4}\,\delta
r^{n-1}.$ (3.6)

Thus, (3.3) follows. Since $\partial C\cap\partial G_{1}=\Gamma,$ the proof of
3.2 follows from (3.6) and (3.3). Similarly,

$\displaystyle\int_{\partial D}\phi(\nu_{D})\mathrm{d}\mathcal{H}^{n-1}=$
$\displaystyle\int_{V^{\prime}}\Big{[}\phi(-\nabla(f+g),1)\mathrm{d}\mathcal{H}^{n-1}+\phi(-\nabla(f-g),1)\Big{]}\mathrm{d}\mathcal{H}^{n-1}$
$\displaystyle\leq$ $\displaystyle 2\int_{V^{\prime}}\phi(-\nabla
f,1)\mathrm{d}\mathcal{H}^{n-1}+2\int_{V^{\prime}}\phi(-\nabla
g,0)\mathrm{d}\mathcal{H}^{n-1}$ $\displaystyle\leq$ $\displaystyle
2\int_{\Gamma}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}+\frac{b_{2}}{2}\,\delta
r^{n-1}$ $\displaystyle\leq$ $\displaystyle
2\int_{K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}+\frac{9b_{2}}{2}\,\delta
r^{n-1}.$

∎

The following result will be used in the proof of Proposition 4.1 with
$K=A_{k}^{(1)}\cap J_{u_{k}}$ and allows to replace $u_{k}$ with $v_{k},$
whose jump set is a reduced boundary of an open set of finite perimeter (see
Corollary 3.3 below). Recall that this property is important to obtain the
surface tension $2\varphi$ in the “interior” jump energy in the functional
$\mathcal{S}.$

######  Lemma 3.2.

Let $U\subset\mathbb{R}^{n}$ be an open set, $K\subset U$ be a
$\mathcal{H}^{n-1}$-rectifiable set and $\delta>0.$ There exists an at most
countable family $\\{C_{i}\\}_{i\geq 1}$ of open sets of finite perimeter such
that

  * (i)

$C_{i}\subset\subset U$ and $\mathcal{H}^{n-1}(\partial
C_{i}\setminus\partial^{*}C_{i})=0$;

  * (ii)

$\mathcal{H}^{n-1}(K\setminus\bigcup_{i}C_{i})<\delta$ and
$|\bigcup_{i}C_{i}|<\delta;$

  * (iii)

for any norm $\phi$ in $\mathbb{R}^{n}$ satisfying (4.1)

$\sum\limits_{i\geq 1}\int_{\partial
C_{i}}\phi(\nu_{C_{i}})\,\mathrm{d}\mathcal{H}^{n-1}<2\int_{K}\phi(\nu_{K})\,\mathrm{d}\mathcal{H}^{n-1}+\delta.$

###### Proof.

First we consider a special case.

Claim. Let $K={\rm graph}(f)$ for some $f\in\mathrm{Lip}(V),$ where
$V\subset\mathbb{R}^{n-1}$ is a bounded open set. Let
$V^{\prime\prime}\subset\subset V^{\prime}\subset\subset V$ be smooth open
sets such that

$\Big{(}1+\frac{1}{b_{1}}\Big{)}\,\int_{V\setminus
V^{\prime\prime}}\phi(-\nabla
f,1)\mathrm{d}x^{\prime}+\mathcal{H}^{n-1}(V\setminus
V^{\prime\prime})<\frac{\delta}{2+2b_{2}}.$ (3.7)

For
$\gamma\in\big{(}0,\frac{\delta}{4[1+\mathcal{H}^{n-1}(V^{\prime})]}\big{)}$
let $g\in\mathrm{Lip}(V;[0,\gamma])$ be such that
$\mathrm{supp}\,(g)=\overline{V^{\prime}},$ $g\equiv\gamma$ in
$V^{\prime\prime}$ and $\|\nabla g\|_{L^{\infty}(V)}\leq 1.$ Then $g=0$ on
$\partial V^{\prime}.$ Moreover, taking $\gamma$ small enough we assume that
the graphs of $f\pm g\big{|}_{V^{\prime}}$ are compactly contained in $U.$ Let
$C$ be the bounded open set whose boundary consists of the graphs of
$f-g:V^{\prime}\to\mathbb{R}$ and $f+g:V^{\prime}\to\mathbb{R}.$ Then
$C\subset\subset U$ and by the area formula, triangle inequality for $\phi,$
(4.1), (3.7) and the inequality $\|\nabla g\|_{\infty}\leq 1$

$\displaystyle\int_{\partial C}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}=$
$\displaystyle\int_{V^{\prime}}\Big{(}\phi(-\nabla(f+g),1)+\phi(-\nabla(f-g),1)\Big{)}\mathrm{d}x^{\prime}$
$\displaystyle\leq$ $\displaystyle 2\int_{V^{\prime}}\phi(-\nabla
f,1)\mathrm{d}x^{\prime}+2\int_{V^{\prime}}\phi(-\nabla
g,0)\mathrm{d}x^{\prime}$ $\displaystyle\leq$ $\displaystyle
2\int_{V^{\prime\prime}}\phi(-\nabla
f,1)\mathrm{d}x^{\prime}+2\int_{V^{\prime}\setminus
V^{\prime\prime}}\phi(-\nabla
f,1)\mathrm{d}x^{\prime}+2\int_{V^{\prime}\setminus
V^{\prime\prime}}\phi(\nabla g,0)\mathrm{d}x^{\prime}$ $\displaystyle\leq$
$\displaystyle 2\int_{V^{\prime}}\phi(-\nabla
f,1)\mathrm{d}x^{\prime}+\delta=2\int_{K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}+\delta.$

Moreover, by (4.1) and (3.7)

$\mathcal{H}^{n-1}(K\setminus C)=\int_{V\setminus V^{\prime}}\sqrt{1+|\nabla
f|^{2}}\mathrm{d}x^{\prime}\leq\frac{1}{b_{1}}\,\int_{V\setminus
V^{\prime\prime}}\phi(-\nabla f,1)\mathrm{d}x^{\prime}<\delta.$

Finally since $0\leq|g|\leq\frac{\delta}{4[1+\mathcal{H}^{n-1}(V^{\prime})]}$
it follows that

$|C|=\int_{W^{\prime}}[f+g-(f-g)]\mathrm{d}x^{\prime}\leq
2\|g\|_{\infty}\mathcal{H}^{n-1}(W^{\prime})<\delta.$

The equality $\mathcal{H}^{n-1}(\partial C\setminus\partial^{*}C)=0$ follows
from the smoothness of $V^{\prime}$.

Now we prove the lemma. By the countable $\mathcal{H}^{n-1}$-rectifiability of
$K$ there exists an at most countable family $\\{\Gamma_{i}\\}$ of Lipschitz
graphs such that $\Gamma_{i}\subset U,$ $\Gamma_{i}\cap\Gamma_{j}=\emptyset$
for $i\neq j,$ and $\mathcal{H}^{n-1}(K\setminus\bigcup_{i}\Gamma_{i})=0.$
Since $\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits\Gamma_{i}$ is Radon, by the
regularity of Radon measures for each $i$ there exists a relatively open
subset $\Gamma_{i}^{\prime}$ of $\Gamma_{i}$ such that
$\Gamma_{i}^{\prime}\cap K\subset\Gamma_{i}\cap K$ and

$\mathcal{H}^{n-1}(\Gamma_{i}^{\prime}\setminus
K)<\frac{\delta}{2^{i+2}(1+b_{2})},\quad i\geq 1.$ (3.8)

For shortness, we assume $\Gamma_{i}=\Gamma_{i}^{\prime}$. Then applying the
claim above with $\delta:=\frac{\delta}{2^{i+1}(1+b_{2})}$ and
$\Gamma=\Gamma_{i}$ we find an open set $C_{i}\subset\subset U$ such that

$|C_{i}|<\frac{\delta}{2^{i+1}(1+b_{2})},\qquad\mathcal{H}^{n-1}(\Gamma_{i}\setminus
C_{i})<\frac{\delta}{2^{i+1}(1+b_{2})}$ (3.9)

and

$\int_{\partial C_{i}}\phi(\nu_{C_{i}})\mathrm{d}\mathcal{H}^{n-1}\leq
2\int_{\Gamma_{i}}\phi(\nu_{\Gamma_{i}})\mathrm{d}\mathcal{H}^{n-1}+\frac{\delta}{2^{i+1}(1+b_{2})}.$
(3.10)

Thus, by the pairwise disjointness of $\\{\Gamma_{i}\\}$

$\displaystyle\mathcal{H}^{n-1}\Big{(}K\setminus\bigcup_{j}C_{j}\Big{)}\leq$
$\displaystyle\mathcal{H}^{n-1}\Big{(}\bigcup_{i}\Big{(}\Gamma_{i}\setminus\bigcup_{j}C_{j}\Big{)}\Big{)}=\sum_{i}\mathcal{H}^{n-1}\Big{(}\Gamma_{i}\setminus\bigcup_{j}C_{j}\Big{)}$
$\displaystyle\leq$
$\displaystyle\sum_{i}\mathcal{H}^{n-1}(\Gamma_{i}\setminus C_{i})<\delta$

and by (3.8) and (3.10),

$\displaystyle\sum\limits_{i}\int_{\partial
C_{i}}\phi(\nu_{C_{i}})\mathrm{d}\mathcal{H}^{n-1}\leq$ $\displaystyle
2\sum\limits_{i}\int_{\Gamma_{i}}\phi(\nu_{\Gamma_{i}})\mathrm{d}\mathcal{H}^{n-1}+\frac{\delta}{2}$
$\displaystyle\leq$ $\displaystyle 2\sum\limits_{i}\int_{\Gamma_{i}\cap
K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}+2\sum\limits_{i}\int_{\Gamma_{i}\setminus
K}\phi(\nu_{\Gamma_{i}})\mathrm{d}\mathcal{H}^{n-1}+\frac{\delta}{2}$
$\displaystyle\leq$ $\displaystyle 2\int_{\cup_{i}\Gamma_{i}\cap
K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}+\sum\limits_{i}\frac{2b_{2}\delta}{2^{i+2}(1+b_{2})}+\frac{\delta}{2}$
$\displaystyle=$ $\displaystyle
2\int_{K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}+\delta.$

Finally, by the estimate for $|C_{i}|$ in (3.9)

$\Big{|}\bigcup_{i}C_{i}\Big{|}\leq\sum\limits_{i}|C_{i}|<\delta.$

∎

######  Corollary 3.3.

Let $U\subset\subset\Omega$ be an open set, $(A,u)\in\mathcal{C}$ and
$\delta>0.$ Then there exists an open set $G\subset\subset U$ of finite
perimeter such that

  * (i)

the configuration $(B,v)$ with $B:=A\setminus G$ and $v:=u\chi_{B\cup S}$
belongs to $\mathcal{C};$

  * (ii)

$|G|<\delta;$

  * (iii)

$\mathcal{H}^{n-1}(U\cap B^{(1)}\cap J_{v})<\delta;$

  * (iv)

for any norm $\phi$ in $\mathbb{R}^{n}$ satisfying (4.1)

$\displaystyle\int_{U\cap\partial^{*}A}\phi(\nu_{A})\mathrm{d}\mathcal{H}^{n-1}+2\int_{U\cap
A^{(1)}\cap
J_{u}}\phi(\nu_{J_{u}})\mathrm{d}\mathcal{H}^{n-1}\geq\int_{U\cap\partial^{*}B}\phi(\nu_{B})\mathrm{d}\mathcal{H}^{n-1}-\delta.$

###### Proof.

Let $\epsilon:=\frac{\delta}{8}.$ Since $\mathcal{H}^{n-1}(U\cap A^{(1)}\cap
J_{u})<+\infty,$ there exists an open set $U^{\prime}\subset\subset U$ such
that

$\mathcal{H}^{n-1}((U\setminus U^{\prime})\cap A^{(1)}\cap J_{u})<\epsilon.$
(3.11)

By Lemma 3.2 applied with $U^{\prime},$ $K:=U^{\prime}\cap A^{(1)}\cap J_{u}$
and $\epsilon$ we find an at most countable family $\\{C_{i}\\}_{i\geq 1}$ of
open sets of finite perimeter such that

  * ($a_{1}$)

$C_{i}\subset\subset U^{\prime}$ and $\mathcal{H}^{n-1}(\partial
C_{i}\setminus\partial^{*}C_{i})=0$;

  * ($a_{2}$)

$\mathcal{H}^{n-1}([U^{\prime}\cap K]\setminus\bigcup_{i}C_{i})<\epsilon$ and
$|\bigcup_{i}C_{i}|<\epsilon;$

  * ($a_{3}$)

$\sum\limits_{i\geq 1}\int_{\partial
C_{i}}\phi(\nu_{C_{i}})\,\mathrm{d}\mathcal{H}^{n-1}<2\int_{U^{\prime}\cap
K}\phi(\nu_{K})\,\mathrm{d}\mathcal{H}^{n-1}+\epsilon.$

Define

$G:=\bigcup_{i\geq 1}C_{i}.$

We claim that $G$ satisfies the assertion of the lemma. Indeed, (i) is obvious
and (ii) follows from ($a_{2}$). By construction, $B^{(1)}\cap
J_{v}=_{\mathcal{H}^{n-1}}B^{(1)}\cap J_{u}$ and hence, by (3.11) and
($a_{2}$) we have

