# Existence of minimizers for a two-phase free boundary problem with coherent
and incoherent interfaces

Randy Llerena  Research Platform MMM “Mathematics-Magnetism-Materials” - Fak.
Mathematik Univ. Wien, A1090 Vienna<EMAIL_ADDRESS>and  Paolo
Piovano  Dipartimento di Matematica, Politecnico di Milano, P.zza Leonardo da
Vinci 32, 20133 Milano, Italy111MUR Excellence Department 2023-2027
<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

A variational model for describing the morphology of two-phase continua by
allowing for the interplay between coherent and incoherent interfaces is
introduced. Coherent interfaces are characterized by the microscopical
arrangement of atoms of the two materials in a homogeneous lattice, with
deformation being the solely stress relief mechanism, while at incoherent
interfaces delamination between the two materials occurs. The model is
designed in the framework of the theory of Stress Driven Rearrangement
Instabilities, which are characterized by the competition between elastic and
surface effects. The existence of energy minimizers is established in the
plane by means of the direct method of the calculus of variations under a
constraint on the number of boundary connected components of the underlying
phase, whose exterior boundary is prescribed to satify a graph assumption, and
of the two-phase composite region. Both the wetting and the dewetting regimes
are included in the analysis.

###### Key words and phrases:

Multiphase boundary, Free boundary problem, surface energy, elastic energy,
coherent interface, incoherent interface, thin films, delamination

###### 2010 Mathematics Subject Classification:

49J10, 49Q15, 49Q20, 35R35 , 74A45, 74G65

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. 3 Mathematical setting and main results
  4. 4 Compactness
  5. 5 Lower semicontinuity
  6. 6 Existence

##  1\. Introduction

In this manuscript we address the problem of providing a mathematical
variational framework for the description of the morphology and the elastic
properties of two-phase continua based on Gibbs’s notion of a sharp phase-
interface dividing them [13, 31, 35]. In the presence of two interacting media
large stresses due to the different crystalline order of the two materials
originate and, besides bulk deformation, various types of morphological
destabilization may occur as a further strain relief mode. These are often
referred to as the family of _stress driven rearrangement instabilities_
(SDRI) [5, 20, 34, 36, 49], which include the roughness of the exposed
crystalline boundaries, the formation of cracks in the bulk materials, the
nucleation of dislocations in the crystalline lattices, and the delamination
(as opposed to the adhesion) at the contact regions with the other material.

Literature provides extensive studies of these phenomena under the assumption
that one phase is a rigid fixed continuous medium underlying, such as the
substrates for epitaxially-strained thin films [14, 17, 27, 39], or
constraining, such as crystal cavities [26] or the containers in capillarity
problems [23], the other phase, which is instead let free, or by modeling the
interactions with other media simply by means of fixed boundary conditions.
There are though settings in which the hierarchy between the phases is not
clear, or a rigidity ranking between them is not easily identifiable, since
the interplay among the deformation and the interface instabilities affecting
all phases is crucial, such as, in the shock-induced transformations and
mechanical twinning [35] or in the deposition of film multilayers [42].

As described in [35] the extension of classical theories of continuum
mechanics to two-phase deformable media is though “not as straightforward as
it might appear”, since combining the accretion and deletion of material
constituents responsible for the moving of the interface between the two
phases and their boundaries, with the framework of elasticity related to bulk
deformation and fractures [15, 30], by quoting [35], “leads to conceptual
difficulties”. A critical modeling issue related to the interface between the
two phases is the interplay between _coherency_ , that is here intended as the
microscopical arrangement of atoms of the two materials in a homogeneous
lattice, with deformation being the solely stress relief mechanism [35], and
_incoherency_ , that instead refers to the debonding occurring between the
atoms of the two materials [13], which results in the composite delamination
at the two-phase interface [40]. This manuscript seems to be, to the best of
our knowledge, the first attempt to provide a mathematical framework able to
simultaneously describe coherent and incoherent interfaces for a two-phase
setting, which we carry out by also keeping in the picture the other features
of SDRI, such as the dichotomy between the _wetting regime_ , that is the
setting in which it is more convenient for a phase to cover the surface of the
other phase with an infinitesimal layer of atoms, and the _dewetting regime_ ,
in which it is preferable to let such surface exposed to the vapor.

The studies for the setting of only coherent interfaces go back to Almgren
[1], who was the first to formulate the problem in $\mathbb{R}^{d}$, $d>1$, in
the context without elasticity for surface tensions proportional among the
various interfaces, by means of _integral currents_ in geometric measure
theory and by singling out a condition ensuring the lower semicontinuity of
the overall surface energy with respect to the $L^{1}$-convergence of the sets
in the partition. Then, Ambrosio and Braides in [2, 3] extended the setting to
also non-proportional surface tensions by introducing a new integral condition
referred to as $BV$-_ellipticity_ , which they show to be both sufficient and
necessary for the lower semicontinuity with respect to the
$L^{1}$-convergence. As such condition is the analogous, for the setting of
_Caccioppoli partitions_ , of Morrey’s _quasi-convexity_ , it is difficult to
check it in practice. In [2, 3] $BV$-ellipticity is proved to coincide with a
simpler to check _triangle inequality condition_ among surface tensions for
the case of partitions in 3 sets (like the setting considered in this paper,
in which one element of the partition is always represented by the _vapor_
outside the two phases), which was then confirmed to be the only case by [10].
Other conditions therefore have been introduced, such as $B$-_convexity_ and
_joint convexity_ , with though $BV$-ellipticity remaining so far the only
condition known to be both necessary and sufficient for lower semicontinuity
apart from specific settings (see [11, 45] for more details). Recently, the
$BV$-ellipticity has been extended in the context of $BD$-_spaces_ in [32].
Finally, we refer to [7] for a variant of the _Ohta–Kawasaki model_ considered
to model thin films of diblock copolymers in the unconfined case, which
represents a recent example in the literature of a two-phase model in the
absence of elasticity and of incoherent and crack interfaces, under a graph
constraint for union of the two phases.

Regarding incoherent interfaces the problem is intrinsically related to the
renowned _segmentation problem_ in image reconstruction that was actually
originally introduced by Mumford and Shah in [46] with a multiphase
formulation, as a partition problem of an original image, with the connceted
contours of the image areas characterized as discontinuity set of an auxiliary
state function. Then, the approaches developed to tackle the problem led to
the study of a single phase setting with the jump set of the state function
representing internal interfaces, proven to satisfy _Ahlfors-type regularity_
result [4, 19]. Such single-phase framework has been then extended to the
context of linear elasticity in fracture mechanics with the state function
being vectorial and representing the bulk displacement of a crystalline
material and the energy replaced by the _Griffith energy_ [15, 30]. The
attempt to recover the original setting of [46] in a rigorous mathematical
formulation (apart from some formulations with piecewise-constant state
functions or numerical investigations) has been then addressed by Bucur,
Fragalà, and Giacomini in [8] and [9] (see also [16] for a related multiphase
boundary problem in the context of reaction-diffusion systems). In [8]
Ahlfors-type regularity is established for _ad hoc_ notions of multiphase
local _almost-quasi minimizers_ of an energy accounting for incoherent
isotropic interfaces and disregarding the contribution of the coherent
portions, while in [9] they introduce a multiphase version of the Mumford-Shah
problem by treating all the reduced phase boundaries as incoherent interfaces
(like the internal jump sets) and by adding an extra (statistical) term, which
induces multi-phase minimizers.

In order to finally include in the model both coherent and incoherent portions
(possibly also on the same interface between the phases), we first restrict to
the two-phase setting (for a multi-phase setting in the context of film
multilayers we refer to the related paper [42] under finalization) and follow
a different direction than the one of [8, 9], which works for $d=2$: We adopt
the approach considered in [36, 37], that was relying on the strategy
developed for the Mumford-Shah problem in [19]. Such approach consists in
first imposing a fixed constraint on the number of connected components for
the boundary of the free phases, in order then to employ adaptations of
Golab’s Theorem [33] for proving the compactness with respect to a proper
selected topology, and then in studying the convergence of the solutions of
the different minimum problems related to different constraints on the
connected components, as such constraints tend to infinity. This second step
has been performed for the one-phase setting in [37] (and for higher dimension
in [38]) by means of density estimates.

Here, we performed the first step in this program, reaching an existence
result analogous to the one in [36]. However, the extension of [36] to the
two-phase setting requires important modification in the model setting, since
characterizing the incoherent interface as the jump portion of the bulk
displacement on the two-phase interface as in [36, 37, 38] appears to be not
feasible, as in our setting the two-phase interfaces need to be considered
much less regular than Lipschitz manifolds like in [36, 37, 38]. To solve this
issue we consider as set variables of the energy not the two phases, to which
we refer to as the _film_ and the _substrate phase_ , but the substrate phase
and the whole region occupied by the composite of both the two phases, and we
characterize the incoherent interfaces as a portion of the boundary
intersection of such variables. As a byproduct of this strategy we do not need
to impose a constraint on the number of boundary components of the film phases
(but only with respect to the substrates and the composite regions), so that
the physical relevant setting of countable separated isolated film islands
forming on top of the substrate is included in our analysis, even though it
was prevented by the formulation in [36]. Moreover, we can also extend [36] to
the presence of adjacent materials for the Griffith-type model with mismatch
strain and delamination [15, 30, 38].

In agreement with the SDRI theory [5, 20, 34] the total energy $\mathcal{F}$
is given by the sum of two contributions, namely the elastic energy
$\mathcal{W}$ and the surface energy $\mathcal{S}$, and it is defined on
triples $(A,S,u)\in\widetilde{\mathcal{C}}$, where $u$ represent the _bulk
displacement_ of the composite material of the two phases, and $A$ and $S$ are
sets whose closures represents the _composite region_ and the _substrate
region_ , respectively, while the _film region_ is given by
$\overline{A}\setminus S^{(1)}$ (for $S^{(1)}$ denoting the points with
density 1 in $S$). More precisely, given
$\Omega:=(-l,l)\times_{\mathbb{R}^{2}}(-L,L)\subset\mathbb{R}^{2}$ as the
region where the composite material is located, which is defined for the two
parameters $l,L>0$ and to which we refer as the _container_ in analogy to the
notation of capillarity problems, we introduce

$\displaystyle\widetilde{\mathcal{C}}:=\Large\\{(A,S,u):\quad$ $A$ and $S$ are
$\mathcal{L}^{2}$-measurable sets with
$S\subset\overline{A}\subset\overline{\Omega}$ such that $\partial
A\cap{\rm{Int}}(S)=\emptyset$, $\partial A$ and $\partial S$ are
$\mathcal{H}^{1}$-rectifiable, $\displaystyle\text{$\mathcal{H}^{1}(\partial
A)+\mathcal{H}^{1}(\partial S)<\infty$, and $u\in
H^{1}_{\mathrm{{loc}}}({\rm{Int}}(A);\mathbb{R}^{2})$}\\}.$

We define $\mathcal{F}:\widetilde{\mathcal{C}}\rightarrow\mathbb{R}$ as

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

for every $(A,S,u)\in\widetilde{\mathcal{C}}$. The elastic energy
$\mathcal{W}(A,u)$ is defined analogously to [21, 36, 37, 38] by

$\mathcal{W}(A,u):=\int_{A}{W\left(x,E\left(u\left(x\right)-E_{0}\left(x\right)\right)\right)dx},$

where the elastic density $W$ is determined by the quadratic form

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

for a fourth-order tensor $\mathbb{C}:\Omega\to\mathbb{M}^{2}_{\rm sym}$, $E$
denotes the symmetric gradient, i.e., $E(v):=\frac{\nabla v+\nabla^{T}v}{2}$
for any $v\in H^{1}_{\mathrm{{loc}}}({\rm{Int}}(A);\mathbb{R}^{2})$,
representing the _strain_ , and $E_{0}$ is the _mismatch strain_
$x\in\Omega\mapsto E_{0}\left(x\right)\in\mathbb{M}^{2}_{\mathrm{{sym}}}$
defined as

$E_{0}:=\begin{cases}E(u_{0})&\text{in }\Omega\setminus S,\\\ 0&\text{in
}S,\end{cases}$

for a fixed $u_{0}\in H^{1}(\Omega;\mathbb{R}^{2})$. The mismatch strain is
included in the SDRI theory to represent the fact that the two phases are
given by possibly different crystalline materials whose free-standing
equilibrium lattice could present a lattice mismatch. In this context notice
that $\mathbb{C}$ is allowed to present discontinuities at the interface
between the two materials (see hypothesis (H3) in Section 3.2).

The surface energy $\mathcal{S}(A,S)$ is given by

$\mathcal{S}(A,S):=\int_{\Omega\cap(\partial A\cup\partial
S)}{\psi(x,\nu(x))\,d\mathcal{H}^{1}(x)},$

where, by denoting with $\nu_{U}(z)$ the normal unit vector pointing outward
to a set $U\subset\mathbb{R}^{2}$ with $\mathcal{H}^{1}$-rectifiable boundary
at a point $x\in\partial U$,

$\nu(x):=\begin{cases}\nu_{A}(x)&\text{if $z\in\partial A\setminus\partial
S$},\\\ \nu_{S}(x)&\text{if $z\in\partial S$,}\end{cases}$

and $\psi:\overline{\Omega}\times\mathbb{R}^{2}\to[0,\infty]$ represents the
_surface tension of the composite_ of the two phases, which we allow to be
anisotropic.

In order to properly define $\psi$ we need to consider the three surface
tensions
$\varphi_{\mathrm{F}},\,\varphi_{\mathrm{S}},\,\varphi_{\mathrm{FS}}:\overline{\Omega}\times\mathbb{R}^{2}\to[0,\infty]$
characterizing the three possible interfaces for the two-phase setting, i.e.,
the interface between the film phase and the vapor, the interface between the
substrate phase and the vapor, and the interface between the film and the
substrate phases. Furthermore, to simultaneously treat both the wetting and
the dewetting regime, we introduce two auxiliary surface tensions, to which we
refer as the _regime surface tensions_ , that we defined as:

$\varphi:=\min\\{\varphi_{\mathrm{S}},\varphi_{\mathrm{F}}+\varphi_{\mathrm{FS}}\\}\qquad\text{and}\qquad\varphi^{\prime}:=\min\\{\varphi_{\mathrm{S}},\varphi_{\mathrm{F}}\\},$

since $\varphi_{\mathrm{S}}$ is the surface tension associated to the
dewetting regime, as the substrate surface remains exposed to the vapor, while
$\varphi_{\mathrm{F}}+\varphi_{\mathrm{FS}}$ and $\varphi_{\mathrm{F}}$ are
both associated to the wetting regime, respectively, to the situation of an
infinitesimal layer of film atoms covering the substrate surface (by being
bonded to the substrate atoms), which is referred to as the _wetting layer_ ,
or of simply a detached _film filament_. We define

$\psi(x,\nu(x)):=\begin{cases}\varphi_{\mathrm{F}}(x,\nu\left(x\right))&x\in\Omega\cap(\partial^{*}A\setminus\partial^{*}{S}),\\\
\varphi(x,\nu\left(x\right))&x\in\Omega\cap\partial^{*}{S}\cap\partial^{*}A,\\\
\varphi_{\mathrm{FS}}(x,\nu\left(x\right))&x\in\Omega\cap(\partial^{*}{S}\setminus\partial
A),\\\
(\varphi_{\mathrm{F}}+\varphi)(x,\nu\left(x\right))&x\in\Omega\cap\partial^{*}{S}\cap\partial
A\cap A^{(1)},\\\
2\varphi_{\mathrm{F}}(x,\nu\left(x\right))&x\in\Omega\cap\partial A\cap
A^{(1)}\cap S^{(0)},\\\
2\varphi^{\prime}(x,\nu\left(x\right))&x\in\Omega\cap\partial A\cap
A^{(0)},\\\
2\varphi_{\mathrm{FS}}(x,\nu\left(x\right))&x\in\Omega\cap(\partial{S}\setminus\partial
A)\cap\left(S^{(1)}\cup S^{(0)}\right)\cap A^{(1)},\\\
2\varphi(x,\nu\left(x\right))&x\in\Omega\cap\partial{S}\cap\partial A\cap
S^{(1)},\end{cases}$ (1.1)

where $\partial^{*}U$ and $U^{(\alpha)}$ denote, when well defined, the
_reduced boundary_ and the set of points of density $\alpha\in[0,1]$ for a set
$U\subset\mathcal{R}^{2}$. We notice that the 8 subregions of the domain
$\Omega\cap(\partial A\cup\partial S)$ in which the definition of $\psi$ is
distinguished are the counterpart of the 5 terms appearing in the surface
energy of [36] for the two-phase setting (see Remark 3.13 for more details).
Each subregion appearing in (1.1) represents, by moving line by line, the
_film free boundary_ , the _substrate free boundary_ , the _film-substrate
coherent interface_ , the _film-substrate incoherent interface_ , the _film
cracks_ , the _exposed filaments_ , the _substrate filaments and cracks_ in
the film-substrate coherent interface, and the _substrate cracks_ in the film-
substrate incoherent interface, respectively.

We observe that the surface tensions associated to the film free boundary, the
coherent substrate-film interface, and the substrate free boundary, are simply
$\varphi_{\mathrm{F}}$, $\varphi_{\mathrm{FS}}$, and (to accommodate both
wetting and dewetting regimes) $\varphi$ respectively, while the surface
tension associated to the incoherent film-substrate interface is chosen to be
$\varphi_{\mathrm{F}}+\varphi$ in analogy with the film-substrate delamination
or delaminated region in [36, 37, 38], since the incoherent interface
coincides with the portion of the film-substrate interface in which there is
no bonding between the film and the substrate surfaces. All remaining 4 terms
are weighted double (in analogy to the lower-semicontinuity results previously
obtained in [14, 21, 27, 36, 37, 38] for the one-phase setting) as they refer
to either material filaments in the void or cracks in the composite bulk. In
particular, we notice that in the substrate bulk region represented by
$S^{(1)}$ we distinguish between substrate cracks in the coherent and in the
incoherent film-substrate interface, that are counted with weight
$2\varphi_{\mathrm{FS}}$ and $2\varphi$, respectively, while in the film bulk
region $A^{(1)}\cap S^{(0)}$ we distinguish between substrate filaments that
are not film cracks counted with $2\varphi_{\mathrm{FS}}$ and film cracks
counted $2\varphi_{\mathrm{F}}$ (see Figure 1).

Figure 1.  The admissible regions for an admissible configuration
$(A,S,u)\in\mathcal{C}_{\textbf{m}}$ (see Definition 3.2) are represented by
indicating the substrate region $\overline{S}$ and the film region
$\overline{A}\setminus S^{(1)}$ with a darker and a lighter gray,
respectively. In particular, the film and the substrate free boundaries (with
the film and substrate filaments) are indicated with a thinner line, while the
film-substrate interface is depicted with a thicker line that is continuous or
dashed to distinguish between its incoherent portions and its coherent
portions (inclusive of substrate cracks and filaments that are not film
cracks), respectively.

The main result of the paper consists in finding a physically relevant family
of admissible configurations in $\widetilde{\mathcal{C}}$, which is denoted by
$\mathcal{C}_{\textbf{m}}$ for $\textbf{m}:=(m_{0},m_{1})\in\mathbb{N}^{2}$,
in which we can prove that, under a _two-phase volume constraint_ ,
$\mathcal{F}$ admits a minimizer. We find such a family
$\mathcal{C}_{\textbf{m}}\subset\widetilde{\mathcal{C}}$ by considering as
admissible configurations $(A,S,u)\in\widetilde{\mathcal{C}}$ the ones for
which (see Definition 3.2 for more details):

  * -

the number of boundary connected components of $S$ and $A$ are fixed to be at
most $m_{0}$ and $m_{1}$, respectively,

  * -

the substrate regions $S$ satisfy an _exterior graph constraint_ consisting in
requiring that $\partial^{*}S\cup(\partial S\cap S^{(0)})$ is the graph of an
upper semicontinuous function with pointwise bounded variation (while
internal, also non-graph-like, substrate cracks are allowed),

as shown in Figure 1. We notice that such an exterior graph constraint allows
to have a more involved description of the substrate regions than the
previously considered graph constraint in the literature for the one-phase
setting [14, 17, 21, 27], which is indeed needed to achieve the compactness
result contained in Theorem 3.11.

