# Solutions for a free-boundary problem modeling multilayer films 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.

In this paper, we move forward from the results of [42] by introducing a
variational model for the study of multilayer films that allows for the
treatment of both coherent and incoherent interfaces between layers. 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 energy effects. The surface of each film layer is assumed to satisfy
the “exterior graph condition” already introduced in [42], for which bulk
cracks are allowed to be of non-graph type. By applying the direct method of
calculus of variations under a constraint on the number of connected
components of the cracks not connected to the surface of the film layers the
existence of energy minimizers is established in dimension 2. As a byproduct
of the analysis the state of art on the variational modeling of single-layered
films deposited on a fixed substrate is advanced by letting the substrate
surface free, by addressing the presence of multiple layers of various
materials, and by including the possibility of delamination between the
various film layers.

###### Key words and phrases:

free boundary problem, surface energy, delamination, elastic energy,
deformable layers, multiphase morphology, multilayer films

###### 2010 Mathematics Subject Classification:

49J10, 49Q15, 49Q20, 35R35 , 74A45, 74A50 , 74G65

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. 3 Mathematical setting and main results
  4. 4 Single-layer films with delamination
  5. 5 Multilayered films

##  1\. Introduction

In this manuscript, we address the problem of modeling the morphology of
multilayered film composites consisting of different crystalline materials
deposited on a substrate. The goal is to advance the literature on the
variational modeling of single-layered films deposited on a fixed substrate
[14, 20, 21, 26] in a twofold direction: on the one hand, by letting the
substrate surface free and by addressing the presence of multiple layers of
various materials, and, on the other hand, by including into the analysis the
possibility of a failure of the film coatings, since, as described in [52] for
the case of some oxide films, the compressive stresses generated during film
growth can lead to the delamination (and the buckling) between different
layers.

Nowadays film-based nanostructures find several applications, in particular
for the manufacturing of electronic and photonic devices, such as for the
creation of their semiconductor components, and of solar and photovoltaic
cells. The great interest that films and, in particular, multilayer films [50,
53, 54], created by vapor deposition of different material constituents,
continue attracting is due to the fact that, as they are self-assembled
heterostructures, their employment represents one of the nanostructure design
methods with most feasibility potential; therefore, any advancement in the
mathematical modeling of film and multilayer film materials can have an
important practical impact for their design control. Examples of multilayer
films that are used for optoelectronic applications are multiple quantum well
structures with alternating compressive and tensile strained layers, and
short-period _quantum-dot superlattices_. Also for the latter, as described in
[53, 54], it is really the superposition of various layers of materials that
allows to reach the highest degree order needed for the applications with
respect to the size, the density and the distribution of the quantum dots.

The adopted strategy consists in combining the implementation to the
multiphase setting of the film models considered in [14, 20, 21, 26], in which
delamination is not taken into account, with the recent results for a two-
phase setting of [42], in which the interfaces between phases are instead
allowed to present both coherent and incoherent portions. With coherency here
we intend a microscopic organization of atoms that can be regarded as a
(possibly deformed) uniform lattice that is homogeneous through the interface,
while with incoherency we refer to the presence of debonding and delamination
at the interface. In this way the extension of the single-layer literature to
the multilayer setting (with possible delamination at each layer interface) is
performed within the theory of stress driven rearrangement instabilities
(SDRI) [6, 19, 31, 51], which was also at the basis of the variational single-
layer models introduced in [48, 49] and analytically validated in [14, 20, 21,
26]. In fact, as in [48, 49] for thin films, and more generally for free
crystals in [35, 36, 37], we model the mismatch between the free-standing
equilibrium lattices of the materials of each pair of film layers and of the
first layer with the substrate by means of the so-called _mismatch strain_ in
the elastic energy. As described by the SDRI theory the lattice mismatch is
responsible for the migration of the atoms of each phase from their
crystalline order, since the lattice mismatch induces large stresses in the
bulk material and, in order to release the related elastic energy, the atoms
move forming corrugations, cracks, and other interface instabilities [6, 31,
19, 51].

In regard to the literature results for settings with phase interfaces
exclusively assumed to be coherent, we refer to the literature on the optimal
shape of partitions in the absence of elastic effects, which was initiated by
Almgren in [1], who formulated the problem in $\mathbb{R}^{d}$, for $d>1$, for
surface tensions proportional at each interface. By working in the framework
of _integral currents_ of geometric measure theory he singled out a condition
referred to as “partitioning regularity”, that ensures 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
expanded the scope in [2, 3] by including also non-proportional surface
tensions and by introducing an integral condition called $BV$-_ellipticity_ ,
which they proved to be both sufficient and necessary for the $L^{1}$-lower
semicontinuity. Afterwards, various other conditions have been introduced and
studied, such as $B$-_convexity_ and _joint convexity_ , in the attempt of
finding a more practical condition than $BV$-_ellipticity_ , as the latter can
be challenging to be verified as it represents the analogous of Morrey’s
_quasi-convexity_ condition in the setting of _Caccioppoli partitions_.
$BV$-ellipticity though remains the only known condition characterizing the
the $L^{1}$-lower semicontinuity apart from specific contexts (we refer to
[11, 45] for more details), and the fact that it coincides with the _triangle-
inequality condition_ , which is simpler to check, for the case with 3 phases
[2, 3]. Finally, in [30] the analogous version of the $BV$-ellipticity
condition in the framework of $BD$-_spaces_ has been studied.