$\mathcal{H}^{n-1}(U\cap B^{(1)}\cap J_{v})\leq\mathcal{H}^{n-1}((U\setminus
U^{\prime})\cap A^{(1)}\cap J_{u})+\mathcal{H}^{n-1}(U^{\prime}\cap
A^{(1)}\cap J_{u}\setminus G)<2\epsilon.$

Finally, since $\partial^{*}B\setminus\partial^{*}A\subset\partial^{*}G,$ by
($a_{3}$)

$\displaystyle\int_{U\cap\partial^{*}B}\phi(\nu_{B})\mathrm{d}\mathcal{H}^{n-1}=$
$\displaystyle\int_{U\cap\partial^{*}B\cap\partial^{*}A}\phi(\nu_{B})\mathrm{d}\mathcal{H}^{n-1}+\int_{U\cap\partial^{*}B\setminus\partial^{*}A}\phi(\nu_{B})\mathrm{d}\mathcal{H}^{n-1}$
$\displaystyle\leq$
$\displaystyle\int_{U\cap\partial^{*}A}\phi(\nu_{A})\mathrm{d}\mathcal{H}^{n-1}+\sum\limits_{i\geq
1}\int_{\partial C_{i}}\phi(\nu_{C_{i}})\,\mathrm{d}\mathcal{H}^{n-1}$
$\displaystyle\leq$
$\displaystyle\int_{U\cap\partial^{*}A}\phi(\nu_{A})\mathrm{d}\mathcal{H}^{n-1}+2\int_{U^{\prime}\cap
K}\phi(\nu_{J_{u}})\,\mathrm{d}\mathcal{H}^{n-1}+\epsilon.$

∎

The next lemma is a counterpart of Lemma 3.2 and relates to the “opening” of
cracks along $\Sigma.$ Notice that in this case the opening should not get out
from $\Omega.$ Thus, we are replacing the jump of $u$ only from one side
(Corollary 3.5) and this is the reason for having $\varphi$ (without factor
$2$) in the jump energy along $\Sigma$ in the functional $\mathcal{S}.$

######  Lemma 3.4.

Let $U\subset\subset{\rm Int}{(\Omega\cup S\cup\Sigma)}$ be an open set,
$\delta\in(0,1)$ and $K\subset U\cap\Sigma$ be any
$\mathcal{H}^{n-1}$-measurable set. Then there exist an open set $C\subset
U\cap\Omega$ of finite perimeter such that

  * (i)

$C\subset\subset U$ and $\mathcal{H}^{n-1}(\partial
C\setminus\partial^{*}C)=0;$

  * (ii)

$\mathcal{H}^{n-1}(K\setminus\partial
C)=\mathcal{H}^{n-1}(K\setminus\overline{C})<\delta$ and $|C|<\delta;$

  * (iii)

$\mathcal{H}^{n-1}(U\cap\Sigma\cap\partial C\setminus K)<\delta;$

  * (iv)

for any norm $\phi$ in $\mathbb{R}^{n}$ satisfying (4.1)

$\int_{\Omega\cap\partial
C}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}\leq\int_{K}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+\delta,$

and

$\int_{\partial C}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}\leq
2\int_{K}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+\delta.$

###### Proof.

Let

$\epsilon:=\frac{\delta}{8(1+b_{2})(1+\mathcal{H}^{n-1}(\Sigma))}.$

We divide the proof into two steps.

Step 1. Let $Q_{r}(x_{0})\subset U$ be a cube centered at $x\in\Sigma$ such
that $\Sigma\cap Q_{r}(x_{0})={\rm graph}(f)$ for some Lipschitz function
$f:V\to\mathbb{R}$ and a cube $V\subset\mathbb{R}^{n-1},$ and assume that
$S\cap Q_{r}(x_{0})$ is a subgraph of $f.$ Let $V^{\prime\prime}\subset\subset
V^{\prime}\subset\subset V$ be open sets such that

$\mathcal{H}^{n-1}(V\setminus
V^{\prime\prime})<\frac{\mathcal{H}^{n-1}(Q_{r}(x_{0})\cap\Sigma)}{1+\mathrm{Lip}(f)}\,\epsilon$

and for
$\gamma\in(0,\frac{\mathcal{H}^{n-1}(Q_{r}(x_{0})\cap\Sigma)}{1+\mathcal{H}^{n-1}(V)}\,\epsilon)$
let $g\in\mathrm{Lip}_{c}(V;[0,\gamma])$ be such that $g\equiv 1$ in
$V^{\prime\prime},$ $\mathrm{supp}\,(g)=V^{\prime}$ and $\mathrm{Lip}(g)<1.$
We may assume that $\gamma$ is so small that the set open set $C,$ whose
boundary lies on the graphs of $f$ and $f+g,$ is compactly contained in
$Q_{r}(x_{0})$ and $C\cap S=\emptyset.$ Then

$\displaystyle\int_{\Omega\cap\partial
C}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}=$
$\displaystyle\int_{V^{\prime}}\phi(-\nabla(f+g),1)\mathrm{d}\mathcal{H}^{n-1}\leq\int_{V^{\prime}}\Big{(}\phi(-\nabla
f,1)+\phi(-\nabla g,0)\Big{)}\mathrm{d}\mathcal{H}^{n-1}$ $\displaystyle\leq$
$\displaystyle\int_{V}\phi(-\nabla
f,1)\mathrm{d}\mathcal{H}^{n-1}+b_{2}\mathrm{Lip}(f)\mathcal{H}^{n-1}(V^{\prime}\setminus
V^{\prime\prime})$ $\displaystyle<$
$\displaystyle\int_{Q_{r}(x_{0})\cap\Sigma}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+b_{2}\mathcal{H}^{n-1}(Q_{r}(x_{0})\cap\Sigma)\epsilon.$

Similarly,

$\displaystyle\int_{\partial C}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}=$
$\displaystyle\int_{V^{\prime}}[\phi(-\nabla(f+g),1)+\phi(-\nabla
f,1)]\mathrm{d}\mathcal{H}^{n-1}$ $\displaystyle\leq$
$\displaystyle\int_{V^{\prime}}\Big{(}2\phi(-\nabla f,1)+\phi(-\nabla
g,0)\Big{)}\mathrm{d}\mathcal{H}^{n-1}$ $\displaystyle<$ $\displaystyle
2\int_{Q_{r}(x_{0})\cap\Sigma}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+b_{2}\mathcal{H}^{n-1}(Q_{r}(x_{0})\cap\Sigma)\epsilon.$

Also by the Fubini’s theorem

$|C|=\int_{V^{\prime}}g\mathrm{d}x^{\prime}\leq\gamma\mathcal{H}^{n-1}(V^{\prime})<\mathcal{H}^{n-1}(Q_{r}(x_{0})\cap\Sigma)\,\epsilon.$

Finally,

$\displaystyle\mathcal{H}^{n-1}(Q_{r}(x_{0})\cap\Sigma\setminus\partial C)=$
$\displaystyle\mathcal{H}^{n-1}(Q_{r}(x_{0})\cap\Sigma\setminus\overline{C})=\int_{V\setminus
V^{\prime}}\sqrt{1+|\nabla f|^{2}}\mathrm{d}\mathcal{H}^{n-1}$
$\displaystyle\leq$
$\displaystyle(1+\mathrm{Lip}(f))\mathcal{H}^{n-1}(V\setminus
V^{\prime})<\mathcal{H}^{n-1}(Q_{r}(x_{0})\cap\Sigma)\,\epsilon.$

Step 2. Since $\Sigma$ is Lipschitz and $K$ is
$\mathcal{H}^{n-1}$-rectifiable, we can find a finite family
$Q_{r_{1},\nu_{1}}(x_{1}),\ldots,Q_{r_{m},\nu_{m}}(x_{m})\subset U$ of
pairwise disjoint cubes centered at $K$ such that

  * ($\rm a_{1}$)

for each $j,$ $\Sigma\cap Q_{r_{j},\nu_{j}}(x_{j})$ is a graph of a Lipschitz
function in $\nu_{j}$ direction;

  * ($\rm a_{2}$)

$\theta(K,x_{j})=\theta(\Sigma,x_{j})=1,$ and the unit normals
$\nu_{K}(x_{j})$ and $\nu_{\Sigma}(x_{j})$ exist and coincide with $\nu_{j};$

  * ($\rm a_{3}$)

$(1-\epsilon)r_{j}^{n-1}<\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma\cap
K)\leq\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma)<(1+\epsilon)r_{j}^{n-1};$

  * ($\rm a_{4}$)

$\mathcal{H}^{n-1}\Big{(}K\setminus\bigcup_{j=1}^{m}Q_{r_{j},\nu_{j}}(x_{j})\Big{)}<\epsilon.$

Note that by ($\rm a_{3}$)

$\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma\setminus K)<2\epsilon
r_{j}^{n-1}<\frac{2\epsilon}{1-\epsilon}\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma).$
(3.12)

By Step 1 for each $j$ we can contruct an open set $C_{j}\subset\subset
Q_{r_{j},\nu_{j}}(x_{j})$ with $C_{j}\cap S=\emptyset$ and

$\displaystyle\int_{\Omega\cap\partial
C_{j}}\phi(\nu_{C_{j}})\mathrm{d}\mathcal{H}^{n-1}<\int_{Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+b_{2}\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma)\epsilon$
(3.13)

and

$\displaystyle\int_{\partial
C_{j}}\phi(\nu_{C_{j}})\mathrm{d}\mathcal{H}^{n-1}<2\int_{Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+b_{2}\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma)\epsilon.$

Moreover,

$|C_{j}|<\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma)\epsilon$ (3.14)

and

$\displaystyle\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma\setminus\partial
C_{j})<\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma)\,\epsilon.$
(3.15)

We claim that $C=\bigcup_{i=1}^{m}C_{j}$ satisfies all assertions of the
lemma.

(i) By construction $C\subset\subset U$ and since each $C_{j}$ is almost
Lipschitz, $\mathcal{H}^{n-1}(\partial C_{j}\setminus\partial^{*}C_{j})=0.$
Hence, by the pairwise disjointness of $\overline{C_{j}},$
$\mathcal{H}^{n-1}(\partial C\setminus\partial^{*}C)=0$ and (i) follows.

(ii) By ($\rm a_{4}$) and (3.15)

$\displaystyle\mathcal{H}^{n-1}(K\setminus\partial C)\leq$
$\displaystyle\mathcal{H}^{n-1}\Big{(}K\setminus\bigcup_{j=1}^{m}Q_{r_{j},\nu_{j}}(x_{j})\Big{)}+\sum\limits_{j=1}^{m}\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma\setminus\partial
C_{j})$ $\displaystyle<$
$\displaystyle\epsilon+\sum\limits_{j=1}^{m}\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma)\,\epsilon\leq(1+\mathcal{H}^{n-1}(\Sigma))\epsilon<\delta.$

Moreover, by (3.14)

$|C|\leq\sum\limits_{j=1}^{n}|C_{j}|\leq\mathcal{H}^{n-1}(\Sigma)\epsilon<\delta.$

(iii) By (3.12)

$\displaystyle\mathcal{H}^{n-1}(U\cap\Sigma\cap\partial C\setminus K)=$
$\displaystyle\sum\limits_{j=1}^{m}\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma\cap\partial
C_{j}\setminus K)$ $\displaystyle\leq$
$\displaystyle\sum\limits_{j=1}^{m}\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma\setminus
K)<\frac{2\epsilon}{1-\epsilon}\mathcal{H}^{n-1}(\Sigma)<\delta.$

(iv) Since $\partial^{*}C\subset\cup_{j}\partial^{*}C_{j},$ by (3.13) we get

$\int_{\Omega\cap\partial
C}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}\leq\sum\limits_{j=1}^{m}\int_{Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+b_{2}\mathcal{H}^{n-1}(\Sigma)\epsilon$

Moreover, by (4.1)

$\int_{Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}\leq\int_{Q_{r_{j},\nu_{j}}(x_{j})\cap
K}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+b_{2}\mathcal{H}^{n-1}(Q_{r_{j},\nu_{j}}(x_{j})\cap\Sigma\setminus
K),$

and thus, by (3.12)

$\int_{\Omega\cap\partial
C}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}\leq\int_{K}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+\Big{(}\tfrac{b_{2}}{1-\epsilon}+b_{2})\mathcal{H}^{n-1}(\Sigma)\epsilon<\int_{K}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+\delta.$

Finally, since $\Sigma\cap\partial C\subset K\cup(\Sigma\cap\partial
C\setminus K),$

$\displaystyle\int_{\partial C}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}=$
$\displaystyle\int_{\Omega\cap\partial
C}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}+\int_{\Sigma\cap\partial
C}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}$ $\displaystyle\leq$
$\displaystyle
2\int_{K}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+3b_{2}\mathcal{H}^{n-1}(\Sigma)\epsilon+b_{2}\mathcal{H}^{n-1}(\Sigma\cap\partial
C\setminus K)$ $\displaystyle<$ $\displaystyle
2\int_{K}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+7b_{2}\mathcal{H}^{n-1}(\Sigma)\epsilon<2\int_{K}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+\delta.$

∎

######  Corollary 3.5.

Let $U\subset\subset{\rm Int}{(\Omega\cup S\cup\Sigma)}$ be an open set,
$(A,u)\in\mathcal{C}$ and $\delta>0.$ Then there exists an open set
$G\subset\Omega$ of finite perimeter such that

  * (i)

$G\subset\subset U$ and $|G|<\delta;$;

  * (ii)

the configuration $(B,v)$ with $B:=A\setminus G$ and $v:=u\chi_{B\cup S}$
belongs to $\mathcal{C};$