Therefore, for any two volume parameters
$\mathbbm{v}_{0},\mathbbm{v}_{1}\in[\mathcal{L}^{2}(\Omega)/2,\mathcal{L}^{2}(\Omega)]$
such that $\mathbbm{v}_{0}\leq\mathbbm{v}_{1}$, we consider the problem:

$\inf_{\begin{subarray}{l}\qquad(A,S,u)\in\mathcal{C}_{\textbf{m}}\\\
\mathcal{L}^{2}(S)=\mathbbm{v}_{0},\,\mathcal{L}^{2}({A})=\mathbbm{v}_{1}\end{subarray}}{\mathcal{F}(A,S,u)},$
(1.2)

which we tackle by employing the _direct method_ of the calculus of
variations, namely by equipping $\widetilde{\mathcal{C}}$ with a properly
chosen topology $\tau_{\mathcal{C}}$ sufficiently weak to establish a
compactness property for energy-equibounded sequences in
$\mathcal{C}_{\textbf{m}}$ and strong enough to prove the lower semicontinuity
of $\mathcal{F}$ in $\mathcal{C}_{\textbf{m}}$. The topology
$\tau_{\mathcal{C}}$ is characterized by the convergence:

$(A_{k},S_{k},u_{k})\xrightarrow[k\to\infty]{\tau_{\mathcal{C}}}(A,S,u)\,\Longleftrightarrow\,\begin{cases}\text{$\sup_{k\in\mathbb{N}}{\mathcal{H}^{1}\left(\partial
A_{k}\right)}<\infty$, $\sup_{k\in\mathbb{N}}{\mathcal{H}^{1}\left(\partial
S_{k}\right)}<\infty$,}\\\ \text{$\mathrm{sdist}\left(\cdot,\partial
A_{k}\right)\xrightarrow[k\to\infty]{}\mathrm{sdist}\left(\cdot,\partial
A\right)$ locally uniformly in $\mathbb{R}^{2}$},\\\
\text{$\mathrm{sdist}\left(\cdot,\partial
S_{k}\right)\xrightarrow[k\to\infty]{}\mathrm{sdist}\left(\cdot,\partial
S\right)$ locally uniformly in $\mathbb{R}^{2}$,}\\\
u_{k}\xrightarrow[k\to\infty]{}u\quad\text{a.e.\ in
$\mathrm{{Int}\left(A\right)}$,}\end{cases}$

where the _signed distance function_ is defined for any
$E\subset\mathbb{R}^{2}$ as follows

${\rm{sdist}}(x,\partial E):=\begin{cases}{\rm dist}(x,E)&\text{if
}x\in\mathbb{R}^{2}\setminus E,\\\ -{\rm dist}(x,E)&\text{if }x\in
E.\end{cases}$

The compactness property shared by energy-equibounded sequences
$(A_{k},S_{k},u_{k})\in\mathcal{C}_{\mathbf{m}}$ that we establish in Theorem
3.11 consists in the existence, up to a subsequence, of a possibly different
sequence
$(\widetilde{A}_{k},\widetilde{S_{k}},\widetilde{u_{k}})\in\mathcal{C}_{\mathbf{m}}$
compact in $\mathcal{C}_{\mathbf{m}}$ with respect to $\tau_{\mathcal{C}}$
such that

$\liminf_{n\rightarrow\infty}{\mathcal{F}(A_{k},S_{k},u_{k})}=\liminf_{n\rightarrow\infty}{\mathcal{F}\left(\widetilde{A}_{k},\widetilde{S_{k}},\widetilde{u_{k}}\right)}.$

This is achieved by both extending to the two-phase setting and to the
situation with the exterior graph constraint the strategy used in [36, Theorem
2.7]. For the latter, we rely on the arguments already used in [14, 27], while
for the former we used the Blaschke-type selection principle proved in [36,
Proposition 3.1] together with the Golab’s Theorem [33, Theorem 2.1] and we
implement to the two-phase setting the construction of [36, Proposition 3.6].
Such construction is needed to take care of those connected components of
$A_{k}$ that separate in the limit in multiple connected components, e.g., in
the case of neckpinches, in order to properly apply _Korn’s inequality_ just
after having introduced extra boundary to create different components also at
the level $A_{k}$ (by passing to the sequence with composite regions
$\widetilde{A}_{k}$). We notice though that the characterization of the
delamination region introduced in this manuscript allows for a simplification
in the arguments used [36, Theorem 2.7], as the surface energy does not
involve the bulk displacements, also yielding an extension of the result by
including the situation with $S\neq\emptyset$ and
$\mathbbm{v}_{1}:=\mathcal{L}^{2}(\Omega)$.

The crucial point in proving the $\tau_{\mathcal{C}}$-lower semicontinuity of
$\mathcal{F}$ is the $\tau_{\mathcal{C}}$-lower semicontinuity of
$\mathcal{S}$, as the $\tau_{\mathcal{C}}$-lower semicontinuity of
$\mathcal{W}$ directly follows by convexity similarly to [27, 36]. In order to
establish the $\tau_{\mathcal{C}}$-lower semicontinuity of $\mathcal{S}$ in
Proposition 5.13 we fix $(A_{k},S_{k},u_{k})\in\mathcal{C}_{\mathbf{m}}$ and
$(A,S,u)\in\mathcal{C}_{\mathbf{m}}$ such that
$(A_{k},S_{k},u_{k})\xrightarrow{\tau_{\mathcal{C}}}(A,S,u)$, and we associate
the positive Radon measures $\mu_{k}$ and $\mu$ in $\mathbb{R}^{2}$ to the
localized energy versions of $\mathcal{S}(A_{k},S_{k},u_{k})$ and
$\mathcal{S}(A,S,u)$, respectively. We have that

$\liminf\limits_{k\to+\infty}\mathcal{S}(A_{k},S_{k},u_{k})\geq\mathcal{S}(A,S,u)\qquad\Longleftrightarrow\qquad\liminf\limits_{k\to+\infty}\mu_{k}(\mathbb{R}^{2})\geq\mu(\mathbb{R}^{2}),$
(1.3)

and since, up to a subsequence, $\mu_{k}$ weakly* converges to some positive
Radon measure $\mu_{0}$, and $\mu$ is absolutely continuous with respect to
$\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}((\partial A\cup\partial
S)\cap\Omega)$, by proving the following estimate involving _Radon-Nikodym
derivatives_ :

$\frac{\mathrm{d}\mu_{0}}{\mathrm{d}\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(\Omega\cap(\partial
A\cup\partial
S))}\geq\frac{\mathrm{d}\mu}{\mathrm{d}\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(\Omega\cap(\partial
A\cup\partial S))}\quad\text{$\mathcal{H}^{1}$ -a.e. on $\Omega\cap(\partial
A\cup\partial S),$}$ (1.4)

which implies that
$\lim\mu_{k}(\mathbb{R}^{2})=\mu_{0}(\mathbb{R}^{2})\geq\mu(\mathbb{R}^{2}),$
in view of (1.3), the $\tau_{\mathcal{C}}$-lower semicontinuity of
$\mathcal{S}$ follows.

The proof of (1.4) is very involved and it is performed by separating
$\Omega\cap(\partial A\cup\partial S)$ in 12 portions on which we apply a
_blow-up technique_ (see, e.g., [29] ) together with _ad hoc_ (apart from the
2 portions in which it turns out that we can use [43, Theorem 20.1]) results,
i.e., Lemmas 5.7-5.12, which can be seen as the counterpart in the two-phase
setting of [36, Lemmas 4.4 and 4.5] (see Table 1 for more details on the 12
blow-ups). In order to prove Lemmas 5.7-5.12, firstly we formalize the notions
of _film islands_ , _composite voids_ , and _substrate grains_ (see Definition
5.6), secondly we prove in Lemma 5.5 that the coherent interface associated to
any configuration $(A,S,u)\in\mathcal{C}_{\textbf{m}}$ can be regarded, up to
an error and a modification of $(A,S,u)$ by passing to the family
$\widetilde{\mathcal{C}}$, as given by a finite number (depending on the
initial configuration $(A,S,u)$) of connected components, and finally we
design induction arguments (with respect to the number of such components) in
which we are able to use the induction hypothesis by “shrinking” islands,
“filling” voids, and “modifying grains in new voids” as depicted in Figures 3,
4, and 6, respectively, by means of employing the _anisotropic minimality of
segments_ [43, Remark 20.3].

The manuscript is organized as follows: in Section 2 we specify the notation
used throughout this paper, in Section 3 we introduce the model under
consideration, present some preliminary results and state the main results,
i.e., the existence of a solution to (1.2) in Theorem 3.10 together with the
compactness result of Theorem 3.11 and the lower semicontinuity result of
Theorem 3.12, in Section 4, we prove Theorem 3.11, in Section 5 we prove
Theorem 3.12, and finally in Section 6 we prove Theorem 3.10.

##  2\. Notation

In this section we collect the relevant notation used throughout the paper by
separating it for different mathematical areas.

### Linear algebra

We consider the orthonormal basis
$\left\\{\mathbf{e}_{\mathbf{1}},\mathbf{e}_{\mathbf{2}}\right\\}=\left\\{(1,0),(0,1)\right\\}$
in $\mathbb{R}^{2}$ and indicate the coordinates of points $x$ in
$\mathbb{R}^{2}$ by $(x_{1},x_{2})$. We indicate by $a\cdot
b:=\sum_{i=1}^{2}a_{i}b_{i}$ the Euclidean scalar product between points $a$
and $b$ in $\mathbb{R}^{2}$, and we denote the corresponding norm by
$|a|:=\sqrt{a\cdot a}$.

Let $\mathbb{M}^{2}$ be the set of $(2\times 2)$-matrices and by
$\mathbb{M}^{2}_{\mathrm{sym}}$ the space of symmetric $(2\times 2)$-matrices.
The space $\mathbb{M}^{2}$ is endowed with Frobenius inner product
$E:F:=\sum_{i,j=1}^{2}E_{ij}F_{ij}$ and, with a slight abuse of notation, we
denote the corresponding norm by $|E|:=\sqrt{E:E}$.

### Topology

Since the model considered in this manuscript is two-dimensional, if not
otherwise stated, all the sets are contained in $\mathbb{R}^{2}$. We denote
the cartesian product in $\mathbb{R}^{2}$ of two sets $A\subset\mathbb{R}$ and
$B\subset\mathbb{R}$ by $A\times_{\mathbb{R}^{2}}B:=\\{(a,b):\,a\in E\text{
and }b\in F\\}$. For any set $E\subset\mathbb{R}^{2}$, we denote by
${\rm{Int}}(E)$, $\overline{E}$ and $\partial E$ interior, the closure and the
topological boundary of $E$, respectively. Furthermore, we denote by $Cl(F)$
the closure of a set $F$ in $E$ with respect to the relative topology in $E$.
For any $r\in\mathbb{R}$ we define $rE:=\left\\{rx:x\in E\right\\}$.

Given $\nu\in\mathbb{S}^{1}$ and $x\in\mathbb{R}^{2}$,
$Q_{\rho,\nu}\left(x\right)$ is the open square of sidelength $2\rho>0$
centered at $x$ and whose sides are either perpendicular or parallel to $\nu$.
Note that if $\nu=\textbf{e}_{\textbf{1}}=\bm{e}_{\bm{2}}$ and $x=0$,
$Q_{\rho}:=Q_{\rho,\nu}(0)=\left(-\rho,\rho\right)\times\left(-\rho,\rho\right)$.
We define by $I_{\rho}$ the symmetric segment
$I_{\rho}:=\left[-\rho,\rho\right]\subset\mathbb{R}$. Furthermore, a
_(parametrized) curve_ in $\mathbb{R}^{2}$ is a continuous function
$r:[a,b]\to\mathbb{R}^{2}$ injective in $(a,b)$ for $a,b\in\mathbb{R}$ with
$a<b$ and its image is referred to as the _support of the curve_. Moreover,
since any continuous bijection between compact sets has a continuous inverse,
the support of any curve is homeomorphic to a closed interval in $\mathbb{R}$
(see [25, Section 3.2]).

Given $y_{0}\in\mathbb{R}^{2}$ and $\rho>0$, we define the _blow-up map_
centered in $y_{0}$ with radius $\rho$ as the function
$\sigma_{\rho,y_{0}}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ by

$\sigma_{\rho,y_{0}}\left(y\right):=\frac{y-y_{0}}{\rho}$

for all $y\in\mathbb{R}^{2}$. Observe that if we apply $\sigma_{\rho,x}$ to
$Q_{\rho,\nu}\left(x\right)$ and $\overline{Q_{\rho,\nu}\left(x\right)}$, we
obtain that
$\sigma_{\rho,x}\left(Q_{\rho,\nu}\left(x\right)\right)=Q_{1,\nu}(0)$ and
$\sigma_{\rho,x}(\overline{Q_{\rho,\nu}\left(x\right)})=\overline{Q_{1,\nu}(0)}$.
When $y_{0}=0$, we write $\sigma_{\rho}$ instead of $\sigma_{\rho,0}$. We
denote by $\pi_{i}$ the projections onto $x_{i}$-axis for $i=1,2$, i.e., the
maps $\pi_{1}:\mathbb{R}^{2}\to\mathbb{R}$ and
$\pi_{2}:\mathbb{R}^{2}\to\mathbb{R}$ are such that
$\pi_{1}(x_{1},x_{2})=x_{1}$ and $\pi_{2}(x_{1},x_{2})=x_{2}$ for every
$x=(x_{1},x_{2})\in\mathbb{R}^{2}$.

Finally, let ${\rm dist}(\cdot,E)$ and ${\rm{sdist}}(\cdot,\partial E)$ be the
_distance_ function from $E$ and the _signed distance_ from $\partial E$
respectively, where we recall that ${\rm{sdist}}(\cdot,\partial E)$ is defined
by

${\rm{sdist}}(x,\partial E):=\begin{cases}{\rm dist}(x,E)&\textrm{if
}x\in\mathbb{R}^{2}\setminus E,\\\ -{\rm dist}(x,E)&\textrm{if }x\in
E\end{cases}$

for every $x\in\mathbb{R}^{2}$.

### Geometric measure theory

We denote by $\mathcal{L}^{2}{(B)}$ the $2$-dimensional Lebesgue measure of
any Lebesgue measurable set $B\subset\mathbb{R}^{2}$ and by $\mathbbm{1}_{B}$
the characteristic function of $B$. For $\alpha\in\left[0,1\right]$ we denote
by $B^{(\alpha)}$ the set of points of density $\alpha$ in $\mathbb{R}^{2}$,
i.e.,

$B^{(\alpha)}:=\left\\{x\in\mathbb{R}^{2}:\lim_{r\to
0}{\frac{\mathcal{L}^{2}({B\cap
B_{r}\left(x\right)})}{\mathcal{L}^{2}({B_{r}\left(x\right)})}}=\alpha\right\\}.$

We denote the distributional derivative of a function $f\in
L^{1}_{\mathrm{loc}}(\mathbb{R}^{2})$ by $Df$ and define it as the operator
$D:C^{\infty}_{c}(\mathbb{R}^{2})\to\mathbb{R}$ such that

$\int_{\mathbb{R}^{2}}{Df\cdot\varphi}=-\int_{\mathbb{R}^{2}}{f\cdot\gradient\varphi\,dx},$

for any $\varphi\in C^{\infty}_{0}(\mathbb{R}^{2})$.

We denote with $\mathcal{H}^{1}$ the 1-dimensional Hausdorff measure. We say
that $K\subset\mathbb{R}^{2}$ is $\mathcal{H}^{1}$-rectifiable if
$0<\mathcal{H}^{1}(K)<+\infty$ and $\theta_{*}(K,x)=\theta^{*}(K,x)=1$ for
$\mathcal{H}^{1}$-a.e. $x\in K$, where

$\theta_{*}(K,x):=\liminf_{r\to 0^{+}}{\frac{\mathcal{H}^{1}(K\cap
B_{r}\left(x\right))}{2r}}\quad\textrm{and}\quad\theta^{*}(K,x):=\limsup_{r\to
0^{+}}{\frac{\mathcal{H}^{1}(K\cap B_{r}\left(x\right))}{2r}}.$

We define sets of finite perimeter as in [4, Definion 3.35] and the reduced
boundary $\partial^{*}E$ of a set $E$ of finite perimeter by

$\partial^{*}E:=\left\\{x\in\mathbb{R}^{2}:\exists\nu_{E}\left(x\right):=-\lim_{r\to
0}{\frac{D\mathbbm{1}_{E}(B_{r}\left(x\right))}{\absolutevalue{D\mathbbm{1}_{E}}(B_{r}\left(x\right))}},\absolutevalue{\nu_{E}\left(x\right)}=1\right\\},$
(2.1)

where we refer to $\nu_{E}\left(x\right)$ as the measure-theoretical unit
normal at $x\in\partial E$.

Let $x\in\mathbb{R}^{2}$, if $\nu_{E}\left(x\right)$ exists
$T_{x,\nu_{E}\left(x\right)}:=\left\\{y\in\mathbb{R}^{2}:y\cdot\nu_{E}\left(x\right)=0\right\\}$
stands for the approximate tangent line and
$H_{x,\nu_{E}\left(x\right)}:=\left\\{y\in\mathbb{R}^{2}:y\cdot\nu_{E}\left(x\right)\leq
0\right\\}$ is its corresponding half space at $x$.

For any set $E\subset\mathbb{R}^{2}$ of finite perimeter, by [43, Corollary
15.8 and Theorem 16.2] it yields that

$\displaystyle\mathcal{H}^{1}(E^{(1/2)}\setminus\partial^{*}E)=0\quad\text{and}\quad\partial^{*}E\subset
E^{(1/2)}.$ (2.2)

Moreover, for any set $E\subset\mathbb{R}^{2}$ of finite perimeter, we have

$\partial E=N\cup\partial^{*}E\cup(E^{(1)}\cup E^{(0)})\cap\partial E,$ (2.3)

where $N$ is a $\mathcal{H}^{1}$-negligible (see [43, Section 16.1]).

We denote by $U_{k,l}$ the (half-open) dyadic square, i.e.,

$U_{k,l}:=[0,2^{-k})\times_{\mathbb{R}^{2}}[0,2^{-k})+2^{-k}l$

for any $k\in\mathbb{N}$ and $l\in\mathbb{Z}\times\mathbb{Z}$. We denote by
$\mathcal{Q}$ the family of dyadic squares and by $\mathcal{N}^{1}$ the net
measure. More precisely, for any Borel set $E$,

$\mathcal{N}^{1}(E):=\liminf_{\delta\to 0}{\mathcal{N}^{1}_{\delta}}(E),$
(2.4)

is the $\mathcal{N}^{1}$-measure of $E$, where

$\mathcal{N}^{1}_{\delta}(E):=\inf\\{\sum_{i\in
I}\mathrm{diam}(U_{i}):\left\\{U_{i}\right\\}_{i\in I}\subset\mathcal{Q}\text{
is a countable disjoint covering of }E\\\ \text{ and
}\mathrm{diam}(U_{i})\leq\delta\\}.$ (2.5)

Note that this measure does not coincide with the Hausdorff measure, however
we have the following equivalence (see [44, Chapter 5])

$\mathcal{H}^{1}(E)\leq\mathcal{N}^{1}(E)\leq
2^{\frac{5}{2}}\mathcal{H}^{1}(E),$ (2.6)

for any Borel set $E\subset\mathbb{R}^{2}$.

### Functions of bounded pointwise variation

Given a function $h:[a,b]\to\mathbb{R}$ we denote the pointwise variation of
$h$ by

$\textrm{Var}\,h:=\sup{\left\\{\sum_{i}^{n}\absolutevalue{h(x_{i})-h(x_{i-1})}:P:=\\{x_{0},\ldots,x_{n}\\}\
\textrm{is a partition of $[a,b]$}\right\\}}.$

We say that $h:(a,b)\to\mathbb{R}$ has finite pointwise variation if
$\textrm{Var}\,h<\infty.$ We recall that for any function $h$ such that
$\textrm{Var}\,h<\infty$, $h$ has at most countable discontinuities and there
exists $h(x^{\pm}):=\lim_{z\to x^{\pm}}h(z)$. In the following given a
function $h:[a,b]\to\mathbb{R}$ with finite pointwise variation, we define

$h^{-}(x):=\min\\{h(x^{+}),h(x^{-})\\}=\liminf_{z\to x}h(z)$

and

$h^{+}(x):=\max\\{h(x^{+}),h(x^{-})\\}=\limsup_{z\to x}h(z).$

In view of [41, Corollary 2.23] and with slightly abuse of notation, the
limits

$h^{+}(a):=\lim_{x\to a^{-}}h(x)\quad\text{and}\quad h^{-}(b):=\lim_{x\to
b^{-}}h(x)$ (2.7)

are finite.

##  3\. Mathematical setting and main results

In this section we present the model introduced in this manuscript with some
preliminaries, and then we state the main results of the paper outlining the
consequences for the related one-phase setting of [36, 37, 38] and the
multiple-phase setting of film multilayers considered in [42].

###  3.1. The two-phase model

Let $\Omega:=(-l,l)\times_{\mathbb{R}^{2}}(-L,L)\subset\mathbb{R}^{2}$ for
positive parameters $l,L\in\mathbb{R}$. We begin by introducing the family
$\mathcal{C}_{\mathbf{m}}$ of admissible configurations and, in particular,
the admissible substrate regions.