Instead, in regard to the settings with only incoherent interfaces, we refer
to the results obtained with respect to the related _Mumford-Shah_ problem for
also the application to _image segmentation_ , which was actually originally
introduced in [46] as a multiphase formulation. In this context, interfaces
represent the contours of the image color areas that can be characterized as
the discontinuity set of an auxiliary state function. We refer to [4, 18] for
existence and _Ahlfors-type regularity_ results in the context of a single
phase, which has been then extended also to the _Griffith model_ in fracture
mechanics in the context of linear elasticity with respect to vectorial state
functions representing the bulk displacement of crystalline materials [15,
28]. Finally, Bucur, Fragalà, and Giacomini addressed the original multiphase
setting of [46] in [8] and [9] by providing a rigorous mathematical
formulation with incoherent interfaces (see also [17] for a related multiphase
boundary problem for reaction-diffusion systems).

In [8] they recover Ahlfors-type regularity results for an _ad hoc_
nonstandard notion of multiphase local almost-quasi minimizers for an energy
accounting for the incoherent portions of each interface and disregarding the
contribution of the remaining coherent portions. Afterwards, in [9] the same
Authors introduced what they refer to as the _multiphase Mumford-Shah problem_
, that is characterized by the sum of possibly different Mumford-Shah-type
energy contributions, each related to a different phase, to which an extra
term (justified on statistical reasons) is added. Such extra term is needed as
otherwise minimizing configurations would present a single phase. However, in
[9] coherent interfaces are not counted in the energy as “no-jump interface
portions” along the reduced phase boundary are weighted in each phase energy
in the same way as the jump portions.

To include in our model the interplay between coherency and incoherency, by
allowing each phase interface to present also both coherent and incoherent
portions, we adopt the strategy initiated in [42] for the setting with a film
phase deposited on a substrate. Since the results in [42] regards $d=2$ and
were achieved under a so-called _exterior graph constraint_ on the substrate
surface, in order to implement those results to multiple film phases, we also
restrict to $d=2$ and we assume on both the substrate surface and the film
profiles the exterior graph constraint. We notice that even in the presence of
such condition internal cracks in each film layer and in the substrate are
allowed to be also of non-graph type.

We denote by $\Omega:=(-l,l)\times(-L,+\infty)$ for positive parameters
$l,L\in\mathbb{R}$ the region where the multilayer film and the substrate are
located, and, given $\alpha\in\mathbb{N}$, we denote a multilayered film
composite with $\alpha$ layers on top of the substrate phase $S_{0}$, which is
also denoted in the following as the _0th layer_ , by $S_{\alpha}$.
Furthermore, for each $j\in\\{0,\dots,\alpha\\}$ we assume that the profile of
each _$j$ th layer_ is parametrizable by a _height function_
$h^{j}:[-l,l]\mapsto[-L,+\infty)$ measuring the thickness of the profile of
_$j$ th composite_ $S_{j}$, i.e., the $j$-layered film composite including all
$i$th layers for $i\in\\{0,\dots,j\\}$, by assuming that $h^{j-1}\leq h^{j}$
for $j\in\\{1,\dots,\alpha\\}$. We also denote by
$K^{j}\subset\overline{{\rm{Int}}(S_{h^{j}})}$, where $S_{h^{j}}$ is the
subgraph of $h^{j}$, the _cracks of the $j$th composite_, which are assumed
such that $K^{j}\cap{\rm{Int}}(S_{h^{j-1}})\subset K^{j-1}$, so that then for
$j\in\\{1,\dots,\alpha\\}$ the _$j$ -composite_ $S_{j}$ coincides with

$S_{h^{j},K^{j}}:=S_{h^{j}}\setminus K^{j},$

and the $j$th film layer coincides with $S_{h^{j},K^{j}}\setminus
S^{(1)}_{h^{j-1},K^{j-1}}$ (see Figure 1). It follows that there is no formal
distinction in the hypotheses taken on the substrate phase $S_{0}$ and the one
taken on each $j$th film composite $S_{h^{j},K^{j}}$ (apart from the fact of
being contained in all of them).