  * (iii)

$\mathcal{H}^{n-1}(\Sigma\cap\partial^{*}G\setminus(\partial^{*}A\cap
J_{u}))+\mathcal{H}^{n-1}(U\cap\Sigma\cap
J_{u}\cap\partial^{*}A\setminus\partial^{*}G)<\delta$

and

$\mathcal{H}^{n-1}(U\cap B^{(1)}\cap
J_{v})<\delta+\mathcal{H}^{n-1}(U\cap\Sigma\cap
J_{v}\cap\partial^{*}B)<\delta;$

  * (iv)

for any norm $\phi$ in $\mathbb{R}^{n}$ satisfying (4.1)

$\int_{U\cap\Omega\cap\partial^{*}A}\phi(\nu_{A})\mathrm{d}\mathcal{H}^{n-1}+2\int_{U\cap
A^{(1)}\cap
J_{u}}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+\int_{U\cap\Sigma\cap\partial^{*}A\cap
J_{u}}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}\\\
\geq\int_{U\cap\Omega\cap\partial^{*}B}\phi(\nu_{B})\mathrm{d}\mathcal{H}^{n-1}-\delta\geq\int_{U\cap\partial^{*}G}\phi(\nu_{G})\mathrm{d}\mathcal{H}^{n-1}-\delta$
(3.16)

and

$\int_{U\cap\Omega\cap\partial^{*}A}\phi(\nu_{A})\mathrm{d}\mathcal{H}^{n-1}+2\int_{U\cap
A^{(1)}\cap
J_{u}}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+2\int_{U\cap\Sigma\cap\partial^{*}A\cap
J_{u}}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}\\\
\geq\int_{U\cap\Omega\cap\partial^{*}B}\phi(\nu_{B})\mathrm{d}\mathcal{H}^{n-1}+\int_{\Sigma\cap\partial^{*}G}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}-\delta\geq\int_{\partial^{*}G}\phi(\nu_{G})\mathrm{d}\mathcal{H}^{n-1}$
(3.17)

###### Proof.

The last inequalities in (3.16) and (3.17) follow from the definition of $B.$

Let $\epsilon:=\frac{\delta}{16(1+b_{2})}.$ Let
$U^{\prime}\subset\subset\Omega\cap U$ be any open set such that

$\mathcal{H}^{n-1}(\Omega\cap U\cap J_{u}\setminus U^{\prime})<\epsilon.$

By Corollary 3.3 applied with $U^{\prime},$ $(A,u)\in\mathcal{C}$ and
$\epsilon$ we find an open set $D^{\prime}\subset\subset U^{\prime}$ of finte
perimeter such that

  * ($\rm a_{1}$)

the configuration $(B^{\prime},v^{\prime})$ with $B^{\prime}:=A\setminus
D^{\prime}$ and $v^{\prime}:=u\chi_{S\cup B^{\prime}}$ belongs to
$\mathcal{C}$;

  * ($\rm a_{2}$)

$|D^{\prime}|<\epsilon;$

  * ($\rm a_{3}$)

$\mathcal{H}^{n-1}(U^{\prime}\cap[B^{\prime}]^{(1)}\cap
J_{v^{\prime}})<\epsilon;$

  * ($\rm a_{4}$)

$\int_{U^{\prime}\cap\partial^{*}A}\phi(\nu_{A})\mathrm{d}\mathcal{H}^{n-1}+2\int_{U^{\prime}\cap
A^{(1)}\cap
J_{u}}\phi(\nu_{J_{u}})\mathrm{d}\mathcal{H}^{n-1}\geq\int_{U^{\prime}\cap\partial^{*}B^{\prime}}\phi(\nu_{B^{\prime}})\mathrm{d}\mathcal{H}^{n-1}-\epsilon.$

Now choose another open set $U^{\prime\prime}\subset\subset U$ such that
$\overline{U^{\prime}}\cap\overline{U^{\prime\prime}}=\emptyset$ and

$\mathcal{H}^{n-1}((U\setminus
U^{\prime\prime})\cap\Sigma\cap\partial^{*}A\cap J_{u})<\epsilon.$

By Lemma 3.4 applied with $U^{\prime\prime},$ $\epsilon$ and
$K:=U^{\prime\prime}\cap\Sigma\cap\partial^{*}A\cap J_{u}$ we find an open set
$C^{\prime}\subset U^{\prime\prime}\cap\Omega$ of finite perimeter such that

  * ($\rm b_{1}$)

$C^{\prime}\subset\subset U^{\prime}$ and $\mathcal{H}^{n-1}(\partial
C^{\prime}\setminus\partial^{*}C^{\prime})=0$;

  * ($\rm b_{2}$)

$\mathcal{H}^{n-1}(K\setminus\partial
C^{\prime})=\mathcal{H}^{n-1}(K\setminus\overline{C^{\prime}})<\epsilon$ and
$|C^{\prime}|<\epsilon;$

  * ($\rm b_{3}$)

$\mathcal{H}^{n-1}(\Sigma\cap\partial C^{\prime}\setminus K)<\epsilon;$

  * ($\rm b_{4}$)

$\int_{\Omega\cap\partial
C^{\prime}}\phi(\nu_{C^{\prime}})\mathrm{d}\mathcal{H}^{n-1}\leq\int_{K}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+\epsilon,$

and

$\int_{\partial
C^{\prime}}\phi(\nu_{C^{\prime}})\mathrm{d}\mathcal{H}^{n-1}\leq
2\int_{K}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}+\epsilon.$

Define

$G:=C^{\prime}\cup D^{\prime}.$

We claim that $G$ satisfies the assertion of the lemma. Indeed, assertions
(i)-(iii) follow from ($\rm a_{1}$)-($\rm a_{3}$) and ($\rm b_{1}$)-($\rm
b_{3}$), whereas (iv) follows from the inclusion
$\partial^{*}B\setminus\partial^{*}A\subset\Omega\cap\partial
C^{\prime}\cup\partial D^{\prime}$ and conditions ($\rm a_{4}$) and ($\rm
b_{4}$). ∎

##  4\. $\tau_{\mathcal{C}}$-lower semicontinuity

In this section we prove Theorem 2.5 by following the arguments of [36,
Proposition 4.1], and in particular by using density estimates for some Radon
measures associated to $\mathcal{F}.$ We start with the following lower bound
for the localized surface energy.

######  Proposition 4.1.

Let $\delta\in(0,1),$
$Q_{r,\nu}(x_{0})\subset\subset\mathrm{Int}(\Omega\cup\Sigma\cup S),$ $r>0,$
$\nu\in\mathbb{S}^{n-1},$ be a cube and $\Gamma\subset Q_{r,\nu}(x_{0})$ be an
$(n-1)$-dimensional Lipschitz graph separating $Q_{r,\nu}(x_{0})$ into two
connected components such that

  * (a1)

$x_{0}\in\Gamma,$ $\nu_{\Gamma}(x_{0})=\nu$ and

$|\nu_{\Gamma}(x)-\nu|<\delta\quad\text{and}\quad|(x-x_{0})\cdot\nu|<\tfrac{\delta
r}{2}\quad\text{for all $x\in\Gamma;$}$

  * (a2)

$\mathcal{H}^{n-1}(Q_{r,\nu}(x_{0})\cap\Gamma)<(1+\delta)r^{n-1}.$

Assume that a sequence $\\{(A_{k},u_{k})\\}\subset\mathcal{C}$ and a
configuration $(A,u)\in\mathcal{C}$ satisfy

  * (a3)

$u_{k}=\xi$ for some $\xi\in(0,1)^{n}\setminus\Xi_{\\{(A_{k},u_{k})\\}}$ (see
Remark 2.1) and

$M:=\sup_{k\geq 1}\mathcal{F}(A_{k},u_{k})<+\infty;$

  * (a4)

$A_{k}\to A$ in $L^{1}(\mathbb{R}^{n});$

  * (a5)

$\mathcal{H}^{n-1}(Q_{r,\nu}(x_{0})\cap\partial^{*}(A\cup S))<\delta r^{n-1}$
and $|(A\cup S)\cap Q_{r,\nu}(x_{0})|>(1-\delta)r^{n};$

  * (a6)

either

$u_{k}\to u\quad\text{a.e.\ in $Q_{r,\nu}(x_{0})$}$

and

$K:=Q_{r,\nu}(x_{0})\cap J_{u}$

or there exists a set of finite perimeter $E\subset Q_{r,\nu}(x_{0})$ such
that

$u_{k}\to u\quad\text{a.e.\ in $Q_{r,\nu}(x_{0})\setminus
E$}\qquad\text{and}\qquad|u_{k}|\to+\infty\quad\text{a.e.\ in
$Q_{r,\nu}(x_{0})\cap E,$}$

and

$K:=Q_{r,\nu}(x_{0})\cap\partial^{*}E$

(see Figure 2).

Figure 2. Set $K$ in Proposition 4.1.

  * (a7)

the set $K$ satisfies

    * (a7.1)

$\nu_{K}(x_{0})=\nu$ and $\theta(K,x_{0})=\theta(\Gamma\cap K,x_{0})=1;$

    * (a7.2)

$\mathcal{H}^{n-1}(K\cap\Gamma)>(1-\delta)r^{n-1};$

    * (a7.3)

$\mathcal{H}^{n-1}(K\setminus\Gamma)<\delta r^{n-1}.$

We also denote by $\phi$ a norm in $\mathbb{R}^{n}$ satisfying

$b_{1}\leq\phi(\nu)\leq b_{2},\quad\nu\in\mathbb{S}^{n-1}.$ (4.1)

Let $C,D\subset\subset Q_{r,\nu}(x_{0})$ be given by Lemma 3.1 applied with
$\delta,$ $\Gamma$ and $K.$ Then there exist $c^{\prime}=c_{b_{2}}^{\prime}>0$
and $k_{\delta}^{\prime}:=k_{\delta}^{\prime}(b_{2})>0$ such that for any
$k>k_{\delta}^{\prime}:$

  * (i)

if $Q_{r,\nu}(x_{0})\subset\subset\Omega,$ then

$\displaystyle\int_{D\cap\partial^{*}A_{k}}\phi(\nu_{A_{k}})\mathrm{d}\mathcal{H}^{n-1}+2\int_{D\cap
A_{k}^{(1)}\cap J_{u_{k}}}$
$\displaystyle\phi(\nu_{J_{u_{k}}})\mathrm{d}\mathcal{H}^{n-1}$
$\displaystyle\geq$ $\displaystyle
2\int_{K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}-c^{\prime}\delta r^{n-1}$
$\displaystyle\geq$ $\displaystyle\int_{\partial
D}\phi(\nu_{D})\mathrm{d}\mathcal{H}^{n-1}-(c^{\prime}+5b_{2})\delta r^{n-1};$
(4.2)

  * (ii)

if $x_{0}\in\Sigma$ and $\Gamma=Q_{r,\nu}(x_{0})\cap\Sigma,$ then

$\displaystyle\int_{C\cap\partial^{*}A_{k}}\phi(\nu_{A_{k}})\mathrm{d}\mathcal{H}^{n-1}+2\int_{C\cap
A_{k}^{(1)}\cap J_{u_{k}}}\phi(\nu_{J_{u_{k}}})$
$\displaystyle\mathrm{d}\mathcal{H}^{n-1}+2\int_{\Sigma\cap\partial
C\cap\partial^{*}A_{k}\cap
J_{u_{k}}}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}$ $\displaystyle\geq$
$\displaystyle
2\int_{K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}-c^{\prime}\delta r^{n-1}$
$\displaystyle\geq$ $\displaystyle\int_{\partial
C}\phi(\nu_{C})\mathrm{d}\mathcal{H}^{n-1}-(c^{\prime}+5b_{2})\delta r^{n-1}.$
(4.3)

The proof of this proposition is left after the proof of Theorem 2.5. In the
proof of lower semicontinuity we only use the following corollary of
Proposition 4.1; the assertions including sets $C,$ $D$ and $E$ will used in
the proof of compactness.

######  Corollary 4.2.

Under assumptions of Proposition 4.1, together with

  * (a6)

$u_{k}\to u\quad\text{a.e.\ in $Q_{r,\nu}(x_{0})$}$ and

$K:=Q_{r,\nu}(x_{0})\cap J_{u},$

there exist $c^{\prime}=c_{b_{2}}^{\prime}>0$ and
$k_{\delta}^{\prime}:=k_{\delta}^{\prime}(b_{2})>0$ such that for any
$k>k_{\delta}^{\prime}:$

  * (i)

if $Q_{r,\nu}(x_{0})\subset\subset\Omega,$ then

$\int_{Q_{r,\nu}(x_{0})\cap\partial^{*}A_{k}}\phi(\nu_{A_{k}})\mathrm{d}\mathcal{H}^{n-1}+2\int_{Q_{r,\nu}(x_{0})\cap
A_{k}^{(1)}\cap J_{u_{k}}}\phi(\nu_{J_{u_{k}}})\mathrm{d}\mathcal{H}^{n-1}\\\
\geq 2\int_{K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}-c^{\prime}\delta
r^{n-1};$

  * (ii)

if $x_{0}\in\Sigma$ and $\Gamma=Q_{r,\nu}(x_{0})\cap\Sigma,$ then

$\int_{Q_{r,\nu}(x_{0})\cap\partial^{*}A_{k}}\phi(\nu_{A_{k}})\mathrm{d}\mathcal{H}^{n-1}+2\int_{Q_{r,\nu}(x_{0})\cap
A_{k}^{(1)}\cap J_{u_{k}}}\phi(\nu_{J_{u_{k}}})\mathrm{d}\mathcal{H}^{n-1}\\\
+2\int_{Q_{r,\nu}(x_{0})\cap\Sigma\cap\partial^{*}A_{k}\cap
J_{u_{k}}}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}\geq
2\int_{K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}-c^{\prime}\delta r^{n-1}.$

###### Proof of Theorem 2.5.

In view of Remark 2.1 we may assume that $u_{k}=\xi$ for some
$\xi\in(0,1)^{n}\setminus\Xi_{\\{(A_{k},u_{k})\\}}.$ Moreover, there is no
loss of generality in assuming $liminf$ in (2.14) is a finite limit. Thus,

$M:=\sup\limits_{k\geq 1}\,\mathcal{F}(A_{k},u_{k})<+\infty.$

In particular, $\\{(A_{k},u_{k})\\}$ satisfies the assumptions (a3) and (a4)
of Proposition 4.1.