Roughly speaking, an admissible substrate region $S\subset\Omega$ is
characterized as the subgraph of a upper semicontinuous height function $h$
with finite pointwise variation to which we subtract a closed
$\mathcal{H}^{1}$-rectifiable set $K$ such that $\mathcal{H}^{1}(K)<\infty$,
which represents the substrate internal cracks. More precisely, we consider
the _family of admissible (substrate) heights_ $\mathrm{AH}(\Omega)$ defined
by

$\displaystyle\mathrm{AH}(\Omega):=\left\\{h:[-l,l]\to[0,L]:h\text{ is upper
semicontinous and }\mathrm{Var}\,h<\infty\right\\}$ (3.1)

and let $S_{h}$ denote the closed subgraph with height
$h\in\mathrm{AH}(\Omega)$, i.e.,

$S_{h}:=\\{(x,y):-l<x<l,y\leq h(x)\\}.$ (3.2)

We then define the _family of admissible (substrate) cracks_
$\text{AK}(\Omega)$ by

$\mathrm{AK}(\Omega):=\\{K\subset\Omega:\text{ $K$ is a closed set in
$\mathbb{R}^{2}$, $\mathcal{H}^{1}$-rectifiable and
}\mathcal{H}^{1}(K)<\infty\\}$ (3.3)

and the _family of pairs of admissible heights and cracks_
$\text{AHK}(\Omega)$ by

$\mathrm{AHK}(\Omega):=\\{(h,K)\in\mathrm{AH}(\Omega)\times\mathrm{AK}(\Omega):\,K\subset{\overline{{\rm{Int}}(S_{h})}}\\}.$
(3.4)

Finally, given $(h,K)\in\text{AHK}(\Omega)$ we refer to the set

$S_{h,K}:={({S_{h}}\setminus K)}\cap\Omega,$ (3.5)

as the _substrate with height $h$ and cracks $K$_, and we define the _family
of admissible substrates_ as

$\mathrm{AS}(\Omega):=\\{S\subset\Omega\,:\,\text{$S=S_{h,K}$ for a pair
$(h,K)\in\mathrm{AHK}(\Omega)$}\\}.$ (3.6)

We observe that

$\partial{S_{h,K}}=\partial{S_{h}}\cup K$ (3.7)

for every $S_{h,K}\in\mathrm{AS}(\Omega)$, so that $\overline{S_{h,K}}=S_{h}$
and ${\rm{Int}}(S_{h,K})={\rm{Int}}(S_{h})\setminus K$. We denote the _jumps
points_ and the _vertical filament points_ of the graph of $h\in{\rm
AH}(\Omega)$ by

$J(h):=\\{x\in(-l,l):h^{-}(x)\neq h^{+}(x)\\}\quad\text{and}\quad
F(h):=\\{x\in(-l,l):h^{+}(x)<h(x)\\},$ (3.8)

respectively. By [41, Corollary 2.23] and thanks to the fact that $h\in{\rm
AH}(\Omega)$, $J(h)$ and $F(h)$ are countable. Moreover, it follows that
$\partial{S_{h}}$ is connected and, $\partial S_{h}$ and $\partial S_{h,K}$
have finite $\mathcal{H}^{1}$-measure. By [25, Lemma 3.12 and Lemma 3.13], for
any $h\in\mathrm{AH}$, $\partial S_{h}$ is rectifiable and applying the
Besicovitch-Marstrand-Mattila Theorem (see [4, Theorem 2.63]),
$\partial{S_{h}}$ is $\mathcal{H}^{1}$-rectifiable, and hence, $\partial
S_{h,K}$ is $\mathcal{H}^{1}$-rectifiable. Furthermore, applying [37,
Proposition A.1] $S_{h}$ and $S_{h,K}$ are sets of finite perimeter.

The following result allows to interchangeably bound the pointwise variation
of a function $h\in{\rm AH}(\Omega)$ from the $\mathcal{H}^{1}$-measure of
$\partial S_{h}$, and vice versa.

######  Lemma 3.1.

Let $h\in{\rm AH}(\Omega)$. Then

$\mathrm{Var}\,h\leq\mathcal{H}^{1}(\partial S_{h})\leq 2l+2\mathrm{Var}\,h,$
(3.9)

where $S_{h}$ is defined as in (3.2).

###### Proof.

The proof is divided in two steps.

Step 1. We prove the left inequality of (3.9). We proceed as in [14, Section
A.2.3]. Let $m\in\mathbb{N}$ and let
$\\{l_{i}:i\in\\{0,\ldots,m\\},l_{0}=-l,l_{m}=l\text{ and }l_{i}<l_{i+1}\\}$
be a partition of $[-l,l]$. Take $i\in\\{0,\ldots,m-1\\}$ and define by
$L_{i}$ the segment connecting $(l_{i},h(l_{i}))$ with $(l_{i+1},h(l_{i+1}))$.
By [25, Lemma 3.12], there exists a parametrization
$r_{i}:[0,1]\to\mathbb{R}^{2}$ of
$\partial{\rm{Int}}(S_{h})\cap((l_{i},l_{i+1})\times_{\mathbb{R}^{2}}[0,L))$,
whose support $\gamma_{i}$ joins the points $(l_{i},h(l_{i}))$ and
$(l_{i+1},h(l_{i+1}))$, furthermore, it follows that

$\absolutevalue{h(l_{i+1})-h(l_{i})}\leq\sqrt{\absolutevalue{l_{i+1}-l_{i}}^{2}+\absolutevalue{h(l_{i+1})-h(l_{i})}^{2}}=\mathcal{H}^{1}(L_{i})\leq\mathcal{H}^{1}(\gamma_{i}).$

Moreover, repeating the same argument for any $i\in\\{0,\ldots,m-1\\}$ we have
that

$\sum_{i=0}^{m-1}\absolutevalue{h(l_{i+1})-h(l_{i})}\leq\sum_{i=0}^{m-1}\mathcal{H}^{1}(\gamma_{i})\leq\mathcal{H}^{1}(\partial
S_{h}),$

where in the last inequality we have used that

$\partial S_{h}=\partial{\rm{Int}}(S_{h})\cup(\partial S_{h}\cap
S_{h}^{(0)})\cup N,$ (3.10)

where $N$ is a $\mathcal{H}^{1}$-negligible set. Taking the supremum aver all
partitions of we obtain the left inequality of (3.9).

Step 2. In this step, we prove the right inequality of (3.9). We observe that

${\rm{Int}}(S_{h})=\\{(x,y):-l<x<l,y<h^{-}(x)\\}$

and so,

$\begin{matrix}r:&[-l,l]&\to&\partial{\rm{Int}}(S_{h})\\\
&x&\mapsto&(x,h^{-}(x))\end{matrix}$

is a parametrization of $\partial{\rm{Int}}(S_{h})$, whose support we denote
by $\gamma$. Therefore, from [41, Definition 4.6 and Remark 4.20] it follows
that

$\mathcal{H}^{1}(\partial{\rm{Int}}(S_{h}))=\sup\left\\{\sum_{i=0}^{m-1}\absolutevalue{r(x_{i+1})-r(x_{i})}\right\\},$
(3.11)

where the supremum is taken over all partitions of $[-l,l]$. By definition of
$r$ and thanks to (3.11), we see that

$\mathcal{H}^{1}(\partial{\rm{Int}}(S_{h}))\leq\sup\left\\{\sum_{i=0}^{m-1}\left(\absolutevalue{x_{i+1}-x_{i}}+\absolutevalue{h^{-}(x_{i+1})-h^{-}(x_{i})}\right)\right\\}\leq
2l+\mathrm{Var}\,h,$

where we used the fact that $\mathrm{Var}\,h^{-}\leq\mathrm{Var}\,h$. Finally,
by (3.10) we have that $\mathcal{H}^{1}(\partial
S_{h})=\mathcal{H}^{1}(\partial{\rm{Int}}(S_{h}))+\mathcal{H}^{1}(\partial
S_{h}\cap S_{h}^{(0)})$ and since $\mathcal{H}^{1}(\partial S_{h}\cap
S_{h}^{(0)})\leq\mathrm{Var}\,h$, by the fact that $\partial S_{h}\cap
S_{h}^{(0)}$ is the union of vertical segments, we can deduce the right
inequality of (3.9). ∎

We now introduce the family of admissible region pairs and configurations.

######  Definition 3.2 (Admissible regions and configurations).

We define the families of admissible pairs $\mathcal{B}(\Omega)$ and of
admissible configurations $\mathcal{C}$ by

$\mathcal{B}(\Omega):=\\{(A,S):A\text{ is
}\mathcal{L}^{2}\text{-measurable,}\,\partial A\textrm{ is
$\mathcal{H}^{1}$-rectifiable, }\mathcal{H}^{1}(\partial A)<\infty,\,\\\
{\text{there exists
}}(h,K)\in\mathrm{AHK}(\Omega),\,S=S_{h,K}\in\mathrm{AS}(\Omega),\,\\\
S_{h,K}\subset\overline{A}\subset\overline{\Omega}\text{ and }\partial
A\cap{\rm{Int}}(S_{h,K})=\emptyset\\},$

and

${{\mathcal{C}}}:=\left\\{\left(A,S,u\right){\in\widetilde{\mathcal{C}}}:{(A,S)\in\mathcal{B}}\right\\},$

respectively.

In the following we also refer to the sets $\overline{A}$, $\overline{S}$, and
$\overline{A}\setminus S^{(1)}$ with respect to an admissible pair
$(A,S)\in\mathcal{B}$ as the _composite region_ , the _substrate region_ , and
the _film region_ of the admissible pair. Moreover, we refer to $S^{(1)}$ and
$A^{(1)}\cap S^{0}$ as the _substrate_ and the _film bulk regions_ ,
respectively.

In Theorem 3.12 we will need to consider a natural extension of the families
$\mathcal{B}$ and $\mathcal{C}$, which we denote as $\widetilde{\mathcal{B}}$
and $\widetilde{\mathcal{C}}$, respectively.

######  Definition 3.3.

We define the families of admissible pairs $\widetilde{\mathcal{B}}(\Omega)$
and of admissible configurations $\widetilde{\mathcal{C}}$ by

$\widetilde{\mathcal{B}}(\Omega):=\\{(A,S):A,S\text{ are
}\mathcal{L}^{2}\text{-measurable,}\,\partial A,\partial S\textrm{ are
$\mathcal{H}^{1}$-rectifiable, }\\\ \mathcal{H}^{1}(\partial
A),\mathcal{H}^{1}(\partial
S)<\infty,\,S\subset\overline{A}\subset\overline{\Omega}\text{ and }\partial
A\cap{\rm{Int}}(S)=\emptyset\\},$

and

$\widetilde{\mathcal{C}}:=\left\\{\left(A,S,u\right):(A,S)\in\widetilde{\mathcal{B}},u\in{H^{1}_{\mathrm{{loc}}}\left({\rm{Int}}(A);\mathbb{R}^{2}\right)}\right\\},$

respectively.

We observe that $\mathcal{B}\subset\widetilde{\mathcal{B}}$, since for any
$(A,S)\in\mathcal{B}$ there exists $(h,K)\in\text{AHK}(\Omega)$ such that
$S=S_{h,K}\in\text{AS}(\Omega)$ and $\partial S_{h,K}$ is
$\mathcal{H}^{1}$-rectifiable, and thus,
$(A,S)\in\widetilde{\mathcal{B}}(\Omega)$.

Notice that, for simplicity, in the absence of ambiguity we omit the
dependence on the set $\Omega$ in the notation
${\widetilde{\mathcal{B}}(\Omega)\,}$ and $\mathcal{B}(\Omega)$ by writing in
the following only ${\widetilde{\mathcal{B}}\,}$ and $\mathcal{B}$,
respectively.

######  Remark 3.4.

We observe that any bounded $\mathcal{L}^{2}$-measurable set
$A\subset\mathbb{R}^{2}$ such that $\mathcal{H}^{1}(\partial A)<\infty$ is a
set of finite perimeter in $\mathbb{R}^{2}$ by [37, Proposition A.1].

We now equip the family $\widetilde{\mathcal{B}}$ with a topology.

######  Definition 3.5 (${\tau_{\mathcal{B}}}$-convergence).

A sequence $\\{(A_{k},S_{k})\\}\subset\widetilde{\mathcal{B}}$
${\tau_{\mathcal{B}}}$-converges to $(A,S)\in\widetilde{\mathcal{B}}$, if

  * -

$\sup_{k\in\mathbb{N}}{\mathcal{H}^{1}\left(\partial
A_{k}\right)}<\infty$,$\sup_{k\in\mathbb{N}}{\mathcal{H}^{1}\left(\partial
S_{k}\right)}<\infty$,

  * -

$\mathrm{sdist}\left(\cdot,\partial
A_{k}\right)\rightarrow\mathrm{sdist}\left(\cdot,\partial A\right)$ locally
uniformly in $\mathbb{R}^{2}$ as $k\rightarrow\infty$,

  * -

$\mathrm{sdist}\left(\cdot,\partial
S_{k}\right)\rightarrow\mathrm{sdist}\left(\cdot,\partial S\right)$ locally
uniformly in $\mathbb{R}^{2}$ as $k\rightarrow\infty$.

It will follow from Lemma 3.9 below that the $\tau_{B}$-convergence is closed
in the subfamily of admissible triples
$\mathcal{B}_{\mathbf{m}}\subset\mathcal{B}$ whose definition depending on the
vector ${\mathbf{m}}=(m_{0},m_{1})\in\mathbb{N}\times\mathbb{N}$ we now
provide.

######  Definition 3.6.

For any $\mathbf{m}:={(m_{0},m_{1})}\in\mathbb{N}\times\mathbb{N}$ the family
$\mathcal{B}_{\mathbf{m}}$ is given by all pairs $(A,S)\in\mathcal{B}$ such
that $\partial A$ and $\partial S$ have at most $m_{1}$ and $m_{0}$-connected
components, respectively. Let us also define

$\mathcal{C}_{\mathbf{m}}:=\left\\{\left(A,{S,}u\right)\in\mathcal{C}:(A,S)\in\mathcal{B}_{\mathbf{m}}\right\\}\subset\mathcal{C}.$
(3.12)

We denote the topology with which we equip the family
$\widetilde{\mathcal{C}}$ by $\tau_{\mathcal{C}}$.

######  Definition 3.7 ($\tau_{\mathcal{C}}$-Convergence).

A sequence
$\\{\left(A_{k},{S_{k}},u_{k}\right)\\}_{k\in\mathbb{N}}\subset\mathcal{C}$ is
said to $\tau_{\mathcal{C}}$-convergence to
$\left(A,S,u\right)\in\mathcal{C}$, denoted as
$\left(A_{k},{S_{k}},u_{k}\right)\xrightarrow{\tau_{\mathcal{C}}}\left(A,S,u\right)$,
if

  * -

$(A_{k},{S_{k}})\xrightarrow{\tau_{\mathcal{B}}}(A,S)$,

  * -

$u_{k}\rightarrow u$ a.e. in $\mathrm{{Int}\left(A\right)}$.

We now state some properties of the topology $\tau_{\mathcal{C}}$.

######  Remark 3.8.

We notice that:

  * (i)

The following assertions are equivalent

    * (i.$1$)

${\rm{sdist}}(\cdot,\partial E_{k})\rightarrow{\rm{sdist}}(\cdot,\partial E)$
locally uniformly in $\mathbb{R}^{2}$.

    * (i.$2$)

$E_{k}\xrightarrow{\mathcal{K}}\overline{E}$ and $\mathbb{R}^{2}\setminus
E_{k}\xrightarrow{\mathcal{K}}\mathbb{R}^{2}\setminus{\rm{Int}}(E)$.

Moreover, these imply that $\partial E_{k}\xrightarrow{\mathcal{K}}\partial
E$.

  * (ii)

If there exist $(h_{k},K_{k})\in\mathrm{AHK}(\Omega)$ and
$(h,K)\in\mathrm{AHK}(\Omega)$ such that $E_{k}=E_{h_{k},K_{k}}\in{\rm
AS}(\Omega)$ and $E=E_{h,K}\in AS$, for every $k\in\mathbb{N}$, we observe
that Item (i.$1$) above is equivalent to

$E_{k}=E_{h_{k},K_{k}}\xrightarrow{\mathcal{K}}E_{h}\quad\text{and}\quad\mathbb{R}^{2}\setminus
E_{k}=\mathbb{R}^{2}\setminus
E_{h_{k},K_{k}}\xrightarrow{\mathcal{K}}(\mathbb{R}^{2}\setminus{\rm{Int}}(E_{h}))\cup
K,$

where $E_{h_{k},K_{k}}$, $E_{h,K}$ are defined as in (3.5) and $E_{h}$ is
defined as in (3.2).

  * (iii)

Let $\\{(E_{k},F_{k})\\}\subset\mathbb{R}^{2}\times\mathbb{R}^{2}$ be a
sequence of bounded sets and let $E,F\subset\mathbb{R}^{2}$ be two bounded
sets such that $\partial E_{k}\xrightarrow{\mathcal{K}}\partial E$ and
$\partial F_{k}\xrightarrow{\mathcal{K}}\partial F$. In view of the Kuratowski
convergence (see [4, Section 6.1], [18, Chapter 4] or [36, Appendix A.1]), we
observe that for every $x\in\partial E\setminus\partial F$ there exist
$r:=r(x)>0$ and $k_{r,x}\in\mathbb{N}$ such that $B(x,r)\cap\partial
F_{k}=\emptyset$ for any $k\geq k_{r,x}$. Similarly, for every $x\in\partial
F\setminus\partial E$ there exists $r^{\prime}:=r^{\prime}(x)>0$ and
$k_{r^{\prime},x}\in\mathbb{N}$ such that $B(x,r)\cap\partial E_{k}=\emptyset$
for any $k\geq k_{r^{\prime},x}$.

From the next result the closedness and the compactness (see Theorem 4.2) of
the family $\mathcal{B}_{\textbf{m}}$ with respect to the topology
$\tau_{\mathcal{B}}$ follows for every
$\textbf{m}:=(m_{0},m_{1})\in\mathbb{N}\times\mathbb{N}$.

######  Lemma 3.9.

Let $\\{E_{k}\\}$ be a sequence of $\mathcal{L}^{2}$-measurable subsets of
$\overline{\Omega}$ having $\mathcal{H}^{1}$-rectifiable boundaries $\partial
E_{k}$ with at most $m$-connected components such that

  * -

$\sup_{k}\mathcal{H}^{1}(\partial E_{k})<\infty$,

  * -

${\rm{sdist}}(\cdot,\partial E_{k})\to{\rm{sdist}}(\cdot,\partial E)$ locally
uniformly in $\mathbb{R}^{2}$ as $k\to\infty$ for a set
$E\subset\mathbb{R}^{2}$.

Then, $\partial E$ is $\mathcal{H}^{1}$-finite, $\mathcal{H}^{1}$-rectifiable,
and with at most $m$-connected components, and $E\subset\overline{\Omega}$ is
$\mathcal{L}^{2}$-measurable. Furthermore, if
$E_{k}=E_{h_{k},K_{k}}\in\mathrm{AS}(\Omega)$ for every $k\in\mathbb{N}$ and
for some $(h_{k},K_{k})\in\mathrm{AHK}(\Omega)$, then

$\sup_{k}(\mathcal{H}^{1}(K_{k})+\mathrm{Var}\,h_{k})<\infty$ (3.13)

and there exists $(h,K)\in\mathrm{AHK}(\Omega)$ such that
$E=E_{h,K}\in\mathrm{AS}(\Omega)$.

###### Proof.

The fact that $\partial E$ is $\mathcal{H}^{1}$-finite,
$\mathcal{H}^{1}$-rectifiable, and with at most $m$-connected components, is a
direct consequence of [36, Lemma 3.2]. Since $\mathcal{H}^{1}(\partial
E)<\infty$, it follows that
$\mathcal{L}^{2}(\overline{E}\setminus{\rm{Int}}(E))=\mathcal{L}^{2}(\partial
E)=0$, by applying [6, Theorem 14.5] to
$E\setminus{\rm{Int}}(E)\subset\overline{E}\setminus{\rm{Int}}(E)$ we infer
that $E\setminus{\rm{Int}}(E)$ is $\mathcal{L}^{2}$-measurable and so,
$\mathcal{L}^{2}(E\setminus{\rm{Int}}(E))=0$. Therefore,
$E=E\setminus{\rm{Int}}(E)\cup{\rm{Int}}(E)$ is $\mathcal{L}^{2}$-measurable.

It remains to prove the last assertion of the statement. Let
$(h_{k},K_{k})\in\mathrm{AHK}(\Omega)$ such that
$E_{k}=E_{h_{k},K_{k}}\in\mathrm{AS}(\Omega)$ for every $k$. We begin by
observing that (3.13) is a direct consequence of (3.7), by applying (3.9) to
$h_{k}$. To conclude the proof we proceed in 2 steps.

Step 1. We claim that $\overline{E}=E_{h}$, where $h$ is the upper
semicontinous function defined by

$h(x_{1}):=\sup\\{\limsup_{k\to\infty}h_{k}(x_{1}^{k}):x^{k}_{1}\to x_{1}\\}.$

Let $x=(x_{1},x_{2})\in\overline{E}$, by Remark 3.8-(i) we observe that there
exists $x_{k}=(x_{1}^{k},x_{2}^{k})\in E_{k}$ such that $x_{k}\to x$. We
deduce that

$x_{2}=\lim_{k\to\infty}x^{k}_{2}\leq\limsup_{k\to\infty}h_{k}(x_{1}^{k})\leq
h(x_{1}),$

and by (3.2) we deduce that $\overline{E}\subset E_{h}$. Now let
$x=(x_{1},x_{2})\in E_{h}$, by definition we observe that

$x_{2}\leq h(x_{1}):=\sup\\{\limsup_{k\to\infty}h_{k}(x^{k}_{1}):x^{k}_{1}\to
x_{1}\\}.$

Let $x_{k}=(x^{k}_{1},x^{k}_{2})\in\Omega$ such that $x^{k}_{1}\to x_{1}$,
$h_{k}(x^{k}_{1})\to h(x_{1})$ and define
$x_{2}^{k}:=\min\\{x_{2},h_{k}(x^{k}_{1})\\}$. It follows that
$x_{k}\in\overline{E_{k}}$ and $x_{k}\to x$, by Kuratowski convergence we have
that $x\in\overline{E}$, therefore, $E_{h}\subset\overline{E}$.