In particular, by writing that $(h^{j},K^{j})\in\text{AHK}(\Omega)$ we assume
that each $j$th layer height function $h^{j}$ is an upper semicontinuous
function with bounded pointwise variation and each $j$th composite crack set
$K^{j}$ is a closed $\mathcal{H}^{1}$-rectifiable set with finite
$\mathcal{H}^{1}$ measure. More precisely, we denote the family
$\mathcal{B}^{\alpha}$ of admissible multilayered film composites in $\Omega$
with $\alpha$ layers (on the substrate layer), as a $(\alpha+1)$-tuple of all
the $j$th composites $S_{h^{j},K^{j}}$ for $j\in\\{0,\dots,\alpha\\}$, namely

$\displaystyle\mathcal{B}^{\alpha}:=\\{(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}}):$
$(h^{j},K^{j})\in\text{AHK}(\Omega)$, $h^{j-1}\leq h^{j}$,
$K^{j}\subset\overline{{\rm{Int}}(S_{h^{j}})}$, $\displaystyle\quad\,\,\text{
$K^{j}\cap{\rm{Int}}(S_{h^{j-1}})\subset K^{j-1}$ for
$j\in\\{1,\dots,\alpha\\}$}\\}.$

Furthermore, by following the SDRI theory [6, 19, 31, 48, 51] and in the
analogy with the single-layer film setting [14, 20, 21, 26, 42], we define the
family of _admissible configurations_ $\mathcal{C}^{\alpha}$ by

$\mathcal{C}^{\alpha}:=\\{(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u):(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})\in\mathcal{B}^{\alpha},\,u\in\mathrm{H}^{1}_{\mathrm{loc}}({\rm{Int}}(S_{h^{\alpha},K^{\alpha}}))\\},$

where the functions $u$ represent the _bulk displacements_ in the multilayered
film composites, and we consider a configurational energy
$\mathcal{F}_{\sigma}^{\alpha}:\mathcal{C}\to[-\infty,\infty]$ given by the
sum of an elastic energy $\mathcal{W}$ and a surface energy
$\mathcal{S}_{\sigma}^{\alpha}$, namely,

$\mathcal{F}_{\sigma}^{\alpha}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u):=\mathcal{W}({S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},}u)+\mathcal{S}^{\alpha}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})$

for any
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}^{\alpha}$.
The parameter $\sigma\in\\{s,t\\}$ is introduced to differentiate the setting
in which each layer is assumed to interact with all the other $\alpha$ layers
of the composite for $\sigma=t$ from the setting in which each $j$th layer
interacts only with the two layers surrounding it (if present), i.e., with the
$(j+1)$th layer located directly above and the $(j-1)$th layer directly
underneath, for $\sigma=s$. Let us refer in the following to
$\mathcal{F}_{\sigma}^{\alpha}$ (and to $\mathcal{S}_{\sigma}^{\alpha}$) as
the _$\alpha$ -layered total and sequential (surface) energy_ for $\sigma=t$
and $\sigma=s$, respectively.

The elastic energy $\mathcal{W}$ is defined in $\mathcal{C}^{\alpha}$ by

$\mathcal{W}({S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},}u):=\int_{S_{h^{\alpha},K^{\alpha}}}W(x,Eu(x)-{E_{0}^{\alpha}}(x))\,dx,$

where the elastic density $W$ denotes the quadratic form

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

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

${E_{0}^{\alpha}}:=\begin{cases}E(u_{0}^{\alpha})&\text{in }\Omega\setminus
S_{h^{\alpha-1}},\\\ E(u_{0}^{i})&\text{in }{\rm{Int}}(S_{h^{i}})\setminus
S_{h^{1-1}}\text{ for $i=1,\ldots,\alpha-1$}\\\ 0&\text{in
}{\rm{Int}}(S_{h^{0},K^{0}}),\end{cases}$

with respect to fixed $\alpha$ functions $u_{0}^{i}\in
H^{1}(\Omega;\mathbb{R}^{2})$ for $i\in\\{1,\dots,\alpha\\}$.

Both the surface energies $\mathcal{S}_{\sigma}^{\alpha}$ for $\sigma=s,t$ are
given as sums of pairwise contributions
$\mathcal{S}^{(i,j)}:\mathcal{B}^{1}\to[0,\infty]$ for $0\leq i<j\leq\alpha$
defined by

$\mathcal{S}^{(i,j)}(S_{h^{j},K^{j}},S_{h^{i},K^{i}}):=\int_{\partial
S_{h^{i},K^{i}}\cup\partial
S_{h^{j},K^{j}}}{\psi_{i,j}(z,\nu)\,d\mathcal{H}^{1}},$ (1.1)

for every admissible multilayered composite
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})\in\mathcal{B}^{\alpha}$,
where $\psi_{i,j}$ denotes the anisotropic surface tension that takes
different definition with respect to the various portions of $\partial
S_{h^{i},K^{i}}\cup\partial S_{h^{j},K^{j}}$. More precisely, in order to
properly define $\psi_{i,j}$ we consider the three surface tensions
$\varphi_{i},\,\varphi_{j},\,\varphi_{ij}:\overline{\Omega}\times\mathbb{R}^{2}\to[0,\infty]$
characterizing the vapor-$i$th layer interface, the vapor-$j$ layer interface,
and the $i$th layer-$j$ layer interface. Moreover, in order to address both
the wetting and dewetting regimes with respect to the materials of each pair
of film layers, we introduce two additional surface tensions for each pair
$(S_{h^{j},K^{j}},S_{h^{i},K^{i}})$, denoted as the _$i,j$ regime surface
tensions_, which are defined as follows:

$\varphi^{1}_{ij}:=\min\\{\varphi_{i},\varphi_{j}+\varphi_{ij}\\}\qquad\text{and}\qquad\varphi^{2}_{ij}:=\min\\{\varphi_{i},\varphi_{j}\\},$

in analogy to the definitions given in [42] for two-phase setting. It follows
that

$\psi_{i,j}(x,\nu(x)):=\begin{cases}\varphi_{j}(x,\nu_{S_{h^{j},K^{j}}}(x))&x\in\Omega\cap(\partial^{*}S_{h^{j},K^{j}}\setminus\partial^{*}S_{h^{i},K^{i}})\\\
\varphi^{1}_{ij}(x,\nu_{S_{h^{j},K^{j}}}(x))&x\in\Omega\cap\partial^{*}S_{h^{i},K^{i}}\cap\partial^{*}S_{h^{j},K^{j}},\\\
\varphi_{ij}(x,\nu_{S_{h^{i},K^{i}}}\left(x\right))&x\in\Omega\cap(\partial^{*}S_{h^{i},K^{i}}\setminus\partial
S_{h^{j},K^{j}})\\\
(\varphi_{j}+\varphi^{1}_{ij})(x,\nu_{S_{h^{j},K^{j}}}(x))&x\in\Omega\cap\partial^{*}S_{h^{i},K^{i}}\cap\partial
S_{h^{j},K^{j}}\cap S_{h^{j},K^{j}}^{(1)},\\\
2\varphi_{j}(x,\nu_{S_{h^{j},K^{j}}}\left(x\right))&x\in\Omega\cap\partial
S_{h^{j},K^{j}}\cap S_{h^{j},K^{j}}^{(1)}\cap S_{h^{i},K^{i}}^{(0)},\\\
2\varphi^{2}_{ij}(x,\nu_{S_{h^{j},K^{j}}}\left(x\right))&x\in\Omega\cap\partial{S_{h^{j},K^{j}}}\cap{S_{h^{j},K^{j}}}^{(0)},\\\
2\varphi_{ij}(x,\nu_{S_{h^{i},K^{i}}}\left(x\right))&\begin{aligned}
x\in\Omega&\cap(\partial S_{h^{i},K^{i}}\setminus\partial S_{h^{j},K^{j}})\\\
&\cap\left(S_{h^{i},K^{i}}^{(1)}\cup S_{h^{i},K^{i}}^{(0)}\right)\cap
S_{h^{j},K^{j}}^{(1)},\end{aligned}\\\
\varphi^{1}_{ij}(x,\nu_{S_{h^{j},K^{j}}}(x))&x\in\Omega\cap\partial
S_{h^{i},K^{i}}\cap\partial S_{h^{j},K^{j}}\cap
S_{h^{i},K^{i}}^{(1)},\end{cases}$

where, given a set $U\subset\mathbb{R}^{2}$, $\nu_{U}$, $\partial^{*}U$, and
$U^{(\alpha)}$ denote, when well defined, the outward pointing normal to
$\partial U$, the _reduced boundary_ , and the set of points of density
$\alpha\in[0,1]$, respectively. Notice that if $\alpha=1$ the energy
$\mathcal{S}^{(0,1)}$ coincides with the surface energy defined in [42] as, by
following the notation of [42] we have that
$\varphi_{0}:=\varphi_{\mathrm{S}},\varphi_{1}:=\varphi_{\mathrm{F}},\varphi_{01}:=\varphi_{\mathrm{FS}}$
and as a consequence $\varphi^{1}_{01}=\varphi$ and
$\varphi^{2}_{0,1}=\varphi^{\prime}$. More precisely, the _$\alpha$ -layered
surface energies_
$\mathcal{S}_{\sigma}^{\alpha}:\mathcal{B}^{\alpha}\to[-\infty,\infty]$ are
given for $\sigma=s,t$ by

$\mathcal{S}^{\alpha}_{s}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}}):=\sum_{j=1}^{\alpha}\mathcal{S}^{(j-1,j)}(S_{h^{j},K^{j}},S_{h^{j-1},K^{j-1}}),$

and

$\mathcal{S}_{t}^{\alpha}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}}):=\sum_{j=1}^{\alpha}\sum_{i=0}^{j-1}\mathcal{S}^{(i,j)}(S_{h^{j},K^{j}},S_{h^{i},K^{i}}),$

respectively. Notice that we also address the more general case in which the
surface density $\psi_{i,j}$ in (1.1) are not interconnected for different
pairs of indexes $0\leq i<j\leq\alpha$ (see Remarks 3.7 and 3.11 for more
details).