Let

$\displaystyle\mu_{k}(B):=$
$\displaystyle\int_{B\cap\Omega\cap\partial^{*}A_{k}}\varphi(x,\nu_{A_{k}})\,\mathrm{d}\mathcal{H}^{n-1}+2\int_{B\cap
A_{k}^{(1)}\cap
J_{u_{k}}}\varphi(x,\nu_{J_{u_{k}}})\,\mathrm{d}\mathcal{H}^{n-1}$
$\displaystyle+\int_{B\cap\Sigma\cap\partial^{*}A_{k}\setminus
J_{u_{k}}}[\beta+\varphi(x,\nu_{\Sigma})]\,\mathrm{d}\mathcal{H}^{n-1}+2\int_{B\cap\Sigma\cap\partial^{*}A_{k}\cap
J_{u_{k}}}\varphi(x,\nu_{\Sigma})\,\mathrm{d}\mathcal{H}^{n-1}$
$\displaystyle+\int_{B\cap\Sigma\setminus\partial^{*}A_{k}}\varphi(x,\nu_{\Sigma})\,\mathrm{d}\mathcal{H}^{n-1}+\int_{B\cap(A\cup
S)}W(x,\mathcal{E}u_{k}-\mathbf{M}_{0})\mathrm{d}x$

and

$\displaystyle\mu(B):=$
$\displaystyle\int_{B\cap\Omega\cap\partial^{*}A}\varphi(x,\nu_{A})\,\mathrm{d}\mathcal{H}^{n-1}+2\int_{B\cap
A^{(1)}\cap J_{u}}\varphi(x,\nu_{J_{u}})\,\mathrm{d}\mathcal{H}^{n-1}$
$\displaystyle+\int_{B\cap\Sigma\cap\partial^{*}A\setminus
J_{u}}[\beta+\varphi(x,\nu_{\Sigma})]\,\mathrm{d}\mathcal{H}^{n-1}+2\int_{B\cap\Sigma\cap\partial^{*}A\cap
J_{u}}\varphi(x,\nu_{\Sigma})\,\mathrm{d}\mathcal{H}^{n-1}$
$\displaystyle+\int_{B\cap\Sigma\setminus\partial^{*}A}\varphi(x,\nu_{\Sigma})\,\mathrm{d}\mathcal{H}^{n-1}+\int_{B\cap(A\cup
S)}W(x,\mathcal{E}u-\mathbf{M}_{0})\mathrm{d}x$

be positive Radon measures in $\mathbb{R}^{n}.$ Notice that

$\mu_{k}(\mathbb{R}^{n})=\mathcal{F}(A_{k},u_{k})+\int_{\Sigma}\varphi(x,\nu_{\Sigma})\,\mathrm{d}\mathcal{H}^{n-1}$
(4.4)

and

$\mu(\mathbb{R}^{n})=\mathcal{F}(A,u)+\int_{\Sigma}\varphi(x,\nu_{\Sigma})\,\mathrm{d}\mathcal{H}^{n-1}.$
(4.5)

In particular,

$\sup\limits_{k\geq 1}\mu_{k}(\mathbb{R}^{n})\leq
M+\int_{\Sigma}\varphi(x,\nu_{\Sigma})\,\mathrm{d}\mathcal{H}^{n-1},$

and thus, there exist a positive Radon measure $\mu_{0}$ in $\mathbb{R}^{n}$
and a not relabelled subsequence $\\{\mu_{k}\\}$ such that
$\mu_{k}\rightharpoonup^{*}\mu_{0}.$ Let us show

$\mu_{0}\geq\mu.$ (4.6)

Note that (2.14) directly follows from (4.6), (4.4), (4.5). By the
nonnegativity of $\mu$ and $\mu_{0},$ and the explicit form of the support of
$\mu,$ to establish (4.6) it suffices to prove the following density
estimates:

$\displaystyle\frac{\mathrm{d}\mu_{0}}{\mathrm{d}\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits[\Omega\cap\partial^{*}A]}\,(x)\geq\varphi(x,\nu_{A}(x))\quad\text{$\mathcal{H}^{n-1}$-a.e.\
$x\in(\Omega\cap\partial^{*}A)\cup(\Sigma\setminus\partial^{*}A),$}$ (4.7a)
$\displaystyle\frac{\mathrm{d}\mu_{0}}{\mathrm{d}\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits[A^{(1)}\cap J_{u}]}\,(x)\geq
2\varphi(x,\nu_{J_{u}}(x))\quad\text{$\mathcal{H}^{n-1}$-a.e.\ $x\in
A^{(1)}\cap J_{u},$}$ (4.7b)
$\displaystyle\frac{\mathrm{d}\mu_{0}}{\mathrm{d}\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits[\Sigma\cap\partial^{*}A\cap
J_{u}]}\,(x)\geq
2\varphi(x,\nu_{\Sigma}(x))\quad\text{$\mathcal{H}^{n-1}$-a.e.\
$x\in\Sigma\cap\partial^{*}A\cap J_{u},$}$ (4.7c)
$\displaystyle\frac{\mathrm{d}\mu_{0}}{\mathrm{d}\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits[\Sigma\cap\partial^{*}A]}\,(x)\geq\beta(x)+\varphi(x,\nu_{\Sigma}(x))\quad\text{$\mathcal{H}^{n-1}$-a.e.\
$x\in\Sigma\cap\partial^{*}A,$}$ (4.7d)
$\displaystyle\frac{\mathrm{d}\mu_{0}}{\mathrm{d}\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits[\Sigma\setminus\partial^{*}A]}\,(x)\geq\varphi(x,\nu_{\Sigma}(x))\quad\text{$\mathcal{H}^{n-1}$-a.e.\
$x\in\Sigma\setminus\partial^{*}A,$}$ (4.7e)
$\displaystyle\frac{\mathrm{d}\mu_{0}}{\mathrm{d}\mathcal{L}^{n}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits[A\cup S]}\,(x)\geq
W(x,\mathcal{E}u(x)-\mathbf{M}_{0}(x))\quad\text{$\mathcal{L}^{n}$-a.e.\ $x\in
A\cup S.$}$ (4.7f)

Proofs of (4.7a), (4.7d) and (4.7e). By assumptions (H1)-(H3), the capillary
functional

$\mathcal{C}(E;U)=\int_{U\cap\partial^{*}E}\varphi(x,\nu_{E})\mathrm{d}\mathcal{H}^{n-1}+\int_{U\cap\Sigma\cap\partial^{*}E}[\beta+\varphi(x,\nu_{\Sigma})]\,\mathrm{d}\mathcal{H}^{n-1}+\int_{U\cap\Sigma\setminus\partial^{*}E}\varphi(x,\nu_{\Sigma})\,\mathrm{d}\mathcal{H}^{n-1}$

is $L^{1}(U)$-lowersemicontinuous in any open set $U\subset\mathbb{R}^{n}$
(see e.g., [1, Theorem 3.4]). As $A_{k}\to A$ and
$\mu_{k}\rightharpoonup^{*}\mu_{0},$ for any ball $B_{r}(x_{0})$ with
$\mu_{0}(\partial B_{r}(x_{0}))=0$ we have

$\displaystyle\mu_{0}(B_{r}(x_{0}))=$
$\displaystyle\lim\limits_{k\to+\infty}\mu_{k}(B_{r}(x_{0}))\geq\liminf\limits_{k\to+\infty}\mathcal{C}(A_{k},B_{r}(x_{0}))\geq\mathcal{C}(A,B_{r}(x_{0})).$

This inequality and the Besicovitch differentiation theorem imply (4.7a),
(4.7d) and (4.7e).

Proof of (4.7b). Fix $\epsilon\in(0,2^{-10})$ and let $K:=A^{(1)}\cap J_{u}.$
By the $\mathcal{H}^{n-1}$-rectifiability of $K$, there exists an at most
countable family $\\{\Gamma_{l}\\}$ of $(n-1)$-dimensional $C^{1}$-graphs such
that

$\mathcal{H}^{n-1}\Big{(}K\setminus\bigcup_{l\geq 1}\Gamma_{l}\Big{)}=0.$

Let $x_{0}\in L$ be such that

  * ($\rm a_{1}$)

$x_{0}\in\Gamma_{l}$ for some $l\geq 1$ so that the generalized unit normal
$\nu_{0}:=\nu_{K}(x_{0})$ to $L$ at $x_{0}$ exists and equals to
$\nu_{\Gamma_{l}}(x_{0})$;

  * ($\rm a_{2}$)

$\theta(K,x_{0})=\theta(\Gamma_{l}\cap K,x_{0})=1;$

  * ($\rm a_{3}$)

$\frac{\mathrm{d}\mu_{0}}{\mathrm{d}\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits K}(x_{0})$ exists;

  * ($\rm a_{4}$)

$\lim\limits_{r\to 0}\frac{1}{r^{n-1}}\int_{Q_{r,\nu_{0}}(x_{0})\cap
K}\varphi(x_{0},\nu_{K})\mathrm{d}\mathcal{H}^{n-1}=\varphi(x_{0},\nu_{0}).$

By the $\mathcal{H}^{n-1}$-rectifiability of $K,$ [3, Theorem 2.63] and
Lebesgue-Besicovitch differentiation theorem, the set of $x_{0}\in K$ for
which at least one of these conditions fails is
$\mathcal{H}^{n-1}$-negligible. Since $\varphi$ is uniformly continuous in
$\overline{\Omega},$ there exists $r_{1,\epsilon}>0$ such that

$|\varphi(x,\nu)-\varphi(y,\nu)|<\epsilon\quad\text{whenever
$|x-y|<r_{1,\epsilon}$ and $\nu\in\mathbb{S}^{n-1}.$}$ (4.8)

Decreasing $r_{1,\epsilon}$ if necessary, we assume that
$Q_{r_{1,\epsilon},\nu_{0}}(x_{0})\subset\subset\Omega.$ Then for any
$r\in(0,r_{1,\epsilon})$

$\mu_{k}(Q_{r,\nu_{0}}(x_{0}))\geq\alpha_{k}(Q_{r,\nu_{0}}(x_{0}))\\\
-\epsilon\Big{(}\mathcal{H}^{n-1}(Q_{r,\nu_{0}}(x_{0})\cap\partial^{*}A_{k})+2\mathcal{H}^{n-1}(Q_{r,\nu_{0}}(x_{0})\cap
A_{k}^{(1)}\cap J_{u_{k}})\Big{)},$ (4.9)

where

$\alpha_{k}(U):=\int_{U\cap\partial^{*}A_{k}}\phi(\nu_{A_{k}})\mathrm{d}\mathcal{H}^{n-1}+2\int_{U\cap
A_{k}^{(1)}\cap J_{u_{k}}}\phi(\nu_{J_{u_{k}}})\mathrm{d}\mathcal{H}^{n-1}$

and $\phi(\nu):=\varphi(x_{0},\nu).$ By assumption (2.8) and the nonnegativity
of the summands of $\mu_{k}$ we have an a priori bound

$\mathcal{H}^{n-1}(Q_{r,\nu_{0}}(x_{0})\cap\partial^{*}A_{k})+2\mathcal{H}^{n-1}(Q_{r,\nu_{0}}(x_{0})\cap
A_{k}^{(1)}\cap J_{u_{k}})\leq\frac{\mu(Q_{r,\nu_{0}}(x_{0}))}{b_{1}},$

and thus, inserting this in (4.9) we get

$\Big{(}1+\frac{\epsilon}{b_{1}}\Big{)}\mu_{k}(Q_{r,\nu_{0}}(x_{0}))\geq\alpha_{k}(Q_{r,\nu_{0}}(x_{0})).$
(4.10)

Now we estimate $\alpha_{k}$ from below using Corollary 4.2 (a). Since
$\Gamma_{l}$ is a $C^{1}$-graph, by ($a_{1}$) there exists
$r_{2,\epsilon}\in(0,r_{1,\epsilon})$ such that

  * •

$\Gamma_{l}$ divides the cube $Q_{r_{2,\epsilon},\nu_{0}}(x_{0})$ into two
connected components;

  * •

$|\nu_{\Gamma_{l}}(x)-\nu_{0}|<\epsilon$ for any $x\in
Q_{r_{2,\epsilon},\nu_{0}}(x_{0})\cap\Gamma_{l};$

  * •

$|(x-x_{0})\cdot\nu_{0}|<\epsilon r/2$ for any $r\in(0,r_{2,\epsilon})$ and
$x\in Q_{r,\nu_{0}}(x_{0})\cap\Gamma_{l};$