Step 2. We claim that ${\rm{Int}}(E)={\rm{Int}}({E_{h,K}})$, where
$K:=\partial E\cap{\overline{{\rm{Int}}(E_{h})}}$. Notice that $K$ is a closed
set in $\mathbb{R}^{2}$ and since $\partial E$ is
$\mathcal{H}^{1}$-rectifiable, we deduce that $K$ is also
$\mathcal{H}^{1}$-rectifiable. On one hand, we see that

${\rm{Int}}(E)={\rm{Int}}(E)\setminus(\partial
E\cap{\overline{{\rm{Int}}(E_{h})}})\subset{\rm{Int}}(E_{h})\setminus(\partial
E\cap{\overline{{\rm{Int}}(E_{h})}})=:{\rm{Int}}(E_{h})\setminus
K={\rm{Int}}({E_{h,K}}),$

where in the first equality we used the fact that ${\rm{Int}}(E)\cap\partial
E=\emptyset$ and in the inclusion we used Step 1 and the fact that
$E\subset\overline{E}=E_{h}$. On the other hand, let
$x\in{\rm{Int}}(E_{h,K})={\rm{Int}}(E_{h})\setminus(\partial
E\cap{\overline{{\rm{Int}}(E_{h})}})$ and assume by contradiction that
$x\notin{\rm{Int}}(E)$. This assumption implies that either $x\in\partial E$
or $x\in\Omega\setminus\overline{E}$, which is a contradiction by the facts
that $\partial E\subset\overline{E}=E_{h}$ and

$x\in{\rm{Int}}(E_{h})\setminus(\partial
E\cap{\overline{{\rm{Int}}(E_{h})}})={\rm{Int}}(E_{h})\setminus\partial
E\subset E_{h}=\overline{E}.$

Finally, observe that $(h,K)\in\mathrm{AHK}(\Omega)$. Thanks to the uniqueness
of Kuratowski convergence, the facts that $\overline{E_{h,K}}=E_{h}$ and
${\rm{Int}}(E_{h,K})={\rm{Int}}(E_{h})\setminus(\partial
E\cap{\overline{{\rm{Int}}(E_{h})}})$, and in view of Remark 3.8-(i) we
conclude from the previous two steps that $E=E_{h,K}$. ∎

The total energy
$\mathcal{F}:\widetilde{\mathcal{C}}\rightarrow\left[0,+\infty\right]$ of
admissible configurations is given as the sum of two contributions, namely the
surface energy $\mathcal{S}$ and the elastic energy $\mathcal{W}$, i.e.,

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

for any $(A,S,u)\in\widetilde{\mathcal{C}}$, where we observe that the surface
energy does not depend on the displacements (as a difference from [36, 37]).
The surface energy $\mathcal{S}$ is defined for any
$(A,S)\in\widetilde{\mathcal{B}}$ by

$\begin{split}{\mathcal{S}}(A,{S})&:=\int_{\Omega\cap(\partial^{*}A\setminus\partial^{*}{S})}{\varphi_{\mathrm{F}}(x,\nu_{A}\left(x\right))\,d\mathcal{H}^{1}}+\int_{\Omega\cap\partial^{*}{S}\cap\partial^{*}A}{\varphi(x,\nu_{A}\left(x\right))\,d\mathcal{H}^{1}}\\\
&\quad+\int_{\Omega\cap(\partial^{*}{S}\setminus\partial A)\cap
A^{(1)}}{\varphi_{\mathrm{FS}}(x,\nu_{S}\left(x\right))\,d\mathcal{H}^{1}}+\int_{\Omega\cap\partial^{*}{S}\cap\partial
A\cap
A^{(1)}}{(\varphi_{\mathrm{F}}+\varphi)(x,\nu_{A}\left(x\right))\,d\mathcal{H}^{1}}\\\
&\quad+\int_{\Omega\cap\partial A\cap A^{(1)}\cap
S^{(0)}}{2\varphi_{\mathrm{F}}(x,\nu_{A}\left(x\right))\,d\mathcal{H}^{1}}+\int_{\Omega\cap\partial
A\cap
A^{(0)}}{2\varphi^{\prime}(x,\nu_{A}\left(x\right))\,d\mathcal{H}^{1}}\\\
&\quad+\int_{\Omega\cap(\partial{S}\setminus\partial A)\cap\left(S^{(1)}\cup
S^{(0)}\right)\cap
A^{(1)}}{2\varphi_{\mathrm{FS}}(x,\nu_{S}\left(x\right))\,d\mathcal{H}^{1}}\\\
&\quad+\int_{\Omega\cap\partial{S}\cap\partial A\cap
S^{(1)}}{2\varphi(x,\nu_{A}\left(x\right))\,d\mathcal{H}^{1}},\end{split}$
(3.14)

where
$\varphi_{\mathrm{F}},\,\varphi_{\mathrm{FS}}:\overline{\Omega}\times\mathbb{R}^{2}\to[0,\infty]$
and, given also the function
$\varphi_{\mathrm{S}}:\overline{\Omega}\times\mathbb{R}^{2}\to[0,\infty]$, we
define the functions $\varphi$ and $\varphi^{\prime}$ in in
$C(\overline{\Omega}\times\mathbb{R}^{2};[0,\infty])$ by

$\varphi:=\min\\{\varphi_{\mathrm{S}},\varphi_{\mathrm{F}}+\varphi_{\mathrm{FS}}\\}\qquad\text{and}\qquad\varphi^{\prime}:=\min\\{\varphi_{\mathrm{F}},\varphi_{\mathrm{S}}\\}.$

Notice that $\varphi_{\mathrm{F}},\varphi_{\mathrm{S}},\varphi_{\mathrm{FS}}$
represent the anisotropic surface tensions of the film/vapor, the
substrate/vapor and the substrate/film interfaces, respectively, while
$\varphi$ and $\varphi^{\prime}$ are referred to as the anisotropic _regime
surface tensions_ and are introduced to include into the analysis the wetting
and dewetting regimes. We refer the Reader to the Introduction for related
explanation and for the motivation for the integral densities choice in
(3.14).

Similarly to [21, 36, 37, 38], by also taking into account that in our setting
the film and substrate regions are given as subsets of the composite regions,
the elastic energy is defined for configurations
$\left(A,S,u\right)\in{\widetilde{\mathcal{C}}}$ by

$\mathcal{W}(A,u):=\int_{A}{W\left(x,E\left(u\left(x\right)-E_{0}\left(x\right)\right)\right)dx},$

where the elastic density $W$ is determined by the quadratic form

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

for a fourth-order tensor $\mathbb{C}:\Omega\to\mathbb{M}^{2}_{\rm sym}$, $E$
denotes the symmetric gradient, i.e., $E(v):=\frac{\gradient
v+\gradient^{T}v}{2}$ for any $v\in H^{1}_{\mathrm{{loc}}}(\Omega)$ and
$E_{0}$ is the mismatch strain $x\in\Omega\mapsto
E_{0}\left(x\right)\in\mathbb{M}^{2}_{\mathrm{{sym}}}$ defined as

$E_{0}:=\begin{cases}E(u_{0})&\text{in }\Omega\setminus S,\\\ 0&\text{in
}S,\end{cases}$

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

###  3.2. Main results

We state here the main results of the paper and the connection to the one-
phase and multiple-phase settings.

Fix $l,L>0$ and consider $\Omega:=(-l,l)\times_{\mathbb{R}^{2}}(-L,L)$. Let
$\varphi:=\min\\{\varphi_{\mathrm{S}},\varphi_{\mathrm{F}}+\varphi_{\mathrm{FS}}\\},\,\varphi^{\prime}:=\min\\{\varphi_{\mathrm{F}},\varphi_{\mathrm{S}}\\}$
for three functions
$\varphi_{\mathrm{F}},\varphi_{\mathrm{S}},\varphi_{\mathrm{FS}}:\overline{\Omega}\times\mathbb{R}^{2}\to[0,\infty]$.
We assume throughout the paper that:

  * (H1)

$\varphi_{\mathrm{F}},\varphi_{\mathrm{FS}},\varphi,\varphi^{\prime}\in
C(\overline{\Omega}\times\mathbb{R}^{2};[0,\infty])$ are Finsler norms such
that

$c_{1}\absolutevalue{\xi}\leq\varphi_{\mathrm{F}}(x,\xi),{\varphi}(x,\xi),\varphi_{\mathrm{FS}}(x,\xi)\leq
c_{2}\absolutevalue{\xi}.$ (3.15)

for every $x\in\overline{\Omega}$ and $\xi\in\mathbb{R}^{2}$ and for two
constants $0<c_{1}\leq c_{2}$.

  * (H2)

We have

$\varphi(x,\xi)\geq|\varphi_{\textrm{FS}}(x,\xi)-\varphi_{\mathrm{F}}(x,\xi)|$
(3.16)

for every $x\in\overline{\Omega}$ and $\xi\in\mathbb{R}^{2}$.

  * (H3)

$\mathbb{C}\in L^{\infty}(\Omega;\mathbb{M}^{2}_{\rm sym})$ and there exists
$c_{3}>0$ such that

$\mathbb{C}\left(x\right)M:M\geq 2c_{3}M:M$ (3.17)

for every $M\in\mathbb{M}_{\mathrm{{sym}}}^{2\times 2}.$

We notice that under assumptions (H1)-(H3), the energy
$\mathcal{F}(A,S,u)\in[0,\infty]$ for every
$(A,S,u)\in\widetilde{\mathcal{C}}$.

The main result of the paper is the following existence result.

######  Theorem 3.10 (Existence of minimizers).

Assume (H1)-(H3) and let
$\mathbbm{v}_{0},\mathbbm{v}_{1}\in[\mathcal{L}^{2}({\Omega}/2),\mathcal{L}^{2}({\Omega}))$
such that $\mathbbm{v}_{0}\leq\mathbbm{v}_{1}$. Then for every
$\mathbf{m}=(m_{0},m_{1})\in\mathbb{N}\times\mathbb{N}$ the volume constrained
minimum problem

$\inf_{(A,S,u)\in\mathcal{C}_{\mathbf{m}},\,\mathcal{L}^{2}({A})=\mathbbm{v}_{1},\,\mathcal{L}^{2}({S_{h,K}})=\mathbbm{v}_{0}}{\mathcal{F}(A,S,u)}$
(3.18)

and the unconstrained minimum problem

$\inf_{(A,S,u)\in\mathcal{C}_{\mathbf{m}}}{\mathcal{F}^{\lambda}(A,S,u)}$
(3.19)

have solution, where
$\mathcal{F}^{\lambda}:\mathcal{C}_{\mathbf{m}}\rightarrow\mathbb{R}$ is
defined as

$\mathcal{F}^{\lambda}(A,S,u):=\mathcal{F}(A,S,u)+\lambda_{1}\absolutevalue{\mathcal{L}^{2}({A})-\mathbbm{v}_{1}}+\lambda_{0}\absolutevalue{\mathcal{L}^{2}({S})-\mathbbm{v}_{0}},$

for any ${\lambda}=(\lambda_{0},\lambda_{1}),$ with $\lambda_{0}$,
$\lambda_{1}>0$.

To prove Theorem 3.10 we apply the direct method of calculus of variations. On
the one hand, in Section 4, we show that any energy equi-bounded sequence
$\\{(A_{k},S_{k},u_{k})\\}\subset\mathcal{C}_{\mathbf{m}}$ satisfy the
following compactness property.

######  Theorem 3.11 (Compactness in $\mathcal{C}_{\mathbf{m}}$).

Assume (H1) and (H3). Let
$\\{(A_{k},{S_{h_{k},K_{k}}},u_{k})\\}_{k\in\mathbb{N}}\subset\mathcal{C}_{\mathbf{m}}$
be such that

$\sup_{k\in\mathbb{N}}{\mathcal{F}(A_{k},{S_{h_{k},K_{k}}},u_{k})}<\infty.$
(3.20)

Then, there exist an admissible configuration
$(A,S,u)\in\mathcal{C}_{\mathbf{m}}$ of finite energy, a subsequence
$\\{\left(A_{k_{n}},S_{h_{k_{n}},K_{k_{n}}},u_{k_{n}}\right)\\}_{n\in\mathbb{N}}$,
a sequence
$\left\\{\left(\widetilde{A}_{n},{\widetilde{S}_{n}},u_{k_{n}}\right)\right\\}_{n\in\mathbb{N}}\subset\mathcal{C}_{\mathbf{m}}$
and a sequence $\\{b_{n}\\}_{n\in\mathbb{N}}$ of piecewise rigid displacements
associated to $\widetilde{A}_{n}$ such that
$\left(\widetilde{A}_{n},{\widetilde{S}_{n}},u_{k_{n}}+b_{n}\right)\xrightarrow{\tau_{\mathcal{C}}}(A,S,u)$,
$\mathcal{L}^{2}(A_{k_{n}})=\mathcal{L}^{2}(\widetilde{A}_{n})$,
$\mathcal{L}^{2}(S_{h_{k_{n}},K_{k_{n}}})=\mathcal{L}^{2}(\widetilde{S}_{n})$
for all $n\in\mathbb{N}$ and

$\liminf_{n\rightarrow\infty}{\mathcal{F}(A_{k_{n}},S_{h_{k_{n}},K_{k_{n}}},u_{k_{n}})}=\liminf_{n\rightarrow\infty}{\mathcal{F}\left(\widetilde{A}_{n},{\widetilde{S}_{n}},u_{k_{n}}+b_{n}\right)}.$
(3.21)

On the other hand, in Section 5 we show that $\mathcal{F}$ is lower
semicontinuous in $\mathcal{C}_{\mathbf{m}}$ with respect to the topology
$\tau_{\mathcal{C}}$.

######  Theorem 3.12 (Lower semicontinuity of $\mathcal{F}$).

Assume (H1)-(H3). Let
$\\{\left(A_{k},S_{h_{k},K_{k}},u_{k}\right)\\}_{k\in\mathbb{N}}\subset\mathcal{C}_{\mathbf{m}}$
and $\left(A,S_{h,K},u\right)\in\mathcal{C}_{\mathbf{m}}$ be such that
$\left(A_{k},S_{h_{k},K_{k}},u_{k}\right)\xrightarrow{\tau_{\mathcal{C}}}\left(A,S_{h,K},u\right)$.
Then

$\mathcal{F}\left(A,S_{h,K},u\right)\leq\liminf_{k\rightarrow\infty}{\mathcal{F}\left(A_{k},S_{h_{k},K_{k}},u_{k}\right)}.$
(3.22)

We now describe the consequences for the one-phase setting of the results
obtained in this manuscript for the two-phase setting.

######  Remark 3.13 (Relation to literature models with fixed substrate).

The energy considered in this paper can be seen as an extension of the
energies previously considered in the literature, e.g., in [36, 27, 26], by
“fixing the substrate regions”. More precisely, if we consider the subfamily
$\mathcal{B}^{\prime}\subset\mathcal{B}$ where

$\mathcal{B}^{\prime}:=\\{(A,S)\in\mathcal{B}:\,\overline{\partial
S\cap\Omega}\text{ is a Lipschitz 1-manifold}\\}$

and
$\mathcal{C}^{\prime}:=\\{(A,S,u)\in\mathcal{C}:\,(A,S)\in\mathcal{B}^{\prime}\\}$,
then the energy $\mathcal{F}^{\prime}$ defined for every
$(A,S)\in\mathcal{B}^{\prime}$ by

$\displaystyle{\mathcal{F}^{\prime}}\left(A,S,u\right)$
$\displaystyle:=\mathcal{F}\left(A,S,u\right)-\int_{\Omega\cap\partial^{*}{S}}{\varphi_{\mathrm{FS}}\left(z,\nu_{S}(z)\right)\,d\mathcal{H}^{1}}-\int_{\Omega\cap(\partial
A\setminus\partial{S})\cap
A^{(0)}}{2{\varphi^{\prime}}(z,\nu_{A}(z))d\mathcal{H}^{1}}$
$\displaystyle=\int_{\Omega\cap\partial^{*}A\setminus{\partial{S}}}{\varphi_{\mathrm{F}}(z,\nu_{A}(z))d\mathcal{H}^{1}}+\int_{\Omega\cap(\partial
A\setminus\partial{S})\cap\cup
A^{(1)}}{2\varphi_{\mathrm{F}}(z,\nu_{A}(z))d\mathcal{H}^{1}}$
$\displaystyle\quad+\int_{\Omega\cap\partial^{*}{S}\cap\partial A\cap
A^{(1)}}{(\varphi_{\mathrm{F}}-\beta)(x,\nu_{A}\left(x\right))\,d\mathcal{H}^{1}}-\int_{\Omega\cap\partial^{*}{S}\cap\partial^{*}A}{\beta(z,\nu_{A}(z))d\mathcal{H}^{1}},$

where $\beta:=\varphi_{\textrm{FS}}-\varphi$, is analogous to the energy
$\mathcal{F}^{\prime}$ of [36, Theorem 2.9] (where the notation $\varphi$ was
referring to $\varphi_{\mathrm{F}}$), which is an extension of the energies of
[27, 26] as described in [36, Remark 2.10]. We notice though that, even in the
situation of a fixed regular substrate, i.e., by considering the family
$\mathcal{C}^{\prime\prime}:=\\{(A,S,u)\in\mathcal{C}:\,S=S_{0}\\}\subset\mathcal{C}^{\prime}$
for a fixed admissible region $S_{0}$ such that $\overline{\partial
S_{0}\cap\Omega}$ is a Lipschitz 1-manifold, the setting considered in this
manuscript allows to include into the analysis the possibility of an
uncountable number of film islands (or film voids) on top of the substrate
(which was instead precluded in [36]), because of the crucial difference
introduced in the setting of this manuscript consisting of always including
the substrate regions inside the admissible region $A$ (with the film region
then being represented by $\overline{A}\setminus S^{(1)}$). We also notice
that the hypotheses on the surface tensions in this manuscript coincide with
the ones in [36] up to the observation that on the right hand-side of (3.16)
one can disregard the absolute value for the setting with a fixed substrate).

We conclude the section by outlining the consequences of the results obtained
in this manuscript for the two-phase setting in the multiple-phase setting of
film multilayers that is the object of investigation in [42].

######  Remark 3.14 (Relation to the setting of film multilayers).

In the parallel paper [42] the authors consider the setting in which also the
film region is subject to a graph constraint, by introducing a family of
admissible regions for the film and the substrate of the form
$\mathcal{B}^{1}:=\\{(S_{h^{1},K^{1}},S_{h^{0},K^{0}})\in\mathrm{AS}(\Omega)\times\mathrm{AS}(\Omega):\,h^{0}\leq
h^{1}\text{ and }\partial
S_{h^{1},K^{1}}\cap{\rm{Int}}(S_{h^{0},K^{0}})=\emptyset\\}\subset\mathcal{B}$
and the related family
$\mathcal{C}^{1}:=\\{(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u):\,(S_{h^{1},K^{1}},S_{h^{0},K^{0}})\in\mathcal{B}^{1}\text{
and }u\in
H^{1}_{\mathrm{loc}}(S_{h^{1},K^{1}};\mathbb{R}^{2})\\}\subset\mathcal{C}$ of
admissible configurations. By implementing in the compactness for both the
substrate and the film the arguments employed in this manuscript for the
substrate, a similar result to Theorem 3.10 is established, thus providing an
existence result for the problem of films resting on deformable substrates in
the presence of delaminations, which can be seen as an extension of [21, 22,
27]. Furthermore, in [42] by then performing also an iteration procedure an
existence result is provided also for the setting of finitely-many film
multilayers.

##  4\. Compactness

In this section, we fix
$\mathbf{m}:=(m_{0},m_{1})\in\mathbb{N}\times\mathbb{N}$ and we prove that the
families $\mathcal{B}_{\mathbf{m}}$ and $\mathcal{C}_{\mathbf{m}}$ are compact
with respect to $\tau_{\mathcal{B}}$ and $\tau_{\mathcal{C}}$ topologies,
respectively.

######  Proposition 4.1.

The following assertions hold:

  1. (i)

For every sequence of closed sets $\\{E_{k}\\}$, there exists
$E\subset\mathbb{R}^{2}$ such that $E_{k}\xrightarrow{\mathcal{K}}E$.

  2. (ii)

For every sequence $\\{E_{k}\\}_{k\in\mathbb{N}}$ of subsets of
$\mathbb{R}^{2}$ there exists a subsequence $\\{E_{k_{l}}\\}_{l\in\mathbb{N}}$
and $E\subset\mathbb{R}^{2}$ such that $\mathrm{{sdist}}(\cdot,\partial
E_{k_{l}})\rightarrow\mathrm{{sdist}}(\cdot,\partial E)$ locally uniformly in
$\mathbb{R}^{2}$.

The proof of Proposition 4.1-(i) and -(ii) can be found in [4, Theorem 6.1]
and [36, Theorem 6.1], respectively (see also [4, Theorem 6.1] for the version
of Item (ii) with the signed-distance convergence replaced by the _Hausdorff-
metric convergence_).

######  Theorem 4.2 (Compactness of $\mathcal{B}_{\mathbf{m}}$).

Let $\\{(A_{k},S_{h_{k},K_{k}})\\}\subset\mathcal{B}_{\mathbf{m}}$ such that

$\sup_{k\in\mathbb{N}}{\mathcal{S}(A_{k},S_{h_{k},K_{k}})}<\infty$

for $(h_{k},K_{k})\in\mathrm{AHK}(\Omega)$. Then, there exist a not relabeled
subsequence $\\{(A_{k},S_{h_{k},K_{k}})\\}\subset\mathcal{B}_{\mathbf{m}}$ and
$(A,S_{h,K})\in\mathcal{B}_{\mathbf{m}}$ such that
$(A_{k},S_{h_{k},K_{k}})\xrightarrow{\tau_{\mathcal{B}}}(A,S_{h,K})$.