It was observed in [35, 42] that the family $\mathcal{B}^{1}$ lacks
compactness with respect to the signed distance convergence. In order to
overcome this issue and being able to apply Gołab’s’s Theorem [31] to recover
compactness, we impose a constraint $m_{j}\in\mathbb{N}$ on the number of
connected components of the cracks of the $j$th composite that are not
connected to the $j$th layer for $j=0,\dots,\alpha$. Therefore, we restrict to
the family of configurations
$\mathcal{C}^{\alpha}_{\bm{m}}\subset\mathcal{C}^{\alpha}$ for which such
constraints hold, where
$\textbf{m}:=(m_{0},\ldots,m_{\alpha})\in\mathbb{N}^{\alpha+1}$.

Figure 1. A multilayered film composite with 2 layers (on the substrate 0th
layer $S_{h^{0},K^{0}}$) associated to an admissible configuration
$(S_{h^{2},K^{2}},S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)\in\mathcal{C}^{2}_{\textbf{m}}$
(see Definition 3.1) is represented by indicating each $j$th layer with a gray
color with decreasing value with respect to the increasing order of the index
$j=0,1,2$. Furthermore, the $j$th layer is indicated with a thinner line with
respect to the increasing order of the index $j=0,1,2$, and for the $0$th and
1st layer we distinguish between their coherent and incoherent portions by
using a dashed or a continuous line, respectively.

The main goal of the paper is to prove that, given $\alpha\in\mathbb{N}$,
$\sigma\in\\{s,t\\}$, and a family of area constraints
$\\{\mathbbm{v}_{j}\\}_{i=0}^{\alpha}\subset[\mathcal{L}^{2}(\Omega)/2,\mathcal{L}^{2}(\Omega)]$
for each $j$th composite, the minimum problem

$\inf_{\tiny\begin{aligned}
&(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}^{\alpha}_{\mathbf{m}},\\\
&\vspace{-0.01cm}\mathcal{L}^{2}({S_{h^{j},K^{j}}})=\mathbbm{v}_{j},\,\text{for
}j=0,\ldots,\alpha\end{aligned}}{\mathcal{F}_{\sigma}^{\alpha}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)}.$
(1.2)

admits a solution.

To do that we employ the _Direct Method_ of Calculus of Variations that
consists in finding a proper topology $\tau_{\mathcal{C}^{\alpha}}$ weak
enough to prove compactness in
$\mathcal{C}^{\alpha}_{\mathbf{m}}\subset\mathcal{C}^{\alpha}$ and strong
enough to have lower semicontinuity of $\mathcal{F}^{\alpha}$ in
$\mathcal{C}^{\alpha}_{\mathbf{m}}$. The topology
$\tau_{\mathcal{C}^{\alpha}}$ that we consider is the one for which the
convergence

$(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})\xrightarrow[k\to\infty]{\tau_{\mathcal{C}^{\alpha}}}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)$

is equivalent to

$\displaystyle\begin{cases}\text{for every $i=0,\ldots,\alpha$,
$\sup_{k\in\mathbb{N}}{\mathcal{H}^{1}\left(\partial
S_{h^{i}_{k},K^{i}_{k}}\right)}<\infty$,}\\\
\text{$\mathrm{sdist}\left(\cdot,\partial
S_{h^{i}_{k},K^{i}_{k}}\right)\xrightarrow[k\to\infty]{}\mathrm{sdist}\left(\cdot,\partial
S_{h^{i},K^{i}}\right)$ locally uniformly in $\mathbb{R}^{2}$ and}\\\
u_{k}\xrightarrow[k\to\infty]{}u\quad\text{a.e.\ in
$\mathrm{{Int}\left(S_{h^{\alpha},K^{\alpha}}\right)}$,}\end{cases}$

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

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

For the compactness we implement in the multilayer setting the compactness
results proven for the substrate in [42], which were based on [14, 20, 21, 26]
(with the difference that instead of lower semicontinuous graph we assume an
upper semicontinuity property). We notice that in order to include incoherency
in the setting of [14, 20, 21, 26] we implement for multilayers the setting of
[42], where in the elements in the $(\alpha+1)$-tuple are not each film
layers, but the $j$th composites. In particular, this allows to include in the
model also the possible presence of a countable island of one material onto
the other layers. In order to establish the lower semicontinuity property we
instead proceed by induction by directly using the lower semicontinuity result
of [42] for the basis of the induction.

We conclude by describing the organization of the paper. In Section 2 we state
the notation and recall fundamental definitions used throughout the paper. In
Section 3 we introduce the model and the main results of the paper. In Section
4 we prove the existence of minimizers for single-layer films with
delamination. Finally, in Section 5 we prove the existence result for the
minimum problem (1.2) with a finite number $\alpha$ of layers over the
substrate 0th layer.

##  2\. Notation

In this section, we collect the relevant notation used throughout the paper.

### 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
$\mathbb{M}^{2}_{\rm sym}$ 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 $\mathbb{R}^{2}$. For any set
$E\subset\mathbb{R}^{2}$, we denote by $E^{c}$ the complement of $E$ and by
${\rm{Int}}(E)$, $\overline{E}$ and $\partial E$ interior, the closure and the
topological boundary of $E$, respectively.

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 $f\in L^{1}_{{\rm
loc}}(\mathbb{R}^{2})$ 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, Definition 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 $x\in\partial E$.