  * •

$\mathcal{H}^{n-1}(Q_{r,\nu_{0}}(x_{0})\cap\Gamma_{l})<(1+\epsilon)r^{n-1}$
for all $r\in(0,r_{2,\epsilon}).$

In particular, for any $r\in(0,r_{2,\epsilon})$ the cube
$Q_{r,\nu_{0}}(x_{0})$ and the $C^{1}$-graph
$\Gamma:=Q_{r,\nu_{0}}(x_{0})\cap\Gamma_{l}$ satisfy the assumptions (a1)-(a2)
of Proposition 4.1. As we mentioned in the beginning of the proof,
$\\{(A_{k},u_{k})\\}$ satisfies (a3)-(a4) of Proposition 4.1. Moreover, by
assumptions $x_{0}\in A^{(1)}$ and ($a_{2}$) there exists
$r_{3,\epsilon}\in(0,r_{2,\epsilon})$ such that

  * •

$P(A,Q_{r,\nu_{0}}(x_{0}))<\epsilon r^{n-1}$ and $|A\cap
Q_{r,\nu_{0}}(x_{0})|>(1-\epsilon)r^{n-1}$ for all $r\in(0,r_{3,\epsilon});$

  * •

$\mathcal{H}^{n-1}(Q_{r,\nu_{0}}(x_{0})\cap
K\cap\Gamma_{l})>(1-\epsilon)r^{n-1}$ for any $r\in(0,r_{3,\epsilon});$

  * •

$\mathcal{H}^{n-1}(Q_{r,\nu_{0}}(x_{0})\cap K\setminus\Gamma_{l})<\delta
r^{n-1}$ for any $r\in(0,r_{3,\epsilon}).$

Thus, assumptions (a5)-(a7) of Proposition 4.1 also hold. Therefore, by
Corollary 4.2 (i) there exists $k_{\epsilon}>0$ and $c^{\prime}>0$ such that

$\alpha_{k}(Q_{r,\nu_{0}}(x_{0}))\geq 2\int_{Q_{r,\nu_{0}}(x_{0})\cap
K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}-c^{\prime}\epsilon r^{n-1}$

for all $k>k_{\epsilon}.$ This and (4.10) yield

$\Big{(}1+\frac{\epsilon}{b_{1}}\Big{)}\mu_{k}(Q_{r,\nu_{0}}(x_{0}))\geq
2\int_{Q_{r,\nu_{0}}(x_{0})\cap
K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}-c^{\prime}\epsilon r^{n-1}.$

Now letting $k\to+\infty$ for a.e. $r\in(0,r_{3,\epsilon})$ we get

$\Big{(}1+\frac{\epsilon}{b_{1}}\Big{)}\mu_{0}(Q_{r,\nu_{0}}(x_{0}))\geq
2\int_{Q_{r,\nu_{0}}(x_{0})\cap
K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}-c^{\prime}\epsilon r^{n-1}.$

Therefore, by ($a_{3}$) and ($a_{4}$)

$\Big{(}1+\frac{\epsilon}{b_{1}}\Big{)}\frac{\mathrm{d}\mu_{0}}{\mathrm{d}\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits
K}(x_{0})=\Big{(}1+\frac{\epsilon}{b_{1}}\Big{)}\lim\limits_{r\to
0^{+}}\frac{\mu_{0}(Q_{r,\nu}(x_{0}))}{r^{n-1}}\geq
2\varphi(x_{0},\nu_{0})-c^{\prime}\epsilon.$

Now letting $\epsilon\to 0$ we obtain (4.7b).

Proof of (4.7c). Let $\epsilon\in(0,2^{-10})$ and let
$L:=\Sigma\cap\partial^{*}A\cap J_{u}.$ Since $\Sigma$ is Lipschitz, $L$ is
$\mathcal{H}^{n-1}$-rectifiable.

Let $x_{0}\in L$ be such that

  * ($\rm b_{1}$)

$\nu_{0}:=\nu_{\Sigma}(x_{0})$ exist and equals to $\nu_{L}(x_{0});$

  * ($\rm b_{2}$)

$\theta(L,x_{0})=\theta(\Sigma,x_{0})=\theta(\partial^{*}A,x_{0})=1;$

  * ($\rm b_{3}$)

$\frac{\mathrm{d}\mu_{0}}{\mathrm{d}\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits L}(x_{0})$ exists.

  * ($\rm b_{4}$)

$\lim\limits_{r\to 0}\frac{1}{r^{n-1}}\int_{Q_{r,\nu_{0}}(x_{0})\cap
L}\varphi(x_{0},\nu_{J_{u}})\mathrm{d}\mathcal{H}^{n-1}=\varphi(x_{0},\nu_{0}).$

By the lipschitzianity of $\Sigma,$ $\mathcal{H}^{n-1}$-rectifiablity of
$\partial^{*}A,$ [3, Theorem 2.63] and Besicovitch differentiation theorem,
the set of $x_{0}\in L$ for which at least one of these conditions fails is
$\mathcal{H}^{n-1}$-negligible.

Let $r_{1,\epsilon}>0$ be such that (4.8) holds and
$Q_{r_{1,\epsilon},\nu_{0}}(x_{0})\subset\subset{\rm Int}{(\Omega\cup
S\cup\Sigma)}.$ Then as in (4.10)

$\displaystyle\Big{(}1+\frac{\epsilon}{b_{1}}\Big{)}\mu_{k}(Q_{r,\nu_{0}}(x_{0}))\geq\gamma_{k}(Q_{r,\nu_{0}}(x_{0}))$

for any $r\in(0,r_{1,\epsilon}),$ where

$\gamma_{k}(U):=\int_{U\cap\Omega\cap\partial^{*}A_{k}}\phi(\nu_{A_{k}})\mathrm{d}\mathcal{H}^{n-1}+2\int_{U\cap
A_{k}^{(1)}\cap
J_{u_{k}}}\phi(\nu_{J_{u_{k}}})\mathrm{d}\mathcal{H}^{n-1}+2\int_{U\cap\Sigma\cap\partial^{*}A_{k}\cap
J_{u_{k}}}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}.$

Since $\Sigma$ is Lipschitz continuous, by ($\rm b_{1}$) and ($\rm b_{2}$)
there exists $r_{2,\epsilon}\in(0,r_{1,\epsilon})$ such that

  * •

$\Sigma$ divides the cube $Q_{r_{2,\epsilon},\nu_{0}}(x_{0})$ into two
connected components;

  * •

$|\nu_{\Sigma}(x)-\nu_{\Sigma}(x_{0})|<\epsilon$ for any $x\in
Q_{r_{2,\epsilon},\nu_{0}}(x_{0})\cap\Sigma;$

  * •

$|(x-x_{0})\cdot\nu_{0}|<\epsilon r/2$ for any $r\in(0,r_{2,\epsilon})$ and
$x\in Q_{r,\nu_{0}}(x_{0})\cap\Sigma;$

  * •

$\mathcal{H}^{n-1}(Q_{r,\nu_{0}}(x_{0})\cap\Sigma)<(1+\epsilon)r^{n-1}$ for
all $r\in(0,r_{2,\epsilon}).$

Moreover, since $x_{0}\in\Sigma\cap\partial^{*}A$ and
$\theta(L,x_{0})=\theta(\partial^{*}A,x_{0})=1,$ there exists
$r_{3,\epsilon}\in(0,r_{2,\epsilon})$ such that

  * •

$\mathcal{H}^{n-1}(Q_{r,\nu_{0}}(x_{0})\cap\Sigma\cap\partial^{*}A)>(1-\epsilon)r^{n-1}$
and
$\mathcal{H}^{n-1}(Q_{r,\nu_{0}}(x_{0})\cap\partial^{*}A\setminus\Sigma)<\delta
r^{n-1}.$

Thus, applying Corollary 4.2 (b) we find $k_{\epsilon}^{\prime\prime}>0$ and
$c^{\prime\prime}>0$ such that

$\gamma_{k}(Q_{r,\nu_{0}}(x_{0}))\geq 2\int_{Q_{r,\nu_{0}}(x_{0})\cap
L}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}-c^{\prime\prime}\delta
r^{n-1}$

for all $k>k_{\epsilon}^{\prime\prime}.$ Therefore,

$\Big{(}1+\frac{\epsilon}{b_{1}}\Big{)}\mu_{k}(Q_{r,\nu_{0}}(x_{0}))\geq
2\int_{Q_{r,\nu_{0}}(x_{0})\cap
L}\phi(\nu_{\Sigma})\mathrm{d}\mathcal{H}^{n-1}-c^{\prime\prime}\delta
r^{n-1}$

and hence, by ($\rm b_{3}$) and ($\rm b_{4}$)

$\frac{\mathrm{d}\mu_{0}}{\mathrm{d}\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=9.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=5.0pt,depth=0.0pt}}\nolimits L}(x_{0})\geq
2\varphi(x_{0},\nu_{0}).$

Proof of (4.7f). By the nonnegativity of $\mu_{k}$ and our assumption
$u_{k}=\xi$ on $\Omega\setminus A_{k}$

$\displaystyle\mu_{k}(B_{r}(x))\geq$
$\displaystyle\int_{B_{r}(x)\cap(A_{k}\cup
S)}W(y,\mathcal{E}u_{k}-\mathbf{M}_{0})\mathrm{d}y$ $\displaystyle=$
$\displaystyle\int_{B_{r}(x)\cap(\Omega\cup
S)}W(y,\mathcal{E}u_{k}-\mathbf{M}_{0})\mathrm{d}y-\int_{B_{r}(x)\cap(\Omega\setminus
A_{k})}W(y,-\mathbf{M}_{0})\mathrm{d}y.$ (4.11)

Since $\mu_{k}\rightharpoonup^{*}\mu_{0},$
$\mathcal{E}u_{k}\rightharpoonup\mathcal{E}u$ in $L^{2}(\Omega\cup S)$ (see
(2.11)) and $A_{k}\to A$ in $L^{1}(\mathbb{R}^{n}),$ letting $k\to+\infty$ in
(4) for any ball $B_{r}(x)$ with $\mu_{0}(\partial B_{r}(x))=0,$ we get

$\displaystyle\mu_{0}(B_{r}(x))=$
$\displaystyle\lim\limits_{k\to+\infty}\mu_{k}(B_{r}(x))$ $\displaystyle\geq$
$\displaystyle\int_{B_{r}(x)\cap(\Omega\cup
S)}W(y,\mathcal{E}u-\mathbf{M}_{0})\mathrm{d}y-\int_{B_{r}(x)\cap(\Omega\setminus
A)}W(y,-\mathbf{M}_{0})\mathrm{d}y$ $\displaystyle=$
$\displaystyle\int_{B_{r}(x)\cap(A\cup
S)}W(y,\mathcal{E}u-\mathbf{M}_{0})\mathrm{d}y,$

where in the equality we used $u=\xi$ in $\Omega\setminus A.$ Now (4.7f)
follows from the Besicovitch differentiation theorem. ∎

######  Remark 4.3.

According to the proof of Theorem 2.5 both $\mathcal{S}$ and $\mathcal{W}$ are
$\tau_{\mathcal{C}}$-lower semicontinuous in $\mathcal{C}.$

Now we prove bounds ((i))-((ii)).

###### Proof of Proposition 4.1.

We only prove (i). The last inequality in ((i)) directly follows from
(3.1)-(3.3). Therefore, we establish only the first estimate. Without loss of
generality, we assume $x_{0}=0,$ $r=1$ and $\nu={\bf e}_{n}.$ By (a1)
$\Gamma\subset(-\frac{1}{2},\frac{1}{2})^{n-1}\times(-\frac{\delta}{2},\frac{\delta}{2}),$
by (a3) and a priori estimates in Remark 2.3

$M_{1}:=\sup\limits_{k\geq 1}\Big{(}\int_{\Omega\cup
S}|\mathcal{E}u_{k}|^{2}\mathrm{d}x+\mathcal{H}^{n-1}(J_{u_{k}})\Big{)}<+\infty.$
(4.12)

We prove ((i)) for $K=Q_{1}\cap\partial^{*}E$ (i.e., in the case
$|u_{k}|\to+\infty$ a.e. in $Q_{1}\cap E$); the other case being similar. For
any open set $G\subset Q_{1}$ define

$\alpha_{k}(G):=\int_{G\cap\partial^{*}A_{k}}\phi(\nu_{A_{k}})\mathrm{d}\mathcal{H}^{n-1}+2\int_{G\cap
A_{k}^{(1)}\cap J_{u_{k}}}\phi(\nu_{J_{u_{k}}})\mathrm{d}\mathcal{H}^{n-1}.$

Step 1. Let

$\Upsilon:=\\{\xi\in\mathbb{S}^{n-1}:\,\,|\xi\cdot e_{n}|\geq 2\delta\\}.$

Then by (a1) for any $\xi\in\Upsilon$ and $x\in Q_{1}\cap\Gamma$

$|\xi\cdot\nu_{\Gamma}(x)|\geq|\xi\cdot{\bf
e}_{n}|-|\xi\cdot(\nu_{\Gamma}(x)-{\bf e}_{n})|>\delta,$

and hence, $Q_{1}\cap\Gamma$ is a graph also in $\xi$-direction, i.e., for any
$y\in\Pi_{\xi}$ the line $\pi_{\xi}^{-1}(y)$ intersects $Q_{1}\cap\Gamma$ at
most at one point.