###### Proof.

For simplicity we denote $S_{k}:=S_{h_{k},K_{k}}$. We begin by observing that
by Proposition 4.1-(ii) there exist $A\subset\mathbb{R}^{2}$ and
$S\subset\mathbb{R}^{2}$ such that $\mathrm{{sdist}}(\cdot,\partial
A_{k_{l}})\rightarrow\mathrm{{sdist}}(\cdot,\partial A)$ and
$\mathrm{{sdist}}\left(\cdot,\partial{S_{{k_{l}}}}\right)\rightarrow\mathrm{{sdist}}(\cdot,\partial{S})$
locally uniformly in $\mathbb{R}^{2}$. Let
$R:=\sup_{k\in\mathbb{N}}{\mathcal{S}(A_{k},S_{h_{k},K_{k}})}$. In view of
Remark 3.4 and by (2.3) we have the following decomposition of $\partial
A_{k}$,

$\displaystyle\partial A_{k}$ $\displaystyle=\partial^{*}A_{k}\cup(\partial
A_{k}\cap(A_{k}^{(0)}\cup A_{k}^{(1)}))\cup N_{k},$

where $N_{k}$ is a $\mathcal{H}^{1}$-negligible set for every
$k\in\mathbb{N}$. Thus, for every $k\in\mathbb{N}$ we observe that

$\displaystyle\partial
A_{k}\setminus\partial{S_{k}}=\partial^{*}A_{k}\setminus\partial{S_{k}}\cup\left((\partial
A_{k}\setminus\partial{S_{k}})\cap\left(A_{k}^{(0)}\cup
A_{k}^{(1)}\right)\right)\cup N^{\prime}_{k},$ (4.1)

where $N^{\prime}_{k}:=N_{k}\setminus\partial{S_{k}}$ is a
$\mathcal{H}^{1}$-negligible set. Since for any $k\in\mathbb{N}$, $S_{k}$ is a
set of finite perimeter, by (2.3) we have that

$\displaystyle\partial S_{k}$ $\displaystyle=\partial^{*}S_{k}\cup(\partial
S_{k}\cap(S_{k}^{(0)}\cup S_{k}^{(1)}))\cup\widetilde{N}_{k},$

where $\widetilde{N}_{k}$ is a $\mathcal{H}^{1}$-negligible set for every
$k\in\mathbb{N}$. Reasoning similarly to (4.1) we have that

$\displaystyle\partial S_{k}\setminus\partial{A_{k}}$
$\displaystyle=\partial^{*}S_{k}\setminus\partial{A_{k}}\cup(\partial
S_{k}\setminus\partial{A_{k}})\cap\left(S_{k}^{(0)}\cup
S_{k}^{(1)}\right)\cup\widetilde{N}^{\prime}_{k},$ (4.2)
$\displaystyle(\partial{S_{k}}\setminus\partial A_{k})\cap A_{k}^{(1)}$
$\displaystyle=\left(\partial^{*}{S_{k}}\setminus\partial
A_{k}\cup\left(\partial{S_{k}}\setminus\partial A_{k}\cap\left(S^{(1)}_{k}\cup
S_{k}^{(0)}\right)\right)\right)\cap
A_{k}^{(1)}\cup\widetilde{N}^{\prime\prime}_{k}$ (4.3)

and

$\displaystyle\partial{S_{k}}\cap\partial{A_{k}}\cap
A_{k}^{(1)}=\left((\partial^{*}{S_{k}}\cap\partial{A_{k}})\cup\left((\partial{S_{k}}\cap\partial{A_{k}})\cap(S^{(1)}_{k}\cup
S_{k}^{(0)})\right)\right)\cap
A_{k}^{(1)}\cup\widetilde{N}^{\prime\prime\prime}_{k},$ (4.4)

where $\widetilde{N}^{\prime}_{k}$, $\widetilde{N}^{\prime\prime}_{k}$ and
$\widetilde{N}^{\prime\prime\prime}_{k}$ are $\mathcal{H}^{1}$-negligible sets
for every $k\in\mathbb{N}$. Furthermore, we can deduce that

$\begin{split}\partial S_{k}\cap\partial
A_{k}&=\left(\partial^{*}S_{k}\cup(\partial S_{k}\cap(S_{k}^{(0)}\cup
S_{k}^{(1)}))\right)\cap\left(\partial^{*}A_{k}\cup(\partial
A_{k}\cap(A_{k}^{(0)}\cup A_{k}^{(1)}))\right)\cup\widehat{N}_{k}\\\
&=\left(\partial S_{k}^{*}\cap\partial^{*}A_{k}\right)\cup\left(\partial
S_{k}\cap\partial^{*}A_{k}\cap(S_{k}^{(0)}\cup
S_{k}^{(1)})\right)\cup\left(\partial^{*}S_{k}\cap\partial
A_{k}\cap(A_{k}^{(0)}\cup A_{k}^{(1)})\right)\\\ &\quad\left(\partial
S_{k}\cap\partial A_{k}\cap((A_{k}^{(0)}\cup A_{k}^{(1)})\cup(S_{k}^{(0)}\cup
S_{k}^{(1)}))\right)\cup\widehat{N}_{k},\end{split}$ (4.5)

where $\widehat{N}_{k}$ is a $\mathcal{H}^{1}$-negligible set, for every
$k\in\mathbb{N}$. By (H1),(H3), (4.1)-(4.5) and thanks to the fact that for
every $k\in\mathbb{N}$, $S_{k}\subset\overline{A_{k}}$ we have that

$\begin{split}&c_{1}\left(\mathcal{H}^{1}(\partial
A_{k})+\mathcal{H}^{1}(\partial{S_{k}}\setminus\partial A_{k})\right)\\\
&\quad\leq
c_{1}\left(\int_{\partial^{*}{A_{k}}\setminus\partial{S_{k}}}{d\mathcal{H}^{1}}+\int_{\partial^{*}{S_{k}}\cap\partial^{*}{A_{k}}}{d\mathcal{H}^{1}}\right.+\int_{(\partial{A_{k}}\setminus\partial{S_{k}})\cap({A_{k}}^{(0)}\cup{A_{k}}^{(1)})}{d\mathcal{H}^{1}}\\\
&\qquad+\int_{(\partial^{*}{S_{k}}\setminus\partial{A_{k}})\cap{A_{k}}^{(1)}}{d\mathcal{H}^{1}}+\int_{\partial{S_{k}}\cap\partial{A_{k}}\cap{S_{k}}^{(1)}}{d\mathcal{H}^{1}}+\int_{\partial{S_{k}}\cap\partial^{*}{A_{k}}\cap{S_{k}}^{(0)}}{d\mathcal{H}^{1}}\\\
&\qquad+\int_{(\partial{S_{k}}\setminus\partial
A_{k})\cap({S_{k}}^{(1)}\cup{S_{k}}^{(0)})\cap{A_{k}}^{(1)}}{d\mathcal{H}^{1}}+\int_{\partial^{*}{S_{k}}\cap\partial
A_{k}\cap{A_{k}}^{(1)}}{d\mathcal{H}^{1}}\\\
&\qquad+\left.\int_{(\partial{S_{k}}\cap\partial
A_{k})\cap{S_{k}}^{(0)}\cap{A_{k}}^{(1)}}{d\mathcal{H}^{1}}\right)\\\
&\quad\leq 2({\mathcal{S}}(A_{k},{S_{k}}))\leq 2R,\end{split}$ (4.6)

for every $k\in\mathbb{N}$, where in the first inequality we used Lemma 3.1
and in the second inequality we used (3.15). It follows that

$\mathcal{H}^{1}(\partial A_{k})\leq\frac{2R}{c_{1}}$ (4.7)

and

$\mathcal{H}^{1}(\partial{S_{k}})=\mathcal{H}^{1}(\partial{S_{k}}\cap\partial
A_{k})+\mathcal{H}^{1}(\partial{S_{k}}\setminus\partial
A_{k})\leq\frac{2R}{c_{1}},$ (4.8)

for any $k\in\mathbb{N}$.

In view of (4.7) and (4.8) we conclude by Lemma 3.9 that
$A\subset\overline{\Omega}$ is $\mathcal{L}^{2}$-measurable, $\partial A$ is
$\mathcal{H}^{1}$-finite, $\mathcal{H}^{1}$-rectifiable, and with at most
$m_{1}$-connected components, and that there exists
$(h,K)\in\mathrm{AHK}(\Omega)$ such that $S=S_{h,K}\in{\rm AS}(\Omega)$ and
$\partial S_{h,K}$ has at most $m_{0}$-connected components. Furthermore, in
view of Remark 3.8-(i), since $\overline{S_{k}}\subset\overline{A_{k}}$ for
any $k\in\mathbb{N}$, we have that $S_{h,K}\subset\overline{A}$.

In order to prove that $(A,S_{h,K})\in\mathcal{B}_{\mathbf{m}}$ it remains to
check that $\partial A\cap{\rm{Int}}(S_{h,k})=\emptyset$, to which the rest of
the proof is devoted. Assume by contradiction that

$\partial A\cap{\rm{Int}}(S_{h,k})\neq\emptyset.$ (4.9)

Then, there exists $x\in\partial A\cap{\rm{Int}}(S_{h,k})$. By Remark 3.8-(i),
there exists $x_{k}\in\partial A_{k}$ such that $x_{k}\to x$ and hence, by
$\tau_{\mathcal{B}}$-convergence we observe that

${\rm{sdist}}(x,\partial S_{h_{k},K_{k}})\to{\rm{sdist}}(x,\partial
S_{h,K})\quad\text{as $k\to\infty.$}$ (4.10)

Since by (4.9) there exists $\varepsilon>0$ such that ${\rm{sdist}}(x,\partial
S_{h,K})=-\varepsilon,$ we can find $k_{0}:=k_{0}(x)$ for which
${\rm{sdist}}(x,\partial S_{h_{k_{0}},K_{k_{0}}})$ is negative. Then,
$x\in{\rm{Int}}(S_{h_{k_{0}},K_{k_{0}}})$ and so, there exists
$\delta\leq\varepsilon/2$ such that

$x_{k_{0}}\in B_{\delta}(x)\subset{\rm{Int}}(S_{h_{k_{0}},K_{k_{0}}}),$

which is an absurd since $\partial
A_{k}\cap{\rm{Int}}(S_{h_{k},K_{k}})=\emptyset$ for every $k\in\mathbb{N}$. ∎

We are now in the position to prove Theorem 3.11. To this end, we implement
the arguments used in [36, Theorem 2.7] to the situation with free-boundary
substrates and in particular the ones contained in Step 1 of the proof of [36,
Theorem 2.7]. In fact, the original setting introduced in this paper with
respect to [36] to model the delaminated interface regions allows to avoid the
further modification of the film admissible regions that was performed in
Steps 2 and 3 of [36, Theorem 2.7].

###### Proof of Theorem 3.11.

Denote $R:=\sup_{k\in\mathbb{N}}{\mathcal{F}(A_{k},S_{h_{k},K_{k}},u_{k})}$.
Without loss of generality (by passing, if necessary, to a not relabeled
subsequence), we assume that

$\liminf_{k\rightarrow\infty}{\mathcal{F}(A_{k},S_{h_{k},K_{k}},u_{k})}=\lim_{k\rightarrow\infty}{\mathcal{F}(A_{k},S_{h_{k},K_{k}},u_{k})}\leq
R.$ (4.11)

Since $\mathcal{W}$ is non-negative, by Theorem 4.2 there exist a subsequence
$\\{(A_{k_{n}},S_{h_{k_{n}},K_{k_{n}}})\\}\subset\mathcal{B}_{\mathbf{m}}$ and
$(A,S)\in\mathcal{B}_{\mathbf{m}}$ such that
$(A_{k_{n}},S_{h_{k_{n}},K_{k_{n}}})\xrightarrow{\tau_{\mathcal{B}}}(A,S)$. As
a consequence of Theorem 4.2, there exists $(h,K)\in\text{AHK}$ such that
$S=S_{h,K}$.

The rest of the proof is devoted to the construction of a sequence
$(\widetilde{A}_{n},{\widetilde{S}_{n}})\subset\mathcal{B}_{\mathbf{m}}$ to
which we can apply [36, Corollary 3.8] (with $P={\rm{Int}}{(A)}$ and
$P_{n}={\rm{Int}}{(\widetilde{A}_{n})}$, respectively) in order to obtain
$u\in H^{1}_{\mathrm{loc}}({\rm{Int}}{(A)};\mathbb{R}^{2})$ such that
$(A,S,u)\in\mathcal{C}_{\mathbf{m}}$ has finite energy, and a sequence
$\\{b_{n}\\}_{n\in\mathbb{N}}$ of piecewise rigid displacements such that
$\left(\widetilde{A}_{n},{\widetilde{S}_{n}},u_{k_{n}}+b_{n}\right)\xrightarrow{\tau_{\mathcal{C}}}(A,S,u)$.
Furthermore, we observe that also Equation (3.21) will be a consequence of
such construction and hence, the assertion will directly follow.

By [36, Proposition 3.6] applied to $A_{k_{n}}$ and $A$ there exist a not
relabeled subsequence $\\{A_{k_{n}}\\}$ and a sequence
$\\{\widetilde{A}_{n}\\}$ with $\mathcal{H}^{1}$-rectifiable boundary
$\partial\widetilde{A}_{n}$ of at most $m_{1}$-connected components such that

$\sup_{n\in\mathbb{N}}\mathcal{H}^{1}(\partial\widetilde{A}_{n})<\infty,$
(4.12)

that satisfy the following properties:

  * (a1)

$\partial A_{k_{n}}\subset\partial\widetilde{A}_{n}$ and
$\displaystyle\lim_{n\to\infty}\mathcal{H}^{1}(\partial\widetilde{A}_{n}\setminus\partial
A_{k_{n}})=0$,

  * (a2)

${\rm{sdist}}(\cdot,\partial\widetilde{A}_{n})\to{\rm{sdist}}(\cdot,\partial
A)$ locally uniformly in $\mathbb{R}^{2}$ as $n\to\infty$,

  * (a3)

if $\\{E_{i}\\}_{i\in I}$ is the family of all connected components of
$\mathrm{{Int}}(A),$ there exist connected components of
$\mathrm{{Int}}(\widetilde{A}_{n})$, which we enumerate as
$\\{E_{i}^{n}\\}_{i\in I}$, such that for every $i$ and $G\subset\subset
E_{i}$ one has that $G\subset\subset E_{i}^{n}$ for all $n$ large (depending
only on $i$ and $G$),

  * (a4)

$\mathcal{L}^{2}(\widetilde{A}_{n})=\mathcal{L}^{2}(A_{k_{n}})$.

Furthermore, from the construction of $\widetilde{A}_{n}$ (namely from the
fact that $\widetilde{A}_{n}$ is constructed by adding extra “internal”
topological boundary to the selected subsequence $A_{k_{n}}$, see [36,
Propositions 3.4 and 3.6]) it follows that

$\widetilde{A}_{n}=A_{k_{n}}\setminus(\partial\widetilde{A}_{n}\setminus\partial
A_{k_{n}})$ (4.13)

with $\partial\widetilde{A}_{n}\setminus\partial A_{k_{n}}$ given by a finite
union of closed $\mathcal{H}^{1}$-rectifiable sets connected to $\partial
A_{k_{n}}$. More precisely, there exist a finite index set $J$ and a family
$\\{\Gamma_{j}\\}_{j\in J}$ of closed $\mathcal{H}^{1}$-rectifiable sets of
$\Omega$ connected to $\partial A_{k_{n}}$ such that

$\partial\widetilde{A}_{n}\setminus\partial A_{k_{n}}=\bigcup_{j\in
J}\Gamma_{j}.$

We define

$\widetilde{K}_{n}:=K_{k_{n}}\cup((\partial\widetilde{A}_{n}\setminus\partial
A_{k_{n}})\cap{\overline{{\rm{Int}}(S_{h_{k_{n}}})}})\subset{\overline{{\rm{Int}}(S_{h_{k_{n}}})}}$

and we observe that $\widetilde{K}_{n}$ is closed and
$\mathcal{H}^{1}$-rectifiable in view of the fact that
$\partial\widetilde{A}_{n}\setminus\partial A_{k_{n}}$ is a closed set in
$\Omega$ and is $\mathcal{H}^{1}$-rectifiable, since
$\partial\widetilde{A}_{n}$ is $\mathcal{H}^{1}$-rectifiable. Therefore,
$(h_{k_{n}},\widetilde{K}_{n})\in\text{AHK}(\Omega)$ and
$\widetilde{S}_{n}:=S_{h_{k_{n}},\widetilde{K}_{n}}\subset\overline{\widetilde{A}_{n}}$.
We claim that $\partial{\widetilde{S}_{n}}$ has at most $m_{0}$-connected
components, so that
$(\widetilde{A}_{n},{\widetilde{S}_{n}})\in\mathcal{B}_{\textbf{m}}$. Indeed,
if for every $j\in J$, $S_{h_{k_{n}},K_{k_{n}}}\cap\Gamma_{j}$ is empty there
is nothing to prove, so we assume that there exists $j\in J$ such that
$S_{h_{k_{n}},K_{k_{n}}}\cap\Gamma_{j}\neq\emptyset$. On one hand if
$\Gamma_{j}\subset S_{h_{k_{n}},K_{k_{n}}}$, thanks to the facts that
$\Gamma_{j}$ is connected to $\partial A_{k_{n}}$ and
$S_{h_{k_{n}},K_{k_{n}}}\subset\overline{A_{k_{n}}}$, we deduce that
$\Gamma_{j}$ needs to be connected to $\partial S_{h_{k_{n}},K_{k_{n}}}$. On
the other hand, if $\Gamma_{j}\cap(A_{k_{n}}\setminus
S_{h_{k_{n}}})\neq\emptyset$, then we can find $x_{1}\in\Gamma_{j}\cap
S_{h_{k_{n}},K_{k_{n}}}$ and $x_{2}\in\Gamma_{j}\cap(A_{k_{n}}\setminus
S_{h_{k_{n}}})$. Since $\Gamma_{j}$ is closed and connected, by [25, Lemma
3.12] there exists a parametrization $r:[0,1]\to\mathbb{R}^{2}$ whose support
$\gamma\subset\Gamma_{j}$ joins the point $x_{1}$ with $x_{2}$. Thus, $\gamma$
crosses $\partial S_{h_{k_{n}},K_{k_{n}}}$ and we conclude that $\Gamma_{j}$
is connected to $\partial S_{h_{k_{n}},K_{k_{n}}}$.

We claim that
$(\widetilde{A}_{n},{\widetilde{S}_{n}})\xrightarrow{\tau_{\mathcal{B}}}(A,S_{h,K})$
as $n\to\infty$. In view of (4.12), (a2) and the fact that by (3.7) and the
previous construction of $\widetilde{K}_{n}$,

${\sup_{n\in\mathbb{N}}\mathcal{H}^{1}(\partial\widetilde{S}_{n})=}\sup_{n\in\mathbb{N}}\mathcal{H}^{1}(\partial
S_{h_{k_{n}},\widetilde{K}_{n}})<\infty,$

it remains to prove that

${\rm{sdist}}(\cdot,\partial
S_{h_{k_{n}},\widetilde{K}_{n}})\to{\rm{sdist}}(\cdot,\partial S_{h,K})$
(4.14)

locally uniformly in $\mathbb{R}^{2}$ as $n\to\infty$. Indeed, by Remark
3.8-(i), it suffices to prove that
$S_{h_{k_{n}},\widetilde{K}_{n}}\xrightarrow{\mathcal{K}}S_{h}$ and that
$\Omega\setminus
S_{h_{k_{n}},\widetilde{K}_{n}}\xrightarrow{\mathcal{K}}\Omega\setminus{\rm{Int}}(S_{h,K})$.
On one hand, by the $\tau_{\mathcal{B}}$-convergence of
$\\{(A_{k_{n}},{S_{{k_{n}}}})\\}$, the fact that
${\overline{\widetilde{S}_{n}}:=}\overline{S_{h_{k_{n}},\widetilde{K}_{n}}}=S_{h_{k_{n}}}$,
and the properties of Kuratowski convergence, it follows that
${\widetilde{S}_{n}:=}S_{h_{k_{n}},\widetilde{K}_{n}}\xrightarrow{\mathcal{K}}S_{h}$.
On the other hand, let $x\in\Omega\setminus{\rm{Int}}(S_{h,K})$, since

${\rm{Int}}(S_{h_{k_{n}},\widetilde{K}_{n}})={\rm{Int}}(S_{h_{k_{n}}})\setminus\widetilde{K}_{n}\subset{\rm{Int}}(S_{h_{k_{n}}})\setminus
K_{k_{n}}={\rm{Int}}(S_{h_{k_{n}},K_{k_{n}}})$

and by the fact that
$\Omega\setminus{\rm{Int}}(S_{h_{k_{n}},K_{k_{n}}})\xrightarrow{\mathcal{K}}\Omega\setminus{\rm{Int}}(S_{h,K})$,
there exists

$x_{n}\in\Omega\setminus{\rm{Int}}(S_{h_{k_{n}},K_{k_{n}}})\subset\Omega\setminus{\rm{Int}}(S_{h_{k_{n}},\widetilde{K}_{n}})$

such that $x_{n}\to x$. Now, we consider a sequence
$x_{n}\in\Omega\setminus{\rm{Int}}(S_{h_{k_{n}},\widetilde{K}_{n}})$
converging to a point $x\in\Omega$. We proceed by contradiction, namely we
assume that $x\in{\rm{Int}}(S_{h,K})$. Therefore, there exists $\epsilon>0$
such that ${\rm{sdist}}(x,\partial S_{h,K})=-\epsilon$, which implies that
${\rm{sdist}}(x,\partial S_{h_{k_{n}},K_{k_{n}}})\to-\epsilon$ as
$n\to\infty$. Thus, there exists $n_{\epsilon}\in\mathbb{N}$, such that
$x_{n}\in B_{\epsilon/2}(x)\subset{\rm{Int}}(S_{h_{k_{n}},K_{k_{n}}})$, for
every $n\geq n_{\epsilon}$. However, notice that

$\begin{split}x_{n}\in\Omega\setminus{\rm{Int}}\left(S_{h_{k_{n}},\widetilde{K}_{n}}\right)&=\Omega\setminus\left({\rm{Int}}({S_{h_{k_{n}}}})\setminus\widetilde{K}_{n}\right)\\\
&=\left(\Omega\setminus{\rm{Int}}\left(S_{h_{k_{n}},K_{k_{n}}}\right)\right)\cup\left(\left(\partial\widetilde{A}_{n}\setminus\partial
A_{k_{n}}\right)\cap{\overline{{\rm{Int}}(S_{h_{k_{n}}})}}\right),\end{split}$
(4.15)

where in the last equality we used the definition of
$\widetilde{K}_{n}:=K_{k_{n}}\cup((\partial\widetilde{A}_{n}\setminus\partial
A_{k_{n}})\cap{\overline{{\rm{Int}}(S_{h_{k_{n}}})}})$ and the fact that
${\rm{Int}}(S_{h_{k_{n}},K_{k_{n}}})={\rm{Int}}(S_{h_{k_{n}}})\setminus
K_{k_{n}}$. Therefore, by (4.15) we deduce that
$x_{n}\in\partial\widetilde{A}_{n}\setminus\partial A_{k_{n}}$ for every
$n\geq n_{\epsilon}$ and hence, $x\in\partial A$ by (a2) and Remark 3.8-(i).
We reached an absurd as it follows that $x\in{\rm{Int}}(S_{h,K})\cap\partial
A=\emptyset$. This conclude the proof of (4.14) and hence, of the claim.