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

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

### 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

$h^{+}(a):=\lim_{x\to a^{-}}h(x)\quad\text{and}\quad h^{-}(b):=\lim_{x\to
b^{-}}h(x)$ (2.4)

are finite.

##  3\. Mathematical setting and main results

###  3.1. Multilayer model

In this section, we introduce the family of admissible regions with finite
number of composite layers and the respective family of admissible
configurations. Let $\Omega:=(-l,l)\times(-L,{\infty})\subset\mathbb{R}^{2}$
for positive parameters $l,L\in\mathbb{R}$.

Analogously to [42], we assume a _graph-crack_ constraint of the composite of
layers, in other words, we consider a _graph constraint_ on the strict
epigraph of the composite of layers, while inside of it, we consider _cracks_
as closed and $\mathcal{H}^{1}$-rectifiable sets of $\Omega$, roughly
speaking, the profile of the composite is given by a function representing its
thickness, plus a countable number of external vertical filaments and internal
cracks. More precisely, we consider the _family of admissible heights_ $\text{
AH}(\Omega)$ defined by

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

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

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

Furthermore, we define the _family of admissible cracks_ $\text{AK}(\Omega)$
by

$\text{ 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

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

Finally, given $(h,K)\in\text{AHK}(\Omega)$ we refer to the region
characterized as the subgraph of the height function $h$ without the internal
cracks of $K$, namely,

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

as the _(generalized) subgraph with height $h$ and cracks $K$_, and we define
the _family of admissible subgraphs_ as

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

We observe that for every $(h,K)\in\text{AHK}(\Omega)$

$\overline{S_{h,K}}=S_{h},\quad{\rm{Int}}(S_{h,K})={\rm{Int}}(S_{h})\setminus
K\quad\text{and}\quad\partial{S_{h,K}}=\partial{S_{h}}\cup K.$ (3.7)

We have that $\partial{S_{h}}$ is connected and, $\partial S_{h}$ and
$\partial S_{h,K}$ have finite $\mathcal{H}^{1}$-measure. By [24, Lemma 3.12
and Lemma 3.13], for any $h\in\text{AH}(\Omega)$, $\partial S_{h}$ 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, by applying
[36, Proposition A.1] $S_{h}$ and $S_{h,K}$ are sets of finite perimeter.

######  Definition 3.1 (Admissible multilayers and admissible configurations).

We define the family of two layers $\mathcal{B}^{1}$ by

$\mathcal{B}^{1}:=\\{(S_{h^{1},K^{1}},S_{h^{0},K^{0}}):\text{ for $i=0,1$
there exists
}(h^{i},K^{i})\in\mathrm{AHK}(\Omega),\,S_{h^{i},K^{i}}\in\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\mathrm{AS}(\Omega)\times\mathrm{AS}(\Omega).$

Let $\alpha\in\mathbb{N}$, we define the family of admissible
$(\alpha+1)$-layers $\mathcal{B}^{\alpha}$ by

$\mathcal{B}^{\alpha}:=\\{(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})\in[\mathrm{AS}(\Omega)]^{\alpha+1}:\,(S_{h^{i},K^{i}},S_{h^{i-1},K^{i-1}})\in\mathcal{B}^{1}\\\
\text{ for every $1\leq i\leq\alpha$}\\},$

where
$[\mathrm{AS}(\Omega)]^{\alpha+1}:=\mathrm{AS}(\Omega)\times\ldots\times\mathrm{AS}(\Omega)$
represents the $(\alpha+1)$ cartesian product of $\mathrm{AS}(\Omega)$.

We define the family of admissible configurations by

$\mathcal{C}^{\alpha}:=\\{(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u):(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})\in\mathcal{B}^{\alpha},\,u\in\mathrm{H}^{1}_{\mathrm{loc}}({\rm{Int}}(S_{h^{\alpha},K^{\alpha}}))\\}.$

######  Remark 3.2.

In view of Definition 3.1, it follows that every
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})\in\mathcal{B}^{\alpha}$
satisfy the following properties:

  * (i)

$h^{i}\leq h^{j}$ and $\partial
S_{h^{j},K^{j}}\cap{\rm{Int}}(S_{h^{i},K^{i}})=\emptyset$ for every $0\leq
i<j\leq\alpha$ and hence,
$(S_{h^{j},K^{j}},S_{h^{i},K^{i}})\in\mathcal{B}^{1}$ for every $0\leq
i<j\leq\alpha$;

  * (ii)

$K^{i}\subset\overline{{\rm{Int}}(S_{h^{i}})}\setminus{\rm{Int}}(S_{h^{i-1},K^{i-1}})$
for every $i=1,\ldots,\alpha$;

  * (iii)

$K^{i}\cap K^{i-2}\cap{\rm{Int}}(S_{h^{i-2}})\subset
K^{i}\cap{\rm{Int}}(S_{h^{i-1}})\subset\partial
S_{h^{i},K^{i}}\cap{\rm{Int}}(S_{h^{i-1}})\subset K^{i-1}$ for every
$i=2,\ldots,\alpha$.