Step 2. Let $D$ be given by Lemma 3.1 and let $U\subset\subset D$ be any open
set such that $U\cap\Gamma\cap K\neq\emptyset.$ Let also
$(B_{k}^{U},v_{k}^{U})$ be given by Corollary 3.3 applied with $U,$
$\delta=\frac{|U|}{k}$ and $(A_{k},u_{k})$. Then for all $k:$

  * ($\rm a_{1}$)

$B_{k}^{U}\subset A_{k},$ $A_{k}\setminus B_{k}^{U}\subset\subset U$ and
$|A_{k}\setminus B_{k}^{U}|<1/k;$

  * ($\rm a_{2}$)

$v_{k}^{U}=u_{k}$ in $B_{k}^{U}\cup S;$

  * ($\rm a_{3}$)

$\mathcal{H}^{n-1}(U\cap[B_{k}^{U}]^{(1)}\cap J_{v_{k}^{U}})<1/k;$

  * ($a_{4}$)

$\alpha_{k}(U)+|U|/k\geq\Lambda_{k}(U),$ where

$\Lambda_{k}(U):=\int_{U\cap\partial^{*}B_{k}^{U}}\phi(\nu_{B_{k}^{U}})\mathrm{d}\mathcal{H}^{n-1}.$

By ($\rm a_{1}$) $B_{k}^{U}\to A$ in $L^{1}(\mathbb{R}^{n})$, and by ($\rm
a_{1}$), ($\rm a_{2}$) and also (a6)

$v_{k}^{U}\to u\,\,\,\,\text{a.e.\ in $U\setminus
E$}\qquad\text{and}\qquad|v_{k}^{U}|\to+\infty\,\,\,\,\text{a.e.\ in $U\cap
E.$}$ (4.13)

Moreover, by (4.12) and ($\rm a_{2}$)

$\sup_{k\geq
1}\Big{(}\int_{U}|\mathcal{E}v_{k}^{U}|^{2}\mathrm{d}x+\mathcal{H}^{n-1}(U\cap
J_{v_{k}^{U}})\Big{)}<+\infty.$

We claim that

$\displaystyle\liminf\limits_{k\to+\infty}\,\Lambda_{k}(U)\geq$
$\displaystyle\frac{2}{\phi^{o}(\xi)}\,\int_{U\cap\Gamma}|\nu_{\Gamma}\cdot\xi|\mathrm{d}\mathcal{H}^{n-1}-2b_{2}P(A,U)-2b_{2}\mathcal{H}^{n-1}(U\cap[\Gamma\setminus\partial^{*}E]).$
(4.14)

for $\mathcal{H}^{n-1}$-a.e. $\xi\in\Upsilon.$

To prove (4.14) we study some properties of one-dimensional slices
$[\widehat{v}_{k}^{U}]_{y}^{\xi}$ of $v_{k}^{U}.$ We closely follow the
arguments of [14, pp. 11-13]; see also [15]. Let $k_{j}:=k_{j}^{U}$ be such
that

$\liminf\limits_{k\to+\infty}\int_{U\cap
J_{v_{k}^{U}}}\phi(\nu_{J_{v_{k}^{U}}})\mathrm{d}\mathcal{H}^{n-1}=\lim\limits_{j\to+\infty}\int_{U\cap
J_{v_{k_{j}}^{U}}}\phi(\nu_{J_{v_{k_{j}}^{U}}})\mathrm{d}\mathcal{H}^{n-1}.$

Applying (2.3) and (2.4) with $v=v_{k_{j}}^{U},$ (2.5) with
$L=J_{v_{k_{j}}^{U}},$ and using (4.12) we find

$\liminf\limits_{j\to+\infty}\int_{\Pi_{\xi}}\Big{[}\mathcal{H}^{0}(J_{[\widehat{v}_{k_{j}}^{U}]_{y}^{\xi}})+\kappa
I_{y,\xi}^{U}(v_{k_{j}}^{U})+\kappa
II_{y,\xi}^{U}(v_{k_{j}}^{U})\Big{]}\mathrm{d}\mathcal{H}^{n-1}(y)<+\infty$
(4.15)

for any $\kappa>0$ and $\mathcal{H}^{n-1}$-a.e. $\xi\in\Upsilon.$ Moreover, by
[14, Lemma 2.7] and (4.13)

$|v_{k}^{U}\cdot\xi|\to+\infty\quad\text{a.e.\ in $U\cap E$}$ (4.16)

for $\mathcal{H}^{n-1}$-a.e. $\xi\in\Upsilon.$ Fix any $\xi\in\Upsilon$
satisfying (4.15) and (4.16) and consider the one-dimensional slices
$[\widehat{v}_{k_{j}}^{U}]_{y}^{\xi}$ and $\widehat{u}_{y}^{\xi}.$ In view of
(4.15) and Fatou’s lemma, for $\mathcal{H}^{n-1}$-a.e. $y\in\pi_{\xi}(U)$

$\liminf\limits_{k\to+\infty}\Big{[}\mathcal{H}^{0}(J_{[\widehat{v}_{k_{j}}^{U}]_{y}^{\xi}})+\kappa
I_{y,\xi}^{U}(v_{k_{j}}^{U})+\kappa
II_{y,\xi}^{U}(v_{k_{j}}^{U})\Big{]}<+\infty.$

Thus, for $\mathcal{H}^{n-1}$-a.e. $y\in\pi_{\xi}(U)$ there exists a
subsequence $\\{k_{j}^{y}\\}\subset\\{k_{j}\\}$ (depending also on $\kappa>0$)
such that

$\liminf\limits_{j\to+\infty}\Big{[}\mathcal{H}^{0}(J_{[\widehat{v}_{k_{j}}^{U}]_{y}^{\xi}})+\kappa
I_{y,\xi}^{U}(v_{k_{j}}^{U})+\kappa II_{y,\xi}^{U}(v_{k_{j}}^{U})\Big{]}\\\
=\lim\limits_{j\to+\infty}\Big{[}\mathcal{H}^{0}(J_{[\widehat{v}_{k_{j}^{y}}^{U}]_{y}^{\xi}})+\kappa
I_{y,\xi}^{U}(v_{k_{j}^{y}}^{U})+\kappa
II_{y,\xi}^{U}(v_{k_{j}^{y}}^{U})\Big{]},$ (4.17)

and by (4.13) and (4.16)

$[\widehat{v}_{k_{j}^{y}}^{U}]_{y}^{\xi}\to\widehat{u}_{y}^{\xi}\quad\text{$\mathcal{L}^{1}$-a.e.\
in $[U\setminus
E]_{y}^{\xi}$}\quad\text{and}\quad\Big{|}[\widehat{v}_{k_{j}^{y}}^{U}]_{y}^{\xi}\Big{|}\to+\infty\quad\text{$\mathcal{L}^{1}$-a.e.\
in $[U\cap E]_{y}^{\xi}.$}$ (4.18)

For $\tau(t)=\arctan(t),$ set
$f_{j}:=\tau\circ[\widehat{v}_{k_{j}^{y}}^{U}]_{y}^{\xi}.$ Then $f_{j}\in
SBV_{\mathrm{loc}}^{2}(U_{y}^{\xi})$ and
$J_{[\widehat{v}_{k_{j}^{y}}^{U}]_{y}^{\xi}}=J_{f_{j}}.$ By (4.17), (4.18) and
[2, Proposition 4.2] we find a not relabelled subsequence
$\\{v_{k_{j}^{y}}^{U}\\}$ such that

$f_{j}\to f_{0}\quad\text{$\mathcal{L}^{1}$-a.e.\ in $U_{y}^{\xi}$ as
$j\to+\infty.$}$

By (4.18)

$\begin{cases}f_{0}=\tau\circ\widehat{u}_{y}^{U}&\text{in $[U\setminus
E]_{y}^{\xi},$}\\\ |f_{0}|=\pi/2&\text{in $[U\cap E]_{y}^{\xi}.$}\end{cases}$

By [2, Proposition 4.2]

$\liminf\limits_{j\to+\infty}\mathcal{H}^{0}\Big{(}J_{[\widehat{v}_{k_{j}^{y}}^{U}]_{y}^{\xi}}\Big{)}=\liminf\limits_{j\to+\infty}\mathcal{H}^{0}(J_{f_{j}})\geq\mathcal{H}^{0}(J_{f_{0}}).$
(4.19)

Thus, $\mathcal{H}^{0}(U_{y}^{\xi}\cap J_{f_{0}})<+\infty$ and hence, $[U\cap
E]_{y}^{\xi}$ consists of finitely many segments in each of which either
$f_{0}\equiv\pi/2$ or $f_{0}\equiv-\pi/2.$

By (4.17) $\mathcal{H}^{0}(J_{f_{j}})$ is uniformly bounded and hence, there
exists a further not relabelled subsequence and $N_{y}\in\mathbb{N}_{0}$ such
that

$\mathcal{H}^{0}(J_{f_{j}})=N_{y}\quad\text{and}\quad
J_{f_{j}}=\\{t_{j}^{1},\ldots,t_{j}^{N_{y}}\\}\subset
U_{y}^{\xi}\quad\text{for all $j.$}$

Then points of $J_{f_{j}}$ converges to $M_{y}\leq N_{y}$ points
$t^{1}<\ldots<t^{M_{y}}.$ Since $II_{y,\xi}^{U}(v_{k_{j}^{y}}^{U})$ is
uniformly bounded, the precise representatives of $f_{j}$ uniformly bounded in
$W_{\mathrm{loc}}^{1,1}(t^{l},t^{l+1})$ so that $f_{j}\to f_{0}$ locally
uniformly in $(t^{l},t^{l+1})$ and
$J_{f_{0}}\subset\\{t^{1},\ldots,t^{M_{y}}\\}.$ Repeating the arguments of
[14, Section 1] we can show that
$t^{1}:=U_{y}^{\xi}\cap[\partial^{*}E]_{y}^{\xi}\in J_{f_{0}}.$

Let us estimate the $\mathcal{H}^{n-1}$-measures of the sets

$\displaystyle Y_{0}:=$ $\displaystyle\\{y\in\Pi_{\xi}\cap\pi_{\xi}(U\cap
K):\,\,N_{y}=0\\},$ $\displaystyle Y_{1}:=$
$\displaystyle\\{y\in\Pi_{\xi}\cap\pi_{\xi}(U\cap K):\,\,N_{y}=1\\},$
$\displaystyle Y_{2}:=$ $\displaystyle\\{y\in\Pi_{\xi}\cap\pi_{\xi}(U\cap
K):\,\,N_{y}\geq 2\\}.$

By (4.19) $\mathcal{H}^{0}(J_{f_{0}})=0$ for any $y\in Y_{0}.$ Hence,
$U\cap\pi_{\xi}^{-1}(y)\cap(\partial^{*}E\cup J_{u})=\emptyset,$ and therefore
$Y_{0}\subset\pi_{\xi}(U\cap\Gamma\setminus\partial^{*}E).$ Then by the
$1$-Lipschitz continuity of the projection $\pi_{\xi}$

$\mathcal{H}^{n-1}(Y_{0})\leq\mathcal{H}^{n-1}(\pi_{\xi}(U\cap[\Gamma\setminus\partial^{*}E]))\leq\mathcal{H}^{n-1}(U\cap[\Gamma\setminus\partial^{*}E]).$
(4.20)

Now consider any $y\in Y_{1}.$ By definition $\pi_{\xi}^{-1}(y)$ intersects
$U\cap J_{v_{k_{j}^{y}}^{U}}$ just once and therefore, by the construction of
$(B_{k}^{U},v_{k}^{U})$ (see the proof of Corollary 3.3) either
$y\in\pi_{\xi}\Big{(}U\cap[B_{k_{j}^{y}}^{U}]^{(1)}\cap J_{u_{k_{j}^{y}}}\cap
J_{v_{k_{j}^{y}}^{U}}\Big{)}$ or $y\in\pi_{\xi}(U\setminus
B_{k_{j}^{y}}^{U}).$ If $y\in\pi_{\xi}(U\setminus B_{k_{j}^{y}}^{U}),$ then
$t_{j}^{1}$ divides the line $U\cap\pi_{\xi}^{-1}(y)$ into two parts one is a
subset of $U\cap B_{k_{j}^{y}}^{U}$ and the other is that of $U\setminus
B_{k_{j}^{y}}^{U}.$ Since $B_{k_{j}^{y}}^{U}\to A$ and
$t^{1}=U_{y}^{\xi}\cap[\partial^{*}E]_{y}^{\xi}\in J_{f_{0}},$ it follows that
$t^{1}\in\partial^{*}A$ and divides $U\cap\pi_{\xi}^{-1}(y)$ into two parts
one belonging to $U\cap A$ other to $U\setminus A.$ In particular,
$y\in\pi_{\xi}(U\cap\partial^{*}A).$ Hence,