By (3.15) and by conditions (a1), (a4) and (4.13), we observe that

$\lim_{n\to\infty}{\absolutevalue{\mathcal{S}(A_{k_{n}},S_{k_{n}})-\mathcal{S}(\widetilde{A}_{n},\widetilde{S}_{n})}}=\lim_{n\to\infty}{\absolutevalue{\mathcal{S}(A_{k_{n}},S_{h_{k_{n}},K_{k_{n}}})-\mathcal{S}(\widetilde{A}_{n},S_{h_{k_{n}},\widetilde{K}_{n}})}}=0,$
(4.16)

and

$\mathcal{W}(A_{k_{n}},u_{k_{n}})=\mathcal{W}(\widetilde{A}_{n},u_{k_{n}}).$
(4.17)

By (3.17), (4.11), (4.13), (4.17), (a3) and thanks to the fact that
$\mathcal{S}$ is non-negative, we obtain that

$\int_{E_{i}^{n}}{\absolutevalue{e(u_{k_{n}})}^{2}dx}\leq\int_{\widetilde{A}_{n}}{\absolutevalue{e(u_{k_{n}})}^{2}dx}\leq\frac{R}{2c_{3}},$

for every $i\in I$ and for $n$ large enough. Therefore, by a diagonal argument
and by [36, Corollary 3.8] (applied to, with the notation of [36], $P=E_{i}$
and $P_{n}=E_{i}^{n}$) up to extracting not relabelled subsequences both for
$\\{u_{{k_{n}}}\\}\subset H^{1}_{\mathrm{loc}}(\Omega;\mathbb{R}^{2})$ and
$\\{E_{i}^{n}\\}_{n}$ there exist $w_{i}\in
H^{1}_{\mathrm{{loc}}}(E_{i},\mathbb{R}^{2})$, and a sequence of rigid
displacements $\\{b_{n}^{i}\\}$ such that
$(u_{k_{n}}+b_{n}^{i})\mathbbm{1}_{E^{n}_{i}}\rightarrow w_{i}$ a.e. in
$E_{i}$. Let $\\{D^{n}_{i}\\}_{i\in\widetilde{I}}$ for an index set
$\widetilde{I}$ be the family of open and connected components of
$\widetilde{A}_{n}\setminus\bigcup_{i\in I}{E_{i}^{n}}$ such that by (a3)
${\rm{Int}}(D^{n}_{i})$ converges to the empty set for every
$i\in\widetilde{I}$. In $D_{i}^{n}$ we consider the null rigid displacement,
and we define

${b}_{n}:=\sum_{i\in I}{b_{n}^{i}}\mathbbm{1}_{E_{i}^{n}}\quad\text{and}\quad
u:=\sum_{i\in I}{w_{i}\mathbbm{1}_{E_{i}}}.$

We have that $u\in H^{1}_{\mathrm{{loc}}}(\mathrm{{Int}}(A);\mathbb{R}^{2})$,
${b}_{n}$ is a rigid displacement associated to $\widetilde{A}_{n}$,
$u_{k_{n}}+{b}_{n}\rightarrow u$ a.e. in $\mathrm{{Int}}(A)$ and hence,
${(A,S,u)=}(A,S_{h,K},u)\in\mathcal{C}_{\mathbf{m}}$ and
${(\widetilde{A}_{n},\widetilde{S}_{n},u_{k_{n}}+{b}_{n}):=}(\widetilde{A}_{n},S_{{h}_{k_{n}},\widetilde{K}_{n}},u_{k_{n}}+{b}_{n})\xrightarrow{\tau_{\mathcal{C}}}(A,S_{h,K},u)$.
Furthermore, as $e(u_{k_{n}}+b_{n})=e(u_{k_{n}})$, from (4.16) and (4.17) it
follows that

$\begin{split}&{\lim_{n\to\infty}{\absolutevalue{\mathcal{F}(A_{k_{n}},S_{h_{k_{n}},K_{k_{n}}},u_{k_{n}})-\mathcal{F}(\widetilde{A}_{n},\widetilde{S}_{n},u_{k_{n}}+b_{n})}}}\\\
&=\lim_{n\to\infty}{\absolutevalue{\mathcal{F}(A_{k_{n}},S_{h_{k_{n}},K_{k_{n}}},u_{k_{n}})-\mathcal{F}(\widetilde{A}_{n},S_{h_{k_{n}},\widetilde{K}_{n}},u_{k_{n}}+b_{n})}}=0,\end{split}$

which implies (3.21) and completes the proof. ∎

##  5\. Lower semicontinuity

In this section we prove that for any fixed
$\mathbf{m}:=(m_{0},m_{1})\in\mathbb{N}\times\mathbb{N}$ the energy
$\mathcal{F}$ is lower semicontinuous in the family of configurations
$\mathcal{C}_{\mathbf{m}}$ with respect to the topology $\tau_{\mathcal{C}}$.
Since $\mathcal{F}$ is given as the sum of the surface energy $\mathcal{S}$
and the elastic energy $\mathcal{W}$, we proceed by proving that both
$\mathcal{S}$ and $\mathcal{W}$ are independently lower semicontinuous with
respect to $\tau_{\mathcal{C}}$.

We begin with $\mathcal{S}$ and we adopt Fonseca-Müller blow-up technique
[29], for which we make use of a localized version ${\mathcal{S}}_{L}$ of the
surface energy, which we can consider with surface tensions constant with
respect to the variable in $\overline{\Omega}$.

######  Definition 5.1.

Let $\phi_{\text{F}},\phi_{\text{S}},\phi_{\text{FS}}$ be three functions and
let
$\phi:=\min\\{\phi_{\mathrm{S}},\phi_{\mathrm{F}}+\phi_{\mathrm{FS}}\\},\,\phi^{\prime}:=\min\\{\phi_{\mathrm{F}},\phi_{\mathrm{S}}\\}$
be such that $\phi_{\text{F}},\,\phi_{\text{FS}},\,\phi\,,\phi^{\prime}\in
C(\mathbb{R}^{2};[0,\infty])$ are Finsler norms and the hypotheses (H1) and
(H2) are satisfied by the functions
$\varphi_{\alpha}{,\varphi}{,\varphi^{\prime}}\in
C(\overline{\Omega}\times\mathbb{R}^{2})$ given for
$\alpha=\text{S},\text{FS},\text{F}$ by
$\varphi_{\alpha}(x,\cdot):=\phi_{\alpha}(\cdot)$,
$\varphi(x,\cdot):=\phi(\cdot)$ and
${\varphi^{\prime}(x,\cdot)=\phi^{\prime}(\cdot)}$ for every
$x\in\overline{\Omega}$. We define the localized surface energy
${\mathcal{S}}_{L}:{\mathcal{B}_{L}}\to[0,+\infty]$ by

$\begin{split}{\mathcal{S}}_{L}\left(A,S,O\right)&:=\int_{O\cap(\partial^{*}A\setminus\partial{S})}{\phi_{\mathrm{F}}(\nu_{A})\,d\mathcal{H}^{1}}+\int_{O\cap\partial^{*}{S}\cap\partial^{*}A}{\phi(\nu_{A})\,d\mathcal{H}^{1}}\\\
&\quad+\int_{O\cap(\partial^{*}{S}\setminus\partial A)\cap
A^{(1)}}{\phi_{\mathrm{FS}}(\nu_{S})\,d\mathcal{H}^{1}}+\int_{O\cap\partial^{*}{S_{h,K}}\cap\partial
A\cap A^{(1)}}{(\phi_{\mathrm{F}}+\phi)(\nu_{A})\,d\mathcal{H}^{1}}\\\
&\quad+\int_{O\cap\partial A\cap A^{(1)}\cap
S^{(0)}}{2\phi_{\mathrm{F}}(\nu_{A})\,d\mathcal{H}^{1}}{+\int_{O\cap\partial
A\cap A^{(0)}}{2\phi^{\prime}(\nu_{A})\,d\mathcal{H}^{1}}}\\\
&\quad+\int_{O\cap(\partial{S}\setminus\partial A)\cap\left(S^{(1)}\cup
S^{(0)}\right)\cap
A^{(1)}}{2\phi_{\mathrm{FS}}(\nu_{S})\,d\mathcal{H}^{1}}+\int_{O\cap\partial{S}\cap\partial
A\cap S^{(1)}}{2{\phi}(\nu_{S})\,d\mathcal{H}^{1}}\end{split}$ (5.1)

for every
$(A,S,O)\in\mathcal{B}_{L}:=\\{(A,S,O)\,:\,(A,S)\in\mathcal{B},\text{ $O$ open
and contained in $\Omega$}\\}$.

We start with some technical results needed in the blow-up argument used in
Theorem 5.13.

######  Lemma 5.2.

Let $Q$ be any open square, $K\subset\overline{Q}$ be a nonempty closed set
and $E_{k}\subset\overline{Q}$ be such that ${\rm{sdist}}(\cdot,\partial
E_{k})\to{\rm dist}(\cdot,K)$ uniformly in $\overline{Q}$ as $k\to\infty$.
Then $E_{k}\xrightarrow{\mathcal{K}}K$ as $k\to\infty$. Analogously, if
${\rm{sdist}}(\cdot,\partial E_{k})\to-{\rm dist}(\cdot,K)$ uniformly in
$\overline{Q}$ as $k\to\infty$, then $\overline{Q}\setminus
E_{k}\xrightarrow{\mathcal{K}}K$ as $k\to\infty$.

The proof of the previous lemma follows from the same arguments of [36, Lemma
4.2].

######  Proposition 5.3.

Let $m\in\mathbb{N}$ and let $E\subset\mathbb{R}^{2}$ be a set such that
$\partial E$ has at most $m$-connected components, is
$\mathcal{H}^{1}$-rectifiable, and satisfies $\mathcal{H}^{1}(\partial
E)<\infty$. Let $x\in\partial E$ be such that the measure-theoretic unit
normal $\nu_{E}\left(x\right)$ of $\partial E$ at $x$ and there exists $R>0$
such that
$\overline{Q_{R,\nu_{E}\left(x\right)}\left(x\right)}\cap\partial{\sigma}_{\rho,x}(E)\xrightarrow{\mathcal{K}}\overline{Q_{R,\nu_{E}\left(x\right)}\left(x\right)}\cap
T_{x}$ as $\rho\to 0^{+}$, where $T_{x}:=T_{x,\nu_{E}(x)}$. Then, the
following assertions hold true:

  1. (a)

If $x\in E^{(1)}\cap\partial E$, then
${\rm{sdist}}(\cdot,\partial{\sigma}_{\rho,x}(E))\to-{\rm dist}(\cdot,T_{x})$
uniformly in $\overline{Q_{R,\nu_{E}\left(x\right)}}$ as $\rho\to 0^{+}$;

  2. (b)

If $x\in E^{(0)}\cap\partial E$, then
${\rm{sdist}}(\cdot,\partial{\sigma}_{\rho,x}(E))\to{\rm dist}(\cdot,T_{x})$
uniformly in $\overline{Q_{R,\nu_{E}\left(x\right)}}$ as $\rho\to 0^{+}$;

  3. (c)

If $x\in\partial^{*}E$ then
${\rm{sdist}}(\cdot,\partial{\sigma}_{\rho,x}(E))\rightarrow{\rm{sdist}}(\cdot,\partial
H_{x})$ uniformly in $\overline{Q_{R,\nu_{E}\left(x\right)}}$ as $\rho\to
0^{+}$.

###### Proof.

The cases (a) and (b) follow directly from [36, Proposition A.5]. It remains
to prove the case (c) to which the remaining of the proof is devoted. Let
$x\in\partial^{*}E$ and $R>0$ be such that

$\overline{Q_{R,\nu_{E}\left(x\right)}\left(x\right)}\cap\partial{\sigma}_{\rho,x}(E)\xrightarrow{\mathcal{K}}\overline{Q_{R,\nu_{E}\left(x\right)}\left(x\right)}\cap
T_{x}\text{ as }\rho\rightarrow 0^{+}.$ (5.2)

Without loss of generality, we assume that
$x=0,\nu_{E}(0)=\mathbf{e}_{\mathbf{2}}$, $H_{x}=H_{0}$ and
$T_{x}=T_{0}=\partial H_{0}$.

Let $\\{\rho_{k}\\}\subset(0,1)$ be such that $\rho_{k}\to 0$ and let
$f_{k}:={\rm{sdist}}(\cdot,\sigma_{\rho_{k}}(\partial E))\big{|}_{Q_{R}}\in
W^{1,\infty}(Q_{R})$. We see that for any $k>0$, $f_{k}$ is $1$-Lipschitz
continuous, moreover by the fact that
$f_{k}(0)={\rm{sdist}}(0,\sigma_{\rho_{k}}(\partial E))=0$, we deduce that
$\\{f_{k}\\}$ is uniformly bounded. By applying Ascoli-Arzelà Theorem, there
exists $f\in W^{1,\infty}(Q_{R})$ and a non-relabeled subsequence
$\left\\{f_{k}\right\\}$ such that $f_{k}\to f$ uniformly in $Q_{R}$. In view
of [36, Proposition A.1], by (5.2) we obtain that $\absolutevalue{f_{k}}={\rm
dist}(\cdot,\sigma_{\rho_{k}}(\partial E))\to{\rm dist}(\cdot,T_{0})$
uniformly in $Q_{R}$ and thus, $\absolutevalue{f(x)}={\rm dist}(x,\partial
H_{0})$ for any $x\in Q_{R}$.

It remains to prove that $f(\cdot)={\rm{sdist}}(\cdot,\partial H_{0})$ in
$Q_{R}$. We proceed by absurd. Assume by contradiction that
$f\neq{\rm{sdist}}(\cdot,\partial H_{0})$ then, either
$f\equiv{\rm{sdist}}(\cdot,\partial(Q_{R}\setminus H_{0}))$ or $f\equiv{\rm
dist}(x,T_{0})$ or $f\equiv-{\rm dist}(x,T_{0})$. Let us first consider the
case in which $f\equiv{\rm{sdist}}(\cdot,\partial(Q_{R}\setminus H_{0}))$. In
view of Remark 3.8-(i), it follows that
$\sigma_{\rho_{k}}(E)\xrightarrow{\mathcal{K}}Q_{R}\setminus{\rm{Int}}(H_{0})$
and so, as a consequence we have that
$\mathcal{L}^{2}\left(\sigma_{\rho_{k}}(E)\triangle(Q_{R}\setminus{\rm{Int}}(H_{0}))\right)\to
0$ as $k\to\infty$, which is in contradiction with the De Giorgi’s structure
theorem of sets of finite perimeter (see [24, Theorem 5.13] or [43, Theorem
15.5]).

In the case $f(\cdot)\equiv{\rm dist}(\cdot,T_{0})$, thanks to the fact that
$f_{k}\to f$ uniformly in $\overline{Q_{R}}$ and by Lemma 5.2, we have that
$\sigma_{\rho_{k}}(E)\cap\overline{Q_{R}}\xrightarrow{\mathcal{K}}T_{0}\cap\overline{Q_{R}}$,
and thus, $\mathbbm{1}_{\sigma_{\rho_{k}}(E)}\to\mathbbm{1}_{T_{0}}$ in
$L^{1}(\overline{Q_{R}})$, which is a contradiction with the fact that
$\mathbbm{1}_{\sigma_{\rho_{k}}(E)}\to\mathbbm{1}_{H_{0}}$ in
$L^{1}_{\mathrm{loc}}(\mathbb{R}^{2})$ by [24, Theorem 5.13].

In the last case in which $f(\cdot)\equiv-{\rm dist}(\cdot,T_{0})$, we proceed
analogously, and by Lemma 5.2 we obtain that
$\overline{Q_{R}}\setminus\sigma_{\rho_{k}}(E)\xrightarrow{\mathcal{K}}T_{0}$
and hence,
$\mathbbm{1}_{(\overline{Q_{1}}\setminus\sigma_{\rho_{k}}(E))}\to\mathbbm{1}_{T_{0}}$
in $L^{1}(\overline{Q_{R}})$. Therefore, we reach a contradiction again by
applying [24, Theorem 5.13] since $x\in\partial^{*}(\mathbb{R}^{2}\setminus
E)$ and so,
$\mathbbm{1}_{(\overline{Q_{R}}\setminus\sigma_{\rho_{k}}(E))}\to\mathbbm{1}_{{\mathbb{R}^{2}\setminus{\rm{Int}}(H_{0})}}$
in $L^{1}_{\mathrm{loc}}(\mathbb{R}^{2})$. ∎

We now introduce the notions of _film free boundary_ , _substrate free
boundary_ , and _film-substrate adhesion interface_ for triples
$(A,h,K)\in\mathcal{B}$, and of _triple junctions_ at the points where they
“meet”.

######  Definition 5.4.

For any admissible pair $(A,S)\in{\widetilde{\mathcal{B}}}$ we denote:

  * •

the _film free boundary_ , the _substrate free_ boundary and the _film-
substrate adhesion interface_ by

$\displaystyle\Gamma_{\mathrm{F}}(A,S)$ $\displaystyle:=\left((\partial
A\setminus\partial S)\cup(\partial S\cap\partial A\cap
A^{(1)}){\cup(\partial{S}\cap\partial^{*}A\cap S^{(0)})}\right)\cap{\Omega},$
$\displaystyle\Gamma_{\mathrm{S}}(A,S)$ $\displaystyle:=\left((\partial
S\cap\partial A)\setminus{(\partial{S}\cap\partial^{*}A\cap
S^{(0)})}\right)\cap{\Omega},$
$\displaystyle\Gamma^{\text{A}}_{\mathrm{FS}}(A,S)$
$\displaystyle:=\left(\left((\partial S\setminus\partial A)\cap
A^{(1)}\right)\right)\cap{\Omega},$

respectively. Notice that the _film-substrate delamination interface_ , that
we define by

$\Gamma^{\text{D}}_{\mathrm{FS}}(A,S):=\left((\partial S\cap\partial A)\cap
A^{(1)}\right)\cap{\Omega},$

is contained both in $\Gamma_{\mathrm{F}}(A,S)$ and in
$\Gamma_{\mathrm{S}}(A,S)$.

  * •

_triple junction_ (by including for simplicity also the “double” junctions at
the boundary) any point

$\displaystyle p\in$ $\displaystyle\left(Cl(\Gamma_{\mathrm{F}}(A,S))\cap
Cl(\Gamma_{\mathrm{S}}(A,S))\cap
Cl(\Gamma^{\text{A}}_{\mathrm{FS}}(A,S))\cap{\Omega}\right)$
$\displaystyle\cup\left(Cl(\Gamma_{\mathrm{F}}(A,S))\cap
Cl(\Gamma_{\mathrm{S}}(A,S))\cap\partial{\rm{Int}}(\overline{S})\cap\partial{\Omega})\right)$
$\displaystyle\cup\left(Cl(\Gamma_{\mathrm{F}}(A,S))\cap
Cl(\Gamma_{\mathrm{FS}}^{A}(A,S))\cap\partial{\Omega})\right),$

where the closures are considered with respect to the relative topology of
$\partial A\cup\partial S$.

The next result allows us to assume that the adhesion interface of any
admissible pair $(A,S)\in\mathcal{B}_{\textbf{m}}$ (without the substrate
internal cracks) can be considered, up to an error and up to passing to the
family $\widetilde{\mathcal{B}}$, to be given by a finite number (depending on
the initial pair $(A,S)$) of connected components.

######  Lemma 5.5.

Let $R$ be an open rectangle with two sides, that are denoted by $T_{1}$ and
$T_{2}$, perpendicular to $\textbf{e}_{\textbf{1}}$. Let
$(A,S_{h,K})\in\mathcal{B}_{\textbf{m}}$ for $(h,K)\in\mathrm{AHK}$ be such
that ${\mathcal{S}}_{L}(A,S_{h,K},R)<\infty$, where ${\mathcal{S}}_{L}$ is the
localized surface energy defined in (5.1). If
$\mathcal{H}^{1}\left(\left(\Gamma^{\mathrm{{A}}}_{\mathrm{FS}}\left(A,S_{h,K}\right)\setminus{\rm{Int}}(S_{h})\right)\cap
R\right)>0$, for every $\eta\in(0,1)$ small enough there exist
$M:=M(A,S_{h,K},\eta)\in\mathbb{N}\cup\\{0\\}$ and
$(\widetilde{A},\widetilde{S})\in\widetilde{\mathcal{B}}$ such that
$(\Gamma^{\mathrm{{A}}}_{\mathrm{FS}}(\widetilde{A},\widetilde{S})\setminus{\rm{Int}}(\overline{\widetilde{S}}))\cap
R$ has at most $M$ connected components and

${\mathcal{S}}_{L}(A,S_{h,K},R)\geq{\mathcal{S}}_{L}(\widetilde{A},\widetilde{S},R)-\eta.$
(5.3)

Furthermore, $(\widetilde{A},\widetilde{S})$ satisfies the following
properties:

  * (i)

If $\overline{\partial S_{h}\cap R}\cap T_{\ell}\neq\emptyset$ for $\ell=1,2$,
then also $\overline{\partial\widetilde{S}\cap R}\cap T_{\ell}\neq\emptyset$
for $\ell=1,2$;

  * (ii)

If there exists a closed connected set $\Lambda\subset\overline{\partial A\cap
R}$ such that $\Lambda\cap T_{\ell}\neq\emptyset$ for $\ell=1,2$, then there
exists a curve with support
$\widetilde{\Lambda}\subset\overline{\partial\widetilde{A}\cap R}$ such that
$\widetilde{\Lambda}\cap T_{\ell}\neq\emptyset$ for $\ell=1,2$.