For any $\alpha\in\mathbb{N}$, motivated in [35, 36, 42] we introduce a notion
of convergence for the families $\mathcal{B}^{\alpha}$ and
$\mathcal{C}^{\alpha}$.

######  Definition 3.3 ($\tau_{\mathcal{B}^{\alpha}}$-Convergence).

A sequence
$\\{(S_{h^{\alpha}_{n},K^{\alpha}_{n}},\ldots,S_{h^{0}_{n},K^{0}_{n}})\\}\subset\mathcal{B}^{\alpha}$
${\tau_{\mathcal{B}}^{\alpha}}$-converges to
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})\in\mathcal{B}^{\alpha}$ if

  * -

$\sup_{k\in\mathbb{N}}\mathcal{H}^{1}(\partial
S_{h^{i}_{k},K^{i}_{k}})<\infty$, for every $i=0,\ldots,\lambda$

  * -

$\text{sdist}\left(\cdot,\partial
S_{h^{i}_{k},K^{i}_{k}}\right)\rightarrow\text{sdist}\left(\cdot,\partial
S_{h^{i},K^{i}}\right)$ locally uniformly in $\mathbb{R}^{2}$ as
$k\rightarrow\infty$ for every $i=0,\ldots,\lambda$,

where

$S_{h^{i}_{k},K^{i}_{k}}:=S_{h^{i}_{k}}\setminus K^{i}_{k}\quad\text{and}\quad
S_{h^{i},K^{i}}:=S_{h^{i}}\setminus K^{i},$

for every $k\in\mathbb{N}$ and $i=0,\ldots,\alpha$.

######  Definition 3.4 ($\tau_{\mathcal{C}^{\alpha}}$-Convergence).

A sequence
$\\{(S_{h^{\alpha}_{n},K^{\alpha}_{n}},\ldots,S_{h^{0}_{n},K^{0}_{n}},u_{n})\\}\subset\mathcal{C}^{\alpha}$
${\tau_{\mathcal{C}^{\alpha}}}$-converges to
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}^{\alpha}$
if

  * -

$(S_{h^{\alpha}_{n},K^{\alpha}_{n}},\ldots,S_{h^{0}_{n},K^{0}_{n}})\xrightarrow{\tau_{\mathcal{B}^{\alpha}}}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})$,

  * -

$u_{n}\rightarrow u$ a.e. in ${{\rm{Int}}(S_{h^{\alpha},K^{\alpha}})}$.

Analogously to [35, 36, 42], we introduce a subfamily of
$\mathcal{B}^{\alpha}$ subject to a restriction on the number of connected
components of the boundary of each composite layer and the subfamily of
$\mathcal{C}^{\alpha}$ with the corresponding configurations.

######  Definition 3.5.

Let $\alpha\in\mathbb{N}$ and let
$\bm{m}:=(m_{0},\ldots,m_{\alpha})\in\mathbb{N}^{\alpha+1}$. We refer to

$\mathcal{B}^{\alpha}_{\bm{m}}:=\\{(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})\in\mathcal{B}^{\alpha}:\,\text{$\partial
S_{h^{i},K^{i}}$ has at most $m_{i}$-connected components}\\\ \text{for
$i=0,\ldots,\alpha$}\\}$

as the families of _admissible multilayers_ and to

$\mathcal{C}^{\alpha}_{\bm{m}}:=\\{(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}:\,(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})\in\mathcal{B}^{\alpha}_{\bm{m}}\\}$

as the family of _admissible configurations_.

In the sequel we fix $\alpha\in\mathbb{N}$. Motivated in the model introduced
in [42], we consider the surface tension between two layers
$\mathcal{S}^{(i,j)}:\mathcal{B}^{\alpha}\to[0,+\infty]$ in the family of
admissible layers $\mathcal{B}^{\alpha}$ by

$\mathcal{S}^{(i,j)}(S_{h^{j},K^{j}},S_{h^{i},K^{i}}):=\int_{\partial
S_{h^{i},K^{i}}\cup\partial
S_{h^{j},K^{j}}}{\psi_{i,j}(z,\nu)\,d\mathcal{H}^{1}},$

where
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})\in\mathcal{B}^{\alpha}$
for $0\leq i<j\leq\alpha$, and the surface tension $\psi_{i,j}$ is defined in
different portions of $\partial S_{h^{i},K^{i}}\cup\partial S_{h^{j},K^{j}}$,
more precisely,

$\psi_{i,j}(x,\nu(x)):=\begin{cases}\varphi_{j}(x,\nu_{S_{h^{j},K^{j}}}(x))&x\in\Omega\cap(\partial^{*}S_{h^{j},K^{j}}\setminus\partial^{*}S_{h^{i},K^{i}})\\\
\varphi^{1}_{ij}(x,\nu_{S_{h^{j},K^{j}}}(x))&x\in\Omega\cap\partial^{*}S_{h^{i},K^{i}}\cap\partial^{*}S_{h^{j},K^{j}},\\\
\varphi_{ij}(x,\nu_{S_{h^{i},K^{i}}}\left(x\right))&x\in\Omega\cap(\partial^{*}S_{h^{i},K^{i}}\setminus\partial
S_{h^{j},K^{j}})\\\
(\varphi_{j}+\varphi^{1}_{ij})(x,\nu_{S_{h^{j},K^{j}}}(x))&x\in\Omega\cap\partial^{*}S_{h^{i},K^{i}}\cap\partial
S_{h^{j},K^{j}}\cap S_{h^{j},K^{j}}^{(1)},\\\
2\varphi_{j}(x,\nu_{S_{h^{j},K^{j}}}\left(x\right))&x\in\Omega\cap\partial
S_{h^{j},K^{j}}\cap S_{h^{j},K^{j}}^{(1)}\cap S_{h^{i},K^{i}}^{(0)},\\\
2\varphi^{2}_{ij}(x,\nu_{S_{h^{j},K^{j}}}\left(x\right))&x\in\Omega\cap\partial{S_{h^{j},K^{j}}}\cap{S_{h^{j},K^{j}}}^{(0)},\\\
2\varphi_{ij}(x,\nu_{S_{h^{i},K^{i}}}\left(x\right))&\begin{aligned}
x\in\Omega&\cap(\partial S_{h^{i},K^{i}}\setminus\partial S_{h^{j},K^{j}})\\\
&\cap\left(S_{h^{i},K^{i}}^{(1)}\cup S_{h^{i},K^{i}}^{(0)}\right)\cap
S_{h^{j},K^{j}}^{(1)},\end{aligned}\\\
\varphi^{1}_{ij}(x,\nu_{S_{h^{j},K^{j}}}(x))&x\in\Omega\cap\partial
S_{h^{i},K^{i}}\cap\partial S_{h^{j},K^{j}}\cap
S_{h^{i},K^{i}}^{(1)},\end{cases}$ (3.8)

where
$\varphi_{j},\,\varphi_{ij}:\overline{\Omega}\times\mathbb{R}^{2}\to[0,\infty]$
and, given also the function
$\varphi_{i}:\overline{\Omega}\times\mathbb{R}^{2}\to[0,\infty]$, we define
the functions $\varphi^{1}_{ij}$ and $\varphi^{2}_{ij}$ in
$C(\overline{\Omega}\times\mathbb{R}^{2};[0,\infty])$ by

$\varphi^{1}_{ij}:=\min\\{\varphi_{i},\varphi_{j}+\varphi_{ij}\\}\qquad\text{and}\qquad\varphi^{2}_{ij}:=\min\\{\varphi_{j},\varphi_{i}\\}.$

In view of [42], for every $0\leq i\leq j\leq\alpha$,
$\varphi_{j},\varphi_{i},\varphi_{ij}$ represent the anisotropic surface
tensions of the film/vapor, the substrate/vapor and the substrate/film
interfaces, respectively, while $\varphi^{1}_{ij}$ and $\varphi^{2}_{ij}$ are
referred to as the anisotropic _regime surface tensions_ and are introduced to
include into the analysis the wetting and dewetting regimes.

######  Remark 3.6.

If $\alpha=1$, we can observe that the surface tension $\mathcal{S}$
considered in [42] coincides with $\mathcal{S}^{(0,1)}$ by considering, with
respect to the notation of [42],
$\varphi_{0}:=\varphi_{\mathrm{S}},\varphi_{1}:=\varphi_{\mathrm{F}},\varphi_{01}:=\varphi_{\mathrm{FS}}$
and as a consequence $\varphi^{1}_{01}=\varphi$ and
$\varphi^{2}_{01}=\varphi^{\prime}$.

Now, we are in the position to define the _$\alpha$ -layered total surface
energy_ $\mathcal{S}^{\alpha}_{t}:\mathcal{B}^{\alpha}\to[0,\infty]$ and the
_$\alpha$ -layered sequential surface energy_
$\mathcal{S}^{\alpha}_{s}:\mathcal{B}^{\alpha}\to[0,\infty]$ by

$\mathcal{S}^{\alpha}_{t}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}}):=\sum_{j=1}^{\alpha}\sum_{i=0}^{j-1}\mathcal{S}^{(i,j)}(S_{h^{j},K^{j}},S_{h^{i},K^{i}})$
(3.9)

and

$\mathcal{S}^{\alpha}_{s}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}}):=\sum_{j=1}^{\alpha}\mathcal{S}^{(j-1,j)}(S_{h^{j},K^{j}},S_{h^{j-1},K^{j-1}}),$
(3.10)

respectively. For simplicity we refer to the energy
$\mathcal{S}^{\alpha}_{\sigma}:\mathcal{B}^{\alpha}\to[0,\infty]$ for
$\sigma\in\\{s,t\\}$ as the _$\alpha$ -layered surface energy_.

######  Remark 3.7.

The terms in the sum of both definitions (3.9) and (3.10) depend on the
surface tensions $\psi_{i,j}$ that are defined in terms of the family $\Phi