$y\in\Big{[}U\cap[B_{k_{j}^{y}}^{U}]^{(1)}\cap J_{u_{k_{j}^{y}}}\cap
J_{v_{k_{j}^{y}}^{U}})\Big{]}_{y}^{\xi}\cup\Big{[}U\cap\partial^{*}A\Big{]}_{y}^{\xi}$

for all $j.$ Thus,

$\displaystyle\mathcal{H}^{n-1}(Y_{1})=$
$\displaystyle\int_{Y_{1}}\mathcal{H}^{0}\left(\bigcap_{j}\Big{(}\Big{[}U\cap[B_{k_{j}^{y}}^{U}]^{(1)}\cap
J_{u_{k_{j}^{y}}}\cap
J_{v_{k_{j}^{y}}^{U}})\Big{]}_{y}^{\xi}\cup\Big{[}U\setminus
B_{k_{j}^{y}}^{U}\Big{]}_{y}^{\xi}\Big{)}\right)\,\mathrm{d}\mathcal{H}^{n-1}(y)$
$\displaystyle\leq$
$\displaystyle\int_{Y_{1}}\lim\limits_{j\to+\infty}\mathcal{H}^{0}\Big{(}\Big{[}U\cap[B_{k_{j}^{y}}^{U}]^{(1)}\cap
J_{u_{k_{j}^{y}}}\cap
J_{v_{k_{j}^{y}}^{U}})\Big{]}_{y}^{\xi}\Big{)}\,\mathrm{d}\mathcal{H}^{n-1}(y)$
$\displaystyle+\int_{Y_{1}}\mathcal{H}^{0}\Big{(}\Big{[}U\cap\partial^{*}A\Big{]}_{y}^{\xi}\Big{)}\,\mathrm{d}\mathcal{H}^{n-1}(y).$

By the choice of $\\{k_{j}^{y}\\}$, the Fatou’s lemma, the second equality in
(2.5) and ($\rm a_{3}$)

$\int_{Y_{1}}\lim\limits_{j\to+\infty}\mathcal{H}^{0}\Big{(}\Big{[}U\cap[B_{k_{j}^{y}}^{U}]^{(1)}\cap
J_{u_{k_{j}^{y}}}\cap
J_{v_{k_{j}^{y}}^{U}})\Big{]}_{y}^{\xi}\Big{)}\,\mathrm{d}\mathcal{H}^{n-1}(y)\\\
\leq\liminf\limits_{k\to+\infty}\mathcal{H}^{n-1}\Big{(}U\cap[B_{k}^{U}]^{(1)}\cap
J_{u_{k}}\cap J_{v_{k}^{U}})\Big{)}=0.$

Similarly,

$\displaystyle\int_{Y_{1}}\mathcal{H}^{0}\Big{(}\Big{[}U\cap\partial^{*}A\Big{]}_{y}^{\xi}\Big{)}\,\mathrm{d}\mathcal{H}^{n-1}(y)\leq$
$\displaystyle P(A,U).$

Thus,

$\mathcal{H}^{n-1}(Y_{1})\leq P(A,U).$ (4.21)

Now using $\Pi_{\xi}\cap\pi_{\xi}(U)=Y_{0}\cup Y_{1}\cup Y_{2},$ from (4.20)
and (4.21) we obtain

$\mathcal{H}^{n-1}([\Pi_{\xi}\cap\pi_{\xi}(U)]\setminus Y_{2})\leq
P(A,U)+\mathcal{H}^{n-1}(U\cap[\Gamma\setminus\partial^{*}E]).$

Moreover, let

$X:=\\{y\in\Pi_{\xi}\cap\pi_{\xi}(U):\,\,\text{$\pi_{\xi}^{-1}(y)\cap\Gamma\cap\partial^{*}E$
is a singleton}\\}.$

Then as above

$\displaystyle\mathcal{H}^{n-1}(Y_{2}\setminus X)\leq$
$\displaystyle\mathcal{H}^{n-1}([\Pi_{\xi}\cap\pi_{\xi}(U)]\setminus
X)\leq\mathcal{H}^{n-1}([U\cap(\Gamma\cup\partial^{*}A)]\setminus\partial^{*}E)$
$\displaystyle\leq$
$\displaystyle\mathcal{H}^{n-1}(U\cap[\Gamma\setminus\partial^{*}E])+P(A,U),$

and therefore,

$\mathcal{H}^{n-1}([\Pi_{\xi}\cap\pi_{\xi}(U)]\setminus[Y_{2}\cap X])\leq
2P(A,U)+2\mathcal{H}^{n-1}(U\cap[\Gamma\setminus\partial^{*}E]).$ (4.22)

By the definition of $X$ and $Y_{2}$ for any $y\in Y_{2}\cap X$ we have
$\mathcal{H}^{0}(J_{f_{0}})=1$ and $N_{y}\geq 2,$ therefore, we can improve
(4.19) as

$\lim\limits_{j\to+\infty}\mathcal{H}^{0}\Big{(}J_{[\widehat{v}_{k_{j}^{y}}^{U}]_{y}^{\xi}}\Big{)}\geq
2=2\mathcal{H}^{0}([U\cap\Gamma]_{y}^{\xi}).$

For such $y$ from (4.17) we get

$\liminf\limits_{j\to+\infty}\Big{[}\mathcal{H}^{0}(J_{[\widehat{v}_{k_{j}}^{U}]_{y}^{\xi}})+\kappa
I_{y,\xi}^{U}(v_{k_{j}}^{U})+\kappa II_{y,\xi}^{U}(v_{k_{j}}^{U})\Big{]}\geq
2\mathcal{H}^{0}([U\cap\Gamma]_{y}^{\xi})$

Now integrating over $X\cap Y_{2}$ and using (4.15) and the Fatou’s lemma we
get

$\liminf\limits_{k\to+\infty}\int_{\Pi_{\xi}}\Big{[}\mathcal{H}^{0}(J_{[\widehat{v}_{k}^{U}]_{y}^{\xi}})+\kappa
I_{y,\xi}^{U}(v_{k}^{U})+\kappa
II_{y,\xi}^{U}(v_{k}^{U})\Big{]}\mathrm{d}\mathcal{H}^{n-1}(y)\geq
2\int_{X\cap
Y_{2}}\mathcal{H}^{0}([U\cap\Gamma]_{y}^{\xi})\mathrm{d}\mathcal{H}^{n-1}(y).$

By the definition of $\Upsilon,$ $\mathcal{H}^{0}([U\cap\Gamma]_{y}^{\xi})=1$
for all $y\in\Pi_{\xi}\cap\pi_{\xi}(U)$ and therefore by (4.22)

$\int_{X\cap
Y_{2}}\mathcal{H}^{0}([U\cap\Gamma]_{y}^{\xi})\mathrm{d}\mathcal{H}^{n-1}(y)\geq\int_{\Pi_{\xi}\cap\pi_{\xi}(U)}\mathcal{H}^{0}([U\cap\Gamma]_{y}^{\xi})\mathrm{d}\mathcal{H}^{n-1}(y)\\\
-P(A,U)-\mathcal{H}^{n-1}(U\cap[\Gamma\setminus\partial^{*}E]).$

Hence,

$\liminf\limits_{k\to+\infty}\int_{\Pi_{\xi}}\Big{[}\mathcal{H}^{0}(J_{[\widehat{v}_{k}^{U}]_{y}^{\xi}})+\kappa
I_{y,\xi}^{U}(v_{k}^{U})+\kappa
II_{y,\xi}^{U}(v_{k}^{U})\Big{]}\mathrm{d}\mathcal{H}^{n-1}(y)\\\ \geq
2\int_{\Pi_{\xi}\cap\pi_{\xi}(U)}\mathcal{H}^{0}([U\cap\Gamma]_{y}^{\xi})\mathrm{d}\mathcal{H}^{n-1}(y)-2P(A,U)-2\mathcal{H}^{n-1}(U\cap[\Gamma\setminus\partial^{*}E]).$

This, (2.5), (2.3), (2.4) as well as (4.12) yield

$\liminf\limits_{k\to+\infty}\int_{U\cap
J_{v_{k}^{U}}}|\nu_{J_{v_{k}^{U}}}\cdot\xi|\mathrm{d}\mathcal{H}^{n-1}+(M_{1}+|U|)\kappa\geq
2\int_{U\cap\Gamma}|\nu_{\Gamma}\cdot\xi|\mathrm{d}\mathcal{H}^{n-1}\\\
-2P(A,U)-2\mathcal{H}^{n-1}(U\cap[\Gamma\setminus\partial^{*}E])$ (4.23)

Let $\phi^{o}$ be the dual norm to $\phi,$ i.e.,

$\phi^{o}(\xi)=\sup\limits_{\phi(\nu)=1}|\xi\cdot\nu|.$

Then $|\xi\cdot\nu|\leq\phi^{o}(\xi)\phi(\nu)$ and hence, by (4.23) and the
arbitrariness of $\kappa$ we get

$\displaystyle\phi^{o}(\xi)\liminf\limits_{k\to+\infty}\int_{U\cap
J_{v_{k}^{U}}}\phi(\nu_{J_{v_{k}^{U}}})\mathrm{d}\mathcal{H}^{n-1}$
$\displaystyle\geq
2\int_{U\cap\Gamma}|\nu_{\Gamma}\cdot\xi|\mathrm{d}\mathcal{H}^{n-1}$
$\displaystyle-2P(A,U)-2\mathcal{H}^{n-1}(U\cap[\Gamma\setminus\partial^{*}E]).$
(4.24)

Now using $\phi^{o}(\xi)\geq 1/b_{2}$ from (4) we get (4.14).

Step 3. Now we prove ((i)).

Substep 3.1. Let

$\mathbb{S}_{\phi^{o}}^{n-1}:=\\{\xi\in\mathbb{R}^{n}:\,\,\phi^{o}(\xi)=1\\}.$

Since $\mathbb{S}_{\phi^{o}}^{n-1}$ is compact,

$\phi(\eta)=\max\limits_{i\geq 1}\eta\cdot\xi_{i}$

for any countable set $\\{\xi_{j}\\}_{j}\subset\mathbb{S}_{\phi^{o}}^{n-1}$
dense in $\mathbb{S}_{\phi^{o}}^{n-1}.$

Fix any such dense set $\\{\xi_{j}\\}_{j}\subset\mathbb{S}_{\phi^{o}}^{n-1}$
that if $\xi=\xi_{j}/|\xi_{j}|\in\Upsilon$, then (4.15) and (4.16) hold with
$\xi.$ By [23, Lemma 6] there exists a finite family $U_{1},\ldots,U_{m}$ of
disjoint open set compactly contained in $D$ such that

$2\int_{D\cap\Gamma}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}\leq\sum\limits_{j=1}^{m}2\int_{U_{j}\cap\Gamma}|\nu_{\Gamma}\cdot\xi_{j}|\mathrm{d}\mathcal{H}^{n-1}+\delta.$
(4.25)

Recalling the definition of $(B_{k}^{U},u_{k}^{U})$ from Step 2, let us define

$B_{k}=\bigcap\limits_{j=1}^{m}B_{k}^{U_{j}}\quad\text{and}\quad
v_{k}:=u_{k}\chi_{B_{k}\cup S}.$

Then by ($\rm a_{2}$) $B_{k}\subset A_{k},$ $A_{k}\setminus
B_{k}\subset\subset D$ and

$|A_{k}\setminus B_{k}|\leq\sum\limits_{j=1}^{m}|U_{j}\cap(A_{k}\setminus
B_{k}^{U_{j}})|\leq\sum\limits_{j=1}^{m}\frac{|U_{j}|}{k}\leq\frac{|D|}{k}.$

Let $\Lambda_{k}(D)$ be defined as in ($\rm a_{4}$) of Step 2 with
$(B_{k},v_{k})$ in place of $(B_{k}^{U},v_{k}^{U})$. Then by the definition of
$(B_{k},v_{k}),$ $\alpha_{k}(D)$ and ($a_{4}$)

$\displaystyle\alpha_{k}(D)-\Lambda_{k}(D)=\sum\limits_{j=1}^{m}\big{(}\alpha_{k}(U_{j})-\Lambda_{k}(U_{j})\big{)}\geq-\sum\limits_{j=1}^{m}\frac{|U_{j}|}{k}\geq-\frac{|D|}{k}.$

Thus,

$\alpha_{k}(D)\geq\Lambda_{k}(D)-\frac{|D|}{k}.$ (4.26)

Substep 3.2. Now we estimate $\Lambda_{k}(D)$ from below. Note that if
$\xi_{j}/|\xi_{j}|\in\Upsilon,$ then since $\phi^{o}(\xi_{j})=1,$ by (4.14)

$\displaystyle 2\int_{U_{j}\cap\Gamma}|\nu_{\Gamma}\cdot\xi_{j}|$
$\displaystyle\mathrm{d}\mathcal{H}^{n-1}\leq\liminf\limits_{k\to+\infty}\Lambda_{k}(U_{j})+2b_{2}P(A,U_{j})+2b_{2}\mathcal{H}^{n-1}(U_{j}\cap[\Gamma\setminus\partial^{*}E]).$
(4.27)

Now assume that $\xi:=\xi_{j}/|\xi_{j}|\notin\Upsilon.$ Then by the definition
of $\Upsilon$ and (a3)

$|\nu_{\Gamma}(x)\cdot\xi|\leq|(\nu_{\Gamma}(x)-{\bf e}_{n})\cdot\xi|+|{\bf
e}_{n}\cdot\xi|<3\delta$

for any $x\in U_{j}\cap\Gamma.$ Thus,

$2\int_{U_{j}\cap\Gamma}|\nu_{\Gamma}\cdot\xi_{j}|\mathrm{d}\mathcal{H}^{n-1}\leq
6\delta\mathcal{H}^{n-1}(U_{j}\cap\Gamma).$ (4.28)

Now by (4.25), (4.27) and (4.28)

$2\int_{D\cap\Gamma}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}\leq\delta+\sum\limits_{j=1,\,j\in\Upsilon}^{m}\liminf\limits_{k\to+\infty}\,\Lambda_{k}(U_{j})+6\delta\sum\limits_{j=1,\,j\notin\Upsilon}^{m}\mathcal{H}^{n-1}(U_{j}\cap\Gamma)\\\
+2b_{2}\sum\limits_{j=1}^{n}\Big{[}P(A,U_{j})+2\mathcal{H}^{n-1}(U_{j}\cap[\Gamma\setminus\partial^{*}E])\Big{]}.$