###### Proof.

Notice that we cannot a priori exclude that
$(\Gamma^{\text{A}}_{\mathrm{FS}}(A,S_{h,K})\setminus{\rm{Int}}(S_{h}))\cap R$
is a totally disconnected set with positive $\mathcal{H}^{1}$-measure (see,
for instance, the Smith-Volterra-Cantor set in [47, Chapter 3]). We denote by
$L(h)$ the set of substrate filaments of the substrate $S_{h,K}$, namely,

$L(h):=\\{(x_{1},x_{2})\in\overline{\Omega}:x_{1}\in(-l,l)\text{ and
}h^{+}(x_{1})<x_{2}\leq h(x_{1})\\}\subset\partial S_{h}.$ (5.4)

Since $h$ is upper semicontinuous, there exist an index set $J_{1}$ and a
countable family of disjoint points $\\{x_{1}^{j}\\}_{j\in
J_{1}}\subset(-l,l)$ such that

$L(h)=\bigcup_{j\in J_{1}}L_{j}(h),$ (5.5)

where
$L_{j}(h):=\\{(x^{j}_{1},x_{2})\in\overline{\Omega}:h^{+}(x^{j}_{1})<x_{2}\leq
h(x^{j}_{1})\\}$ for every $j\in J_{1}$. In the following three steps, in view
of the outer regularity of Borel measures, we construct an admissible pair
$(\widetilde{A},\widetilde{S})\subset\widetilde{\mathcal{B}}$ by modifying
some portions of $\partial S_{h}$ and $\partial A$. More precisely, in the
first step, we construct an admissible height $h^{1}$ by eliminating a family
of “small” filaments of $L(h)$ so that $L(h^{1})$ consists of only a finite
number of filaments, we accordingly modify $A$ in an admissible region $A^{1}$
containing $L(h)\setminus L(h^{1})$, and we define $K^{1}:=K$. In the second
step, we construct $S^{2}$ by modifying $S_{h^{1},K^{1}}$ and we introduce an
admissible region $A^{2}$ in such a way that
$(\Gamma^{\text{A}}_{\text{FS}}(A^{2},S^{2})\setminus{\rm{Int}}(\overline{S^{2}}))\cap
R$ is a countable union of connected components, and (i) and (ii) hold true.
In the third step, by eliminating some components of
$\Gamma^{\text{A}}_{\text{FS}}(A^{2},S^{2})\setminus{\rm{Int}}(\overline{S^{2}}))\cap
R$ we define an admissible pair
$(\widetilde{A},\widetilde{S})\in\widetilde{\mathcal{B}}$ for which
$(\Gamma^{\text{A}}_{\mathrm{FS}}(\widetilde{A},\widetilde{S})\setminus{\rm{Int}}(\overline{\widetilde{S}}))\cap
R$ has at most $M$-connected components, (i) and (ii) are preserved, and (5.3)
holds true.

Step 1 (Modification of substrate filaments). We modify $(A,S_{h,K})$ in
$(A^{0},S_{h^{1},K^{1}})$ to have a finite number of substrate filaments. We
denote by $J_{2}\subset J_{1}$ the set of indexes $j\in J_{1}$ such that
$\mathcal{H}^{1}(L_{j}\setminus{\rm{Int}}(A))=0$ and $L_{j}$ is not connected
to $\partial A$, and we denote by $F_{J_{2}}(h)$ the set of the
$x_{1}$-coordinates corresponding to the points in each vertical segment
$L_{j}$ for $j\in J_{2}$, i.e., $F_{J_{2}}(h):=\\{x^{j}_{1}\\}_{j\in J_{2}}$,
where $x^{j}_{1}\in{[-l,l]}$ is such that
$(x^{j}_{1},h(x^{j}_{1}))\in{L_{j}(h)}$ and $h^{+}(x^{j}_{1})<h(x^{j}_{1})$
for every $j\in J_{2}$. We define as ${h^{0}}$ the modification of $h$, given
by

$\begin{matrix}h^{0}:&{[-l,l]}&\to&{[0,L]}&\\\
&x&\mapsto&h^{0}(x)&:=\begin{cases}h(x)&\text{if }x\in{[-l,l]}\setminus
F_{J_{2}}(h),\\\ h^{+}(x)&\text{if }x\in F_{J_{2}}(h),\end{cases}\end{matrix}$

and observe that by construction $h^{0}\in\text{AH}$ and

$\mathcal{H}^{1}(\partial S_{h^{0}})\leq\mathcal{H}^{1}(\partial S_{h}),$
(5.6)

where $S_{h^{0}}$ is defined as in (3.2). We notice that the triple
$(A,S_{h^{0},K^{0}})\in\mathcal{B}_{\textbf{m}}$, where $K^{0}:=K$. As a
consequence of the construction and of the non-negativity of
$\phi_{\text{FS}}$, it follows that

${\mathcal{S}}_{L}(A,S_{h,K},R)\geq{\mathcal{S}}_{L}(A,S_{h^{0},K^{0}},R).$
(5.7)

We notice that by (5.5) we have that

$\mathcal{H}^{1}(L(h^{0}))=\mathcal{H}^{1}(L(h))-\sum_{j\in
J_{2}}\mathcal{H}^{1}(L_{j}(h))=\sum_{j\in J_{3}}\mathcal{H}^{1}(L_{j}(h)),$

where $J_{3}:=J_{1}\setminus J_{2}$ and $L(h^{0})$ is the set of substrate
filaments of $S_{h^{0},K^{0}}$ defined as in (5.4). Therefore, for a fix
$\widetilde{\eta}>0$ there is
$j^{\prime}_{1}:=j^{\prime}_{1}(\widetilde{\eta})\in J_{3}$ such that

$\sum_{j=j^{\prime}_{1}+1,j\in
J_{3}}^{\infty}\mathcal{H}^{1}(L_{j}(h))\leq\widetilde{\eta},$ (5.8)

and we define

$A^{1}:=\left(A\setminus\left(\bigcup_{j=j^{\prime}_{1}+1,j\in
J_{3}}^{\infty}(L_{j}(h)\cap{\rm{Int}}(A))\right)\right)\cup\bigcup_{j=j^{\prime}_{1}+1,j\in
J_{3}}^{\infty}(L_{j}(h)\cap(\Omega\setminus\overline{{\rm{Int}}(A)})).$

Furthermore, we denote by $h^{1}$ the modification of $h^{0}$, defined by

$\begin{matrix}h^{1}:&{[-l,l]}&\to&{[0,L]}&\\\
&x&\mapsto&h^{1}(x)&:=\begin{cases}h^{0}(x)&\text{if }x\in{[-l,l]}\setminus
F_{J_{4}}(h^{0}),\\\ {h^{0}}^{+}(x)&\text{if }x\in
F_{J_{4}}(h^{0}),\end{cases}\end{matrix}$

where $J_{4}:=\\{j\in J_{3}:j\geq j^{\prime}_{1}+1\\}$ and
$F_{J_{4}}(h^{0}):=\\{x^{j}_{1}\\}_{j\in J_{4}}$ such that
$(x^{j}_{1},h(x^{j}_{1}))\in{L_{j}(h)}$ for every $j\in J_{4}$. We define
$K^{1}:=K^{0}$, we notice that $(h^{1},K^{1})\in\text{AHK}$ and
$S_{h^{1},K^{1}}\subset\overline{A^{1}}$, thus
$(A^{1},S_{h^{1},K^{1}})\in\mathcal{B}_{\textbf{m}}$ and since $L_{j}(h)$ is
connected to $\partial A$ for every $j\in J_{4}\subset J_{3}$, we deduce that
$\partial A^{1}$ has at most $m_{1}$-connected components. Finally, we observe
that

${\mathcal{S}}_{L}(A,S_{h,K},R)-{\mathcal{S}}_{L}(A^{1},S_{h^{1},K^{1}},R)\geq-\sum_{j\in
J_{4}}2\int_{L_{j}(h)}{\phi_{\text{F}}(\nu_{L_{j}(h)})d\mathcal{H}^{1}}\geq-2c_{2}\widetilde{\eta},$
(5.9)

where we used the non-negativeness of $\phi$, $\phi_{\text{F}S}$ and (3.15),
and we observe that $(A^{1},S_{h^{1},K^{1}})\in\mathcal{B}_{\textbf{m}}$ has a
finite number of substrate filaments, more precisely, we denote by
$j^{\prime}\in\mathbb{N}$ the cardinality of the index set $J:=J_{3}\setminus
J_{4}=\\{j\in J_{3}:j\leq j^{\prime}_{1}\\}$ and we have that $L(h^{1})$ is
the union of $j^{\prime}$ filaments, i.e.,

$L(h^{1})=\bigcup_{j\in J}L_{j}(h),$ (5.10)

where $L(h^{1})$ is defined as in (5.4) with respect to the substrate
$S_{h^{1},K^{1}}\in\text{AS}(\Omega)$.

Step 2 (Modification of the substrate free boundary). Without loss of
generality in the following we assume that
$\partial{\rm{Int}}(S_{h^{1}})\cap\overline{L(h^{1})}\subset R$. Since
$\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\partial{S_{h^{1},K^{1}}}$ is a
finite Borel measure and

$\Gamma^{\text{A}}:=\left(\Gamma^{\text{A}}_{\mathrm{FS}}(A^{1},S_{h^{1},K^{1}})\setminus{\rm{Int}}(S_{h^{1}})\right)\cap
R$

is a Borel set, by the outer regularity of measures (see [43, Theorem 2.10]),
there exists an open set $O=O(\widetilde{\eta})\subset R$ such that
$\Gamma^{\text{A}}\subset O\cap\partial{S_{h^{1},K^{1}}}$ and

$\begin{split}\mathcal{H}^{1}\left(\widehat{\Lambda}\right)&=\mathcal{H}^{1}\left((O\cap\partial^{*}{S_{h^{1},K^{1}}})\setminus\Gamma^{\text{A}}\right)\leq\mathcal{H}^{1}\left((O\cap\partial{S_{h^{1},K^{1}}})\setminus\Gamma^{\text{A}}\right)<\frac{2^{-5/2}}{j^{\prime}+1}\widetilde{\eta},\end{split}$
(5.11)

where
$\widehat{\Lambda}:=(O\cap\partial{\rm{Int}}(\overline{S_{h^{1},K^{1}}}))\setminus\Gamma^{\text{A}}$
and $j^{\prime}$ is defined in the Step 1 as the number of filaments of
$S_{h^{1}}$. Moreover, by using the notation introduced in (3.2) and the fact
that $h^{1}\in\text{AH}(\Omega)$ we conclude that

$\partial{\rm{Int}}\left(\overline{S_{h^{1},K^{1}}}\right)=\partial{\rm{Int}}(S_{h^{1}})=\partial
S_{{h^{1}}^{+}},$ (5.12)

and hence, since $\partial\mathrm{{\rm{Int}}}(\overline{S_{h^{1},K^{1}}})$ is
a connected and compact set in $\mathbb{R}^{2}$ with finite
$\mathcal{H}^{1}$-measure, by [25, Lemma 3.12] there exists a parametrization
$r:[0,1]\to\mathbb{R}^{2}$ of
$\partial\mathrm{{\rm{Int}}}(\overline{S_{h^{1},K^{1}}})$ whose support
$\gamma$ joins the points $(-l,{h^{1}}(-l^{+}))$ with $(l,{h^{1}}(l^{-}))$.

Notice that by Step 1, $\partial{\rm{Int}}(S_{h^{1}})\cap\overline{L(h^{1})}$
is the union of $j^{\prime}$-points that we can order by labeling them with
$p_{1},\dots,p_{j^{\prime}}$. Furthermore, we denote with $p_{0}:=r(t_{0})$
and $p_{j^{\prime}+1}:=r(t_{1})$, where
$t_{0}:=\inf\\{t\in[0,1]:r(t)\in\partial R\\}$ and
$t_{1}:=\sup\\{t\in[0,1]:r(t)\in\partial R\\}$, and we consider the family
$\\{R^{i}\\}_{i=1}^{j^{\prime}+1}$ of the strips $R^{i}$ defined as the open
regions of $R$ contained between the vertical lines passing though the points
$p_{i-1}$ and $p_{i}$.

Since $\widehat{\Lambda}$ is a Borel measurable set, by (2.6) and (5.11) we
have that

$\mathcal{N}^{1}(\widehat{\Lambda})\leq
2^{\frac{5}{2}}\mathcal{H}^{1}(\widehat{\Lambda})<\frac{\widetilde{\eta}}{j^{\prime}+1},$
(5.13)

where $\mathcal{N}^{1}$ is the net measure defined in (2.4). Therefore, there
exists $\delta>0$ such that we can find a family of disjoint dyadic squares
$\\{U_{n}\\}_{n\in\mathbb{N}}\subset\mathcal{Q}$ such that
$\widehat{\Lambda}\subset\bigcup_{n\in\mathbb{N}}U_{n}$,
$\mathrm{diam}(U_{n})\leq\delta$ for any ${n\in\mathbb{N}}$ and

$\sum_{n\in\mathbb{N}}{\mathrm{diam}(U_{n})}\leq\mathcal{N}^{1}(\widehat{\Lambda})+\frac{\widetilde{\eta}}{j^{\prime}+1}<\frac{2}{j^{\prime}+1}\widetilde{\eta}.$
(5.14)

Without loss of generality we assume that ($\widehat{\Lambda}\neq\emptyset$
and that) $U_{n}\cap\widehat{\Lambda}$ is non-empty for every
$n\in\mathbb{N}$. Let $\\{U_{n}^{i}\\}\subset\\{U_{n}\\}$ be the subfamily of
dyadic squares such that $U_{n}^{i}\cap R^{i}\cap\gamma\neq\emptyset$ for
every $n\in\mathbb{N}$. Furthermore, we assume for simplicity that
${\rm{Int}}(U_{n}^{i})\subset R^{i}$. We begin by modifying the pair
$(A^{1},S_{h^{1},K^{1}})$ in the strip $R^{1}$, by denoting the modification
by $(A^{2}_{1},S^{2}_{1})$. We characterize $A^{2}_{1}$ and $S^{2}_{1}$ as

$A^{2}_{1}:=\left(A^{1}\cup\bigcup_{n\in\mathbb{N}}{U_{n}^{1}}\right)\setminus\bigcup_{n\in\mathbb{N}}(\partial
U_{n}^{1}\setminus{U_{n}^{1}})$ (5.15)

and

$S^{2}_{1}:=\left(S_{h^{1},K^{1}}\setminus\bigcup_{n\in\mathbb{N}}{\rm{Int}}(U_{n}^{1})\right)\cup\bigcup_{n\in\mathbb{N}}(\partial
U^{1}_{n}\cap(R^{1}\setminus S_{h^{1}})),$ (5.16)

respectively. By construction, it follows that
$S^{2}_{1}\subset\overline{A^{2}_{1}}$, $\partial A^{2}_{1}$ and $\partial
S^{1}_{1}$ have finite $\mathcal{H}^{1}$ measure and are
$\mathcal{H}^{1}$-rectifiable, and $\partial
A^{2}_{1}\cap{\rm{Int}}(\overline{S^{2}_{1}})=\emptyset$ and hence,
$(A^{2}_{1},S^{2}_{1})\in\widetilde{\mathcal{B}}$. Furthermore, we have that

$\begin{split}&{\mathcal{S}}_{L}\left(A^{1},S_{h^{1},K^{1}},R^{1}\right)-{\mathcal{S}}_{L}\left(A_{1}^{2},S_{1}^{2},R^{1}\right)\\\
&\geq-2\int_{\bigcup_{n\in\mathbb{N}}\partial
U^{1}_{n}}\phi_{\text{F}}(\nu_{U^{1}_{n}})+\phi(\nu_{U^{1}_{n}})+\phi_{\text{FS}}(\nu_{U^{1}_{n}})\,d\mathcal{H}^{1}\\\
&\geq-\sum_{n\in\mathbb{N}}2\int_{\partial
U^{1}_{n}}\phi_{\text{F}}(\nu_{U^{1}_{n}})+\phi(\nu_{U^{1}_{n}})+\phi_{\text{FS}}(\nu_{U^{1}_{n}})\,d\mathcal{H}^{1}\\\
&\geq-\sum_{n\in\mathbb{N}}24\,c_{2}\text{diam}(U^{1}_{n})\geq-\frac{48}{j^{\prime}+1}c_{2}\widetilde{\eta},\end{split}$
(5.17)

where in the first inequality we used the non-negativeness of
$\phi_{\mathrm{F}},\phi_{\mathrm{FS}}$ and $\phi$, in the second inequality we
used the subadditivity of measures, in the third inequality we used (3.15),
and in the last inequality we used (5.14). We notice that
$\Gamma^{\text{A}}_{\text{FS}}(A^{2}_{1},S^{2}_{1})\cap\partial{\rm{Int}}(\overline{S^{2}_{1}})\cap
R^{1}$ is a countable union of connected sets because by construction every
connected component of
$\Gamma^{\text{A}}_{\text{FS}}(A^{2}_{1},S^{2}_{1})\cap\partial{\rm{Int}}(\overline{S^{2}_{1}})\cap
R^{1}$ is connected to an element in the family of sets $\\{\partial
U^{1}_{n}\cap U^{1}_{n}\\}_{n\in\mathbb{N}}$.

Now, we modify $(A_{1}^{2},S_{1}^{2})$ in a new configuration
$(A^{3}_{1},S^{3}_{1})$ in order to prove Assertion (ii). To this end let
$\Lambda\subset\overline{\partial A\cap R}$ be a closed connected set such
that $\Lambda\cap T_{\ell}\neq\emptyset$ for $\ell=1,2$. By [25, Lemma 3.12]
there exists a parametrization $r_{1}:[0,1]\to\mathbb{R}^{2}$ whose support
$\gamma^{1}\subset\Lambda$ joins $T_{1}$ with $T_{2}$. We define
$\gamma^{1}_{1}:=\gamma^{1}\cap\overline{R^{1}}$ and we observe that
$(\gamma^{1}_{1}\setminus{\rm{Int}}(U^{1}_{n}))\cup\partial U^{1}_{n}$ is a
connected set. Let $Z_{1}\subset\mathbb{N}$ be the set of indexes $n$ such
that $\gamma_{1}\cap(\partial U^{1}_{n}\cap U^{1}_{n})\neq\emptyset$. If
$Z_{1}=\emptyset$, then we define $A^{3}_{1}:=A^{2}_{1}$ and
$S^{3}_{1}:=S^{2}_{1}$. If $Z_{1}\neq\emptyset$, then we modify
$\gamma_{1}^{1}$ in $\bigcup_{n\in Z_{1}}U^{1}_{n}$ by defining a new set
$\Lambda_{1}$. More precisely, by using the fact that dyadic squares by
definition do not intersect each other, we fix $n\in Z_{1}$ and we replace
with a set $\Lambda^{1}_{n}$ (see (5.24) below) the portion of $\gamma^{1}$
passing through $U^{1}_{n}$. To this end, let us denote the closures of the
left and bottom sides of $U^{1}_{n}$ by $L^{1}_{n}$ and $L^{2}_{n}$,
respectively, and proceed by defining $\Lambda^{1}_{n}$ in different way with
respect to following three cases (see Figure 2):

Case 1:

$\bm{\gamma^{1}_{1}\cap L^{1}_{n}\neq\emptyset}$ and $\bm{\gamma_{1}\cap
L^{2}_{n}=\emptyset}$. Since $L^{1}_{n}$ is closed, we deduce that
$\gamma^{1}_{1}\cap L^{1}_{n}$ is closed. Therefore, there exist
$a^{2}_{n}:=\max\\{a\in\mathbb{R};(l_{n}^{1},a)\in\gamma^{1}_{1}\cap
L^{1}_{n}\\}$ and
$b^{2}_{n}:=\min\\{b\in\mathbb{R};(l_{n}^{1},b)\in\gamma^{1}_{1}\cap
L^{1}_{n}\\}$, where $l^{1}_{n}$ is the element in the singleton
$\pi_{1}(L^{1}_{n})$. Since $\gamma_{1}$ is parametrized by $r_{1}$, there
exist $t_{n}^{1},t_{n}^{2}\in[0,1]$ such that
$p_{n,1}^{1}:=(l^{1}_{n},a^{2}_{n})=r_{1}(t_{n}^{1})$ and
$p_{n,1}^{2}:=(l^{1}_{n},b^{2}_{n})=r_{1}(t_{n}^{2})$, and by the continuity
of $r_{1}$ there exists $q_{n,1}^{1}\in\gamma^{1}_{1}\setminus U^{1}_{n}$ such
that ${\rm
dist}(p_{n,1}^{1},q_{n,1}^{1})\leq\frac{\mathrm{diam}(U^{1}_{n})}{2}$. If
$\pi_{1}(q_{n,1}^{1})=l^{1}_{n}$, then we define
$\widetilde{q}_{n,1}^{1}:=(l_{n}^{1}-\varepsilon,\pi_{2}(q_{n,1}^{1})-\varepsilon)$
for a $\varepsilon>0$ small enough such that ${\rm
dist}(q_{n,1}^{1},\widetilde{q}_{n,1}^{1})\leq\frac{\mathrm{diam}(U^{1}_{n})}{2}$,
otherwise we let $\widetilde{q}_{n,1}^{1}:=q_{n,1}^{1}$. We denote by
$\widetilde{L}_{n,1}^{1}$ the segment connecting $q_{n,1}^{1}$ with
$\widetilde{q}_{n,1}^{1}$, we denote by $\widetilde{L}_{n,1}^{2}$ the segment
connecting $\widetilde{q}_{n,1}^{1}$ with $p_{n,1}^{2}$, and we denote by
$\widetilde{L}_{n,1}^{3}$ the segment connecting $p_{n,1}^{1}$ with the vertex
of $U^{1}_{n}$ in $\partial U^{1}_{n}\setminus U^{1}_{n}$. Let
$\Lambda^{1}_{n}:=\widetilde{L}_{n,1}^{1}\cup\widetilde{L}_{n,1}^{2}{\cup\widetilde{L}_{n,1}^{3}}$
and observe that by construction it follows that