Since set function $Q\mapsto\Lambda_{k}(Q)$ is additive and non-increasing and
the family $\\{U_{j}\\}$ is pairwise disjoint,

$\sum\limits_{j=1,\,j\in\Upsilon}^{m}\liminf\limits_{k\to+\infty}\,\Lambda_{k}(U_{j})\leq\liminf\limits_{k\to+\infty}\Lambda_{k}(\cup_{j}U_{j})\leq\liminf\limits_{k\to+\infty}\Lambda_{k}(D).$

Moreover, by (a2)

$\sum\limits_{j=1,\,j\notin\Upsilon}^{m}\mathcal{H}^{n-1}(U_{j}\cap\Gamma)\leq\mathcal{H}^{n-1}(Q_{1}\cap\Gamma)<1+\delta,$

and by (a2), (a5), (a7.2) and (a7.3)

$\sum\limits_{j=1}^{n}\Big{[}P(A,U_{j})+\mathcal{H}^{n-1}(Q_{1}\cap[\Gamma\setminus\partial^{*}E])\Big{]}\\\
\leq
P(A,Q_{1})+\mathcal{H}^{n-1}(Q_{1}\cap\Gamma)-\mathcal{H}^{n-1}(Q_{1}\cap\Gamma\cap\partial^{*}E)\leq
6\delta.$

Then

$\displaystyle
2\int_{D\cap\Gamma}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}\leq$
$\displaystyle\delta+\liminf\limits_{k\to+\infty}\Lambda_{k}(D)+6\delta(1+\delta)+6b_{2}\delta,$

and hence,

$\displaystyle\liminf\limits_{k\to+\infty}\Lambda_{k}(D)\geq
2\int_{D\cap\Gamma}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}-c_{0}\delta,$
(4.29)

where

$c_{0}:=13+6b_{2}$

depends only on $b_{2}.$

Substep 3.3. From (4.26) and (4.29) there exist $k_{0}:=k_{0}(\delta,b_{2})>0$
such that

$\Lambda_{k}(D)\geq
2\int_{D\cap\Gamma}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}-2c_{0}\delta$
(4.30)

for all $k>k_{0}.$ Since $|D|<|Q|=1,$ one has $|D|/k<c_{0}\delta$ provided
$k>\frac{1}{c_{0}\delta}.$ Let

$k_{\delta}^{\prime}:=\max\Big{\\{}k_{0},\frac{1}{c_{0}\delta}\Big{\\}}.$

Observe that

$\displaystyle\int_{D\cap\Gamma}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}\geq\int_{D\cap\Gamma\cap\partial^{*}E}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}.$

Moreover, by (a7.3)

$\int_{D\cap\partial^{*}E\setminus\Gamma}\phi(\nu_{E})\mathrm{d}\mathcal{H}^{n-1}\leq
b_{2}\delta$

and by Lemma 3.1 (ii)

$\mathcal{H}^{n-1}(Q_{1}\cap\partial^{*}E\setminus D)<2\delta,$

and therefore,

$\int_{D\cap\Gamma}\phi(\nu_{\Gamma})\mathrm{d}\mathcal{H}^{n-1}\geq\int_{Q_{1}\cap\partial^{*}E}\phi(\nu_{E})\mathrm{d}\mathcal{H}^{n-1}-3b_{2}\delta\qquad\text{for
all $k>k_{\delta}^{\prime}$}.$

Combining these estimates with (4.26) and (4.30) we deduce

$\alpha_{k}(D)\geq
2\int_{K}\phi(\nu_{K})\mathrm{d}\mathcal{H}^{n-1}-(2c_{0}+6b_{2})\delta.$

Hence, $c^{\prime}:=c_{b_{2}}^{\prime}=(2c_{0}+6b_{2})$ satisfies the
assertion. ∎

###  4.1. Lower semicontinuity of $\mathcal{F}_{p}$ and $\mathcal{F}_{\rm
Dir}$

We conclude this section by showing that the functionals $\mathcal{F}_{p}$ and
$\mathcal{F}_{\rm Dir}$ in Theorems 2.8 and 2.9, respectively, are lower
semicontinuous with respect to the $\tau$-convergence defined in (2.18).
Indeed, the proof of the $\tau$-lower semicontinuity of $\mathcal{S}$ in
$\mathcal{C}_{p}$ and $\mathcal{C}_{\rm Dir}$ is exactly the same as the
$\tau_{\mathcal{C}}$-lower semicontinuity of $\mathcal{S}$ in $\mathcal{C}$
(see the proof of Theorem 2.5). To prove the $\tau$-lower semicontinuity of
$\mathcal{W}_{p}$ and $\mathcal{W}_{\rm Dir}$ we notice that according to the
proof of the density estimate (4.7f), we only need the convexity of
$W_{p}(x,\cdot)$ and the weak convergence of $\mathcal{E}u_{k}$ to
$\mathcal{E}u$ in $L^{p}({\rm Int}{(\Omega\cup S\cup\Sigma)});$ the first
condition is already stated in the assumption (a1) of $W_{p}$ and the second
condition follows from the lower bound in (a2) and the compactness result [14,
Theorem 1.1].

##  5\. Compactness in $\mathcal{C}$

In this section we prove Theorem 2.6. Note that if $\\{(A_{k},u_{k})\\}$ is an
energy-equibounded sequence, then by a priori estimates (see Remark 2.3) we
can find a set of finite perimeter $A\subset\Omega$ such that, up to a
subsequence, $A_{k}\to A$ in $L^{1}(\mathbb{R}^{n}).$ Moreover, since each
connected component $S_{i}$ of $S$ is Lipschitz, the convergence of $u_{k}$ in
$S_{i}$ can be obtained by adding rigid displacements in $S_{i}.$ However,
since the rigid displacements for $S_{i}$ may differ from those for $S_{j},$
$j\neq i,$ we need to create extra jumps for the resulting displacement field.
Hence, as in [36] we need to partition $A_{k}$ to compensate those jumps. The
following proposition provides such a partition up to some error.

######  Proposition 5.1.

Let $(A_{k},u_{k}),(A,u)\in\mathcal{C}$ be admissible configurations, $S^{i}$
for $i\in\\{1,\ldots,m\\}$ be a nonempty union of some connected components of
$S$ such that $S^{i}\cap S^{j}=\emptyset$ and $S=\bigcup_{i=1}^{m}S^{i},$
$\\{a_{k}^{1}\\},\ldots,\\{a_{k}^{m}\\}$ be sequences of rigid displacements,
$u^{1},\ldots,u^{m}\in GSBD^{2}({\rm Int}{(\Omega\cup S\cup\Sigma)})$ and
$F^{1},\ldots,F^{m}\subset A$ be pairwise disjoint sets of finite perimeter.
Assume that

  * •

$\sup_{k}\mathcal{F}(A_{k},u_{k})<+\infty$ and $A_{k}\to A$ in
$L^{1}(\mathbb{R}^{n});$

  * •

for any $i\in\\{1,\ldots,m\\}$ one has $u_{k}-a_{k}^{i}\to u^{i}$ a.e. in
$S^{i}\cup F^{i}$ and $|u_{k}-a_{k}^{i}|\to+\infty$ a.e. $(S\setminus
S^{i})\cup(A\setminus F^{i}).$

Then for any $\delta\in(0,\frac{1}{8}\,\min_{i\neq
j}\\{1,\mathrm{dist}(S^{i},S^{j})\\})$ there exist a (not relabelled)
subsequence $\\{(A_{k},u_{k})\\},$ $k_{\delta}>0,$ $s_{\delta}\in(0,\delta)$
and a sequence $\\{G_{k}^{\delta}\\}\subset BV(\Omega;\\{0,1\\})$ such that

$\displaystyle\mathcal{H}^{n-1}([A_{k}\setminus
A]^{(1)}\cap\\{\mathrm{dist}(\cdot,S)=s_{\delta}\\})<c^{*}\delta,$ (5.1a)
$\displaystyle\mathcal{H}^{n-1}(\\{\mathrm{dist}(\cdot,S)<s_{\delta}\\}\cap\partial^{*}A)<c^{*}\delta,$
(5.1b) $\displaystyle|G_{k}^{\delta}|<c^{*}\sqrt{\delta}\sum_{0\leq i\leq
m}P(F^{i}),$ (5.1c) $\displaystyle P(G_{k}^{\delta})\leq c^{*}\sum_{0\leq
i\leq m}P(F^{i}),$ (5.1d)

and the sequence $\\{(B_{k}^{\delta},v_{k}^{\delta})\\},$ defined as

$B_{k}^{\delta}:=A_{k}\setminus G_{k}^{\delta}$ (5.2)

and

$v_{k}^{\delta}:=\begin{cases}u_{k}-a_{k}^{i}&\text{in
$S^{i}\cup[F^{i}\setminus
G_{k}^{\delta}]\cup[R_{\delta}^{i}\cap(B_{k}^{\delta}\setminus A)]$ for
$i=1,\ldots,m,$}\\\ u_{0}&\text{in $B_{k}^{\delta}\cap F^{0},$}\\\
\xi&\text{in $(\Omega\setminus
B_{k}^{\delta})\cup(B_{k}^{\delta}\setminus[A\cup\bigcup_{i=1}^{m}R_{\delta}^{i}]),$}\end{cases}$
(5.3)

where $\xi\in(0,1)^{n},$

$R_{\delta}^{i}:=\\{x\in\Omega:\,\,\mathrm{dist}(x,S^{i})<s_{\delta}\\},\qquad
F^{0}:=A\setminus\bigcup_{i=1}^{m}F^{i},$

satisfies

$\mathcal{S}(A_{k},u_{k})\geq\mathcal{S}(B_{k}^{\delta},v_{k}^{\delta})-c^{*}\sqrt{\delta}\Big{[}1+P(A_{k})+\mathcal{H}^{n-1}(J_{u_{k}})+\sum\limits_{i=0}^{m}\mathcal{H}^{n-1}(\partial^{*}F^{i})\Big{]}$
(5.4)

for all $k>k_{\delta}.$ Here constant $c^{*}>0$ depends only on $n,$ $b_{1}$
and $b_{2}.$

Figure 3. The partition of $A=\bigcup_{i\geq 0}F^{i}$ and the construction of
$B_{k}^{\delta}:=A_{k}\setminus G_{k}^{\delta}$ in Proposition 5.1. The set
$G_{\delta,k}$ is a finite union of holes along the boundaries
$F^{i}\cup\bigcup_{j\neq i}S^{j}$ in which $u_{k}-a_{k}^{i}$ converges. Note
that the sets $\\{F^{i}\setminus G_{k}^{\delta}\\}_{i=0}^{m}$ partition
$B_{k}^{\delta}.$ Since $F^{0}$ is a “hanging” component of $A,$ i.e., not
linked to the substrate, and hence, it is reasonable to assume that the
elastic energy in $F^{0}$ is $0.$ Then we define the displacement fields
$v_{k}^{\delta}$ as follows: in $S^{i}\cup(F^{i}\setminus G_{k}^{\delta})$ for
$i=1,\ldots,m$ we set $v_{k}^{\delta}:=u_{k}-a_{k}^{i}$ and in $F^{0}\setminus
G_{k}^{\delta}$ we write $v_{k}^{\delta}:=u_{0}.$ Finally, since
$A_{k}\setminus A$ may present large trace portions along $\partial S$ on
which $v_{k}^{\delta}$ forms a jump, we need to change the values of
$v_{k}^{\delta}$ in $R_{\delta}^{i}\setminus A$ near $S^{i}.$

We postpone the proof of Proposition 5.1 after the proof of Theorem 2.6.

###### Proof of Theorem 2.6.

Since $S$ is Lipschitz open set with finitely many connected components,
applying the Poincaré-Korn inequality and the Rellich-Kondrachov compactness
theorem we find a not relabelled subsequence $\\{(A_{k},u_{k})\\},$ a
partition $\\{S^{i}\\}_{i=1}^{m}$ of $S$ and $m$ sequences
$\\{a_{k}^{1}\\},\ldots,\\{a_{k}^{m}\\}$ of rigid displacements such that

  * ($\rm a_{1}$)

each $S^{i}$ is the union of some connected components of $S$ and
$S=\bigcup_{i=1}^{m}S^{i};$

  * ($\rm a_{2}$)

for each $i\in\\{1,\ldots,m\\}$ there exists $w^{i}\in H^{1}(S^{i})$ such that
$u_{k}-a_{k}^{i}$ converges to $w^{i}$ weakly in $H^{1}(S^{i})$ and a.e. in
$S^{i};$

  * ($\rm a_{3}$)

if $i\neq j,$ then $|a_{k}^{i}-a_{k}^{j}|\to+\infty$ a.e. in $\mathbb{R}^{n}.$

We may also assume $A_{k}\to A$ in $L^{1}(\mathbb{R}^{n})$ for some $A\in
BV(\Omega;\\{0,1\\}).$ Since $\mathcal{E}v=\mathcal{E}(v+a)$ for any rigid
displacement $a$, by Remark 2.3 we have

$\sup_{k\geq
1}\Big{(}P(A_{k})+\mathcal{H}^{n-1}(J_{(u_{k}-a_{k}^{i})\chi_{A_{k}\cup
S}})+\int_{A_{k}\cup