$\mathcal{H}^{1}(\Lambda^{1}_{n})=\mathcal{H}^{1}(\widetilde{L}_{n,1}^{1}\cup\widetilde{L}_{n,1}^{2}{\cup\widetilde{L}_{n,1}^{3}})\leq{2\,}\text{diam}(U^{1}_{n}).$
(5.18)

Case 2:

$\bm{\gamma^{1}_{1}\cap L^{1}_{n}=\emptyset}$ and $\bm{\gamma_{1}\cap
L^{2}_{n}\neq\emptyset}$. By arguing analogously to Case 1 there exist
$t_{n}^{1},t_{n}^{2}\in[0,1]$ such that
$p_{n,2}^{1}:=(a^{1}_{n},l^{2}_{n})=r_{1}(t_{n}^{1})$ and
$p_{n,2}^{2}:=(b^{1}_{n},l^{2}_{n})=r_{1}(t_{n}^{2})$, where
$\pi_{2}(L^{2}_{n})=\\{l^{2}_{n}\\}$,
$a^{1}_{n}:=\max\\{a\in\mathbb{R};(a,l_{n}^{2})\in\gamma^{1}_{1}\cap
L^{2}_{n}\\}$, and
$b^{1}_{n}:=\min\\{b\in\mathbb{R};(b,l_{n}^{2})\in\gamma^{1}_{1}\cap
L^{2}_{n}\\}$. By the continuity of $r_{1}$ there exists
$q_{n,2}^{1}\in\gamma^{1}_{1}\setminus U^{1}_{n}$ such that ${\rm
dist}(p_{n,2}^{1},q_{n,2}^{1})\leq\frac{\mathrm{diam}(U^{1}_{n})}{2}$. If
$\pi_{2}(q_{n,2}^{1})=l^{2}_{n}$, then we define
$\widetilde{q}_{n,2}^{1}:=(\pi_{1}(q_{n,2}^{1})-\varepsilon,l_{n}^{2}-\varepsilon)$
for $\varepsilon>0$ small enough such that ${\rm
dist}(q_{n,2}^{1},\widetilde{q}_{n,2}^{1})\leq\frac{\mathrm{diam}(U^{1}_{n})}{2}$,
otherwise we let $\widetilde{q}_{n,2}^{1}:=q_{n,2}^{1}$. We denote by
$\widetilde{L}_{n,2}^{1}$ the segment connecting $\widetilde{q}_{n,2}^{1}$
with $p_{n,2}^{1}$, we denote by $\widetilde{L}_{n,2}^{2}$ the segment
connecting $\widetilde{q}_{n,2}^{1}$ with $p_{n,2}^{2}$ and we denote by
$\widetilde{L}_{n,2}^{3}$ the segment connecting $p_{n,2}^{1}$ with with the
vertex of $U^{1}_{n}$ in $\partial U^{1}_{n}\setminus U^{1}_{n}$. Let
$\Lambda^{1}_{n}:=\widetilde{L}_{n,2}^{1}\cup\widetilde{L}_{n,2}^{2}{\cup\widetilde{L}_{n,2}^{3}}$
and observe that by construction it follows that

$\mathcal{H}^{1}(\Lambda^{1}_{n})=\mathcal{H}^{1}(\widetilde{L}_{n,2}^{1}\cup\widetilde{L}_{n,2}^{2}{\cup\widetilde{L}_{n,2}^{3}})\leq{2\,}\text{diam}(U^{1}_{n}).$
(5.19)

Case 3:

$\bm{\gamma^{1}_{1}\cap L^{1}_{n}\neq\emptyset}$ and $\bm{\gamma_{1}\cap
L^{2}_{n}\neq\emptyset}$. We define
$p^{k}_{n,\ell},q^{1}_{n,\ell},\widetilde{q}_{n,\ell}^{1}$,
$\widetilde{L}_{n,\ell}^{\alpha}$ for $k=1,2$, $\alpha=1,2,3$ as in Case 1 for
$\ell=1$ and as in Case 2 for $\ell=2$. Furthermore, we denote by
${\widetilde{L}_{n}^{4}}$ the segment connecting $p^{2}_{n,1}$ with
$p^{2}_{n,2}$. Let
$\Lambda^{1}_{n}:=\widetilde{L}_{n,1}^{1}\cup\widetilde{L}_{n,1}^{2}\cup\widetilde{L}_{n,2}^{1}\cup\widetilde{L}_{n,2}^{2}{\cup\widetilde{L}^{3}_{n,1}\cup\widetilde{L}^{3}_{n,2}\cup\widetilde{L}_{n}^{4}}$
and observe that by construction it follows that

$\displaystyle\mathcal{H}^{1}(\Lambda^{1}_{n})$
$\displaystyle\leq\mathcal{H}^{1}(\widetilde{L}_{n,1}^{1}\cup\widetilde{L}_{n,1}^{2}{\cup\widetilde{L}^{3}_{n,1}})+\mathcal{H}^{1}(\widetilde{L}_{n,2}^{1}\cup\widetilde{L}_{n,2}^{2}{\cup\widetilde{L}^{3}_{n,2}})+{\mathcal{H}^{1}(\widetilde{L}^{4}_{n})}$
$\displaystyle\leq{5}\,\text{diam}(U^{1}_{n}).$ (5.20)

Figure 2. The three squares appearing in the illustration represent dyadic
squares $U^{1}_{n}$ of the first strip $R^{1}$ in the three different cases,
namely, by moving from the left to the right, Cases 1, 2, and 3, that are
considered in Step 2 of the proof of Lemma 5.5. On the left the initial pair
$(A^{2}_{1},S^{2}_{1})$ is represented, while on the right the pair
$(A^{3}_{1},S^{3}_{1})$ that is obtained after the modification described in
such step is depicted.

Let

$\Gamma_{1}:=\overline{\left(\Gamma^{\text{A}}_{\text{FS}}(\overline{A^{2}_{1}},S^{2}_{1})\cap\partial{\rm{Int}}(\overline{S^{2}_{1}})\cap
R^{1}\right)\setminus\overline{\bigcup_{n\in Z_{1}}(U^{1}_{n})}}.$

We now observe that the previous construction of $\bigcup_{n\in
Z_{1}}\Lambda^{1}_{n}$ does not divide $\Gamma_{1}$ in an uncountable number
of connected components. More precisely, we claim that for every given a
connected component $\widehat{\Gamma}$ of $\Gamma_{1}$,
$\widehat{\Gamma}\setminus\overline{\bigcup_{n\in Z_{1}}\Lambda^{1}_{n}}$ is a
countable union of disjoint connected sets. To prove this claim, we notice
that, since $\overline{\widehat{\Gamma}}\subset\gamma$ and $\gamma$ is
parameterized by $r$, also $\overline{\widehat{\Gamma}}$ is parametrizable and
hence, there exists a continuous injective map
$\widehat{r}:[0,1]\to\mathbb{R}^{2}$ whose support coincides with
$\overline{\widehat{\Gamma}}$. This in particular proves that $\widehat{r}$ is
a homeomorphism between $[0,1]$ and $\overline{\widehat{\Gamma}}$. The claim
then follows from the fact that
$\overline{\widehat{\Gamma}}\setminus\overline{\bigcup_{n\in
Z_{1}}\Lambda^{1}_{n}}$ is open with respect to the relative topology of
$\overline{\widehat{\Gamma}}$ and [47, Proposition 8 in Part 1].

We are now in the position to define $A^{3}_{1}$ and $S^{3}_{1}$ as follows

$A^{3}_{1}:=\overline{A^{2}_{1}}\setminus\left(\left(\gamma^{1}\setminus\bigcup_{n\in
Z_{1}}U_{n}^{1}\right)\cup\bigcup_{n\in\mathbb{N}}(\partial
U_{n}^{1}\setminus{U_{n}^{1}})\cup\bigcup_{n\in
Z_{1}}(\overline{A^{2}_{1}}\cap\Lambda^{1}_{n})\right)\\\ \cup\bigcup_{n\in
Z_{1}}((R^{1}\setminus{\rm{Int}}({A^{2}_{1}}))\cap\Lambda^{1}_{n})$ (5.21)

and

$S^{3}_{1}:=S^{2}_{1}\setminus\left(\bigcup_{n\in
Z_{1}}(\overline{A^{2}_{1}}\cap\Lambda^{1}_{n})\right).$ (5.22)

Therefore, we have that

$\begin{split}{\mathcal{S}}_{L}\left(A_{1}^{2},S_{1}^{2},R^{1}\right)-{\mathcal{S}}_{L}\left(A_{1}^{3},S_{1}^{3},R^{1}\right)&\geq-2\int_{\bigcup_{n\in
Z_{1}}\Lambda^{1}_{n}}\phi_{\text{F}}(\nu_{A_{3}^{2}})+\phi(\nu_{A_{3}^{2}})\,d\mathcal{H}^{1}\\\
&\geq-\sum_{n\in
Z_{1}}2\int_{\Lambda^{1}_{n}}\phi_{\text{F}}(\nu_{U^{1}_{n}})+\phi(\nu_{U^{1}_{n}})\,d\mathcal{H}^{1}\\\
&\geq-\sum_{n\in
Z_{1}}{20}\,c_{2}\text{diam}(U^{1}_{n})\geq-\frac{{40}}{j^{\prime}+1}c_{2}\widetilde{\eta},\end{split}$
(5.23)

where in the first inequality we used the non-negativeness of
$\phi_{\mathrm{F}},\phi_{\mathrm{FS}}$ and $\phi$, in the second inequality we
used the subadditivity of measures, in the third inequality we used (3.15) and
(5.18)–( Case 3: ), and in the last inequality we used (5.14).

Moreover, by construction we have that

$\Lambda_{1}:=\left(\gamma^{1}\setminus\left(\bigcup_{n\in
Z_{1}}U_{n}^{1}\right)\right)\cup\bigcup_{n\in
Z_{1}}\Lambda^{1}_{n}\subset\partial A^{3}_{1}$ (5.24)

is closed, connected, and joins $T_{1}$ with $T_{2}$.

We now modify the pair ${(A^{3}_{1},S^{3}_{1})}$ and $\Lambda_{1}$ in the
strip $R^{2}$, first by employing the same construction of (5.15) and (5.16)
to obtain a configuration ${(A^{2}_{2},S^{2}_{2})}\in\widetilde{\mathcal{B}}$,
and then by employing the same construction of (5.21) and (5.22) to modify the
pair ${(A^{2}_{2},S^{2}_{2})}$, by denoting the final modified pair with
${(A^{3}_{2},S^{3}_{2})}\in\widetilde{\mathcal{B}}$. We then define

$\Lambda_{2}:=\left(\Lambda_{1}\setminus\left(\bigcup_{n\in
Z_{2}}U_{n}^{2}\right)\right)\cup\bigcup_{n\in
Z_{2}}\Lambda^{2}_{n}\subset\partial A^{3}_{2}.$ (5.25)

By iterating the same procedure on the strips $R^{i}$ for
$i=3,\ldots,j^{\prime}+1$ we obtain the pair
$(A^{2},S^{2}):=(A^{3}_{j^{\prime}+1},S^{3}_{j^{\prime}+1})\in\widetilde{\mathcal{B}}$
and we define $\Lambda_{j^{\prime}+1}$ as in (5.25) by replacing all the index
1 and 2 with $j^{\prime}$ and $j^{\prime}+1$, respectively. We observe that by
[25, Lemma 3.12] there exists a support
$\widetilde{\Lambda}\subset\Lambda_{j^{\prime}+1}$ of a curve joining $T_{1}$
with $T_{2}$.

Furthermore, as done in (5.23) for i=1,

${\mathcal{S}}_{L}\left(A_{i}^{2},S_{i}^{2},R^{i}\right)-{\mathcal{S}}_{L}\left(A_{i}^{3},S_{i}^{3},R^{i}\right)\geq-\frac{{40}}{j^{\prime}+1}c_{2}\widetilde{\eta}$
(5.26)

for every $i=1,\ldots,j^{\prime}+1$. Therefore, by iteration it follows that

$\begin{split}{\mathcal{S}}_{L}\left(A^{1},S_{h^{1},K^{1}},R\right)-{\mathcal{S}}_{L}\left(A^{2},S^{2},R\right)&\geq\sum_{i=1}^{j^{\prime}+1}\left({\mathcal{S}}_{L}\left(A^{1},S_{h^{1},K^{1}},R^{i}\right){-{\mathcal{S}}_{L}\left(A_{i}^{2},S_{i}^{2},R^{i}\right)}\right.\\\
&\qquad\left.{+{\mathcal{S}}_{L}\left(A_{i}^{2},S_{i}^{2},R^{i}\right)-{\mathcal{S}}_{L}\left(A_{i}^{3},S_{i}^{3},R^{i}\right)}\right)\\\
&\geq-\sum_{i=1}^{j^{\prime}+1}\frac{{88}}{j^{\prime}+1}c_{2}\widetilde{\eta}\\\
&\geq-{88}c_{2}\widetilde{\eta},\end{split}$ (5.27)

where in the second inequality we used (5.17) and (5.26). We notice that
$\Gamma^{\text{A}}_{\text{FS}}(A^{2},S^{2})\cap\partial{\rm{Int}}(\overline{S^{2}})\cap
R$ is a countable union of connected sets because by construction every
connected component of
$\Gamma^{\text{A}}_{\text{FS}}(A^{2},S^{2})\cap\partial{\rm{Int}}(\overline{S^{2}})\cap
R$ is connected to an element in the family of sets $\\{\partial U_{n}\cap
U_{n}\\}_{n\in\mathbb{N}}$. Therefore,
$(\Gamma^{\text{A}}_{\text{FS}}(A^{2},S^{2})\setminus{\rm{Int}}(\overline{S^{2}}))\cap
R$ is equal to a countable union of connected sets. More precisely, there
exists a family of connected and disjoint sets
$\\{\widetilde{\Gamma}_{i}\\}_{i\in\mathbb{N}}$ such that

$(\Gamma_{\text{FS}}(A^{2},S^{2})\setminus{\rm{Int}}(\overline{S^{2}}))\cap
R=\bigcup_{i\in\mathbb{N}}\widetilde{\Gamma}_{i},$ (5.28)

and hence,

$\mathcal{H}^{1}\left((\Gamma_{\text{FS}}(A^{2},S^{2})\setminus{\rm{Int}}(\overline{S^{2}})\cap
R\right)=\sum_{i\in\mathbb{N}}\mathcal{H}^{1}\left(\widetilde{\Gamma}_{i}\right).$
(5.29)

We conclude this step by observing that Assertion (i) follows by the
construction of $S^{2}$, while Assertion (ii) holds with the defined set
$\widetilde{\Lambda}$.

Step 3 (From countable to a finite number of components). Since
$\mathcal{H}^{1}((\Gamma_{\text{FS}}(A^{2},S^{2})\setminus{\rm{Int}}(\overline{S^{2}}))\cap
R)\leq\mathcal{H}^{1}(\partial S^{2})<\infty$, by (5.29) there exists
$i_{0}:=i_{0}(\widetilde{\eta})\in\mathbb{N}$ such that

$\sum_{i=i_{0}+1}^{\infty}\mathcal{H}^{1}(\widetilde{\Gamma}_{i})\leq\widetilde{\eta}.$
(5.30)

Notice that $(\widetilde{A},\widetilde{S})\in\widetilde{\mathcal{B}}$, where
$\widetilde{S}:=S^{2}$ and
$\widetilde{A}:=A^{2}\setminus\bigcup_{i\in\widetilde{I}}\widetilde{\Gamma}_{i}$
with $\widetilde{I}:=\\{i\in\mathbb{N}:i\geq i_{0}+1\\}$. Furthermore, it
follows from Steps 1 and 2 that

$\begin{split}{\mathcal{S}}_{L}(A,S_{h,K},R)-{\mathcal{S}}_{L}(\widetilde{A},\widetilde{S},R)&={\mathcal{S}}_{L}(A,S_{h,K},R)\pm{\mathcal{S}}_{L}(A^{2},S^{2},R)-{\mathcal{S}}_{L}(\widetilde{A},\widetilde{S},R)\\\
&\geq-{88}c_{2}\widetilde{\eta}-2\sum_{i=i_{0}+1}^{\infty}\int_{\widetilde{\Gamma}_{i}}\phi_{\text{F}}(\nu_{\widetilde{\Gamma}_{i}})+\phi(\nu_{\widetilde{\Gamma}_{i}})\,d\mathcal{H}^{1}\\\
&\geq-{92}c_{2}\widetilde{\eta},\end{split}$ (5.31)

where in the first inequality we used (5.9) and (5.27), and the definition of
$\widetilde{A}$ and the non-negativeness of $\phi_{\text{FS}}$ and, in the
second inequality we used (3.15) and (5.30). We conclude this step by defining
$M\in\mathbb{N}\cup\\{0\\}$ as the cardinality of
$\mathbb{N}\setminus\widetilde{I}$, and we notice by construction that
$(\Gamma^{\text{A}}_{\mathrm{FS}}(\widetilde{A},\widetilde{S})\setminus{\rm{Int}}(\overline{\widetilde{S}}))\cap
R$ has at most $M$-connected components.

Finally, we observe that Assertion (i) is a direct consequence of the
construction in Steps 1 and 2, while Assertion (ii) follows from the
definition of $\widetilde{\Lambda}$ in Step 2 (which is not modified in Step 3
since
$\widetilde{\Lambda}\cap(\Gamma^{\text{A}}_{\text{FS}}(A^{2},S^{2})\setminus{\rm{Int}}(\overline{S^{2}}))=\emptyset$).
The proof of this lemma is concluded by taking
$\widetilde{\eta}:=\frac{\eta}{{92}c_{2}}$ in (5.31) with
$\eta\in(0,\min\\{1,{92}c_{2}\\})$. ∎

We formalize below the notions of _film islands_ , _composite voids_ , and
_substrate grains_ for any admissible pair $(A,S_{h,K})\in\mathcal{B}$.

######  Definition 5.6.

Let $R\subset\Omega$ be an open rectangle and let $(A,S)\in\mathcal{B}$. We
refer to:

  * •

any closed component $V{\subset\overline{R}}$ of
$\Omega\setminus{\rm{Int}}(A)$ such that $\partial
V\cap(\Gamma_{\mathrm{S}}(A,S)\cup\Gamma^{\text{D}}_{\mathrm{FS}}(A,S))$ is
not empty and it consists in one and only one connected component as an
_extended_ (as we also include “connected delamination regions”) _composite
void_ of the configuration $(A,S)$ (or sometime for simplicity of the film
region or of $A$).

  * •

any open connected component $P{\subset R}$ of ${\rm
Int}(A\setminus\overline{S})$ such that $\partial
P\cap\Gamma^{\text{A}}_{\mathrm{FS}}(A,S)$ is not empty and it consists in one
and only one connected component as a _film island_ of the configuration
$(A,S)$ (or sometime for simplicity of the film region or of $A$), and to a
film island $P={\rm Int}(A\setminus\overline{S})\cap R$ of $(A,S)$ as the
_full island_ of $A$.

  * •

any open connected component $G{\subset R}$ of ${\rm Int}(S)$ such that
$\partial G\cap\Gamma^{\text{A}}_{\mathrm{FS}}(A,S)$ is not empty and it
consists in one and only one connected component as a _substrate grain_ of the
configuration $(A,S)$ (or sometime for simplicity of the substrate region or
of $S$), and to a substrate grain $G={\rm Int}(S)\cap R$ of $(A,S)$ as the
_full grain_ of $S$.

The following results can be seen as analogous of [36, Lemmas 4.4 and 4.5]
though in our more involved setting of three free interfaces (see Table 1),
where we have to distinguish among the blow-ups at:

  * -

the substrate free boundary (Lemma 5.7),

  * -

the film-substrate incoherent (delaminated) interface (Lemma 5.8),

  * -

the substrate cracks in the film-substrate incoherent interface (Lemma 5.9),

  * -

the filaments of both the substrate and the film (Lemma 5.10),

  * -

the substrate filaments on the film free boundary (Lemma 5.11),

  * -

the delaminated substrate filaments in the film (Lemma 5.12).

The strategy employed in these proofs is based on reducing to the situation of
a finite number of connected components for the film-substrate coherent
interfaces by Lemma 5.5 and then on designing induction arguments (with
respect to the index of such components) in which we “shrink” islands, “fill”
voids, and modify “grains” in new voids (see Figures 3, 4, and 6,
respectively) by means of the _minimality of segments_ (see [43, Remark
20.3]).

We begin by addressing the setting of the substrate free boundary.

######  Lemma 5.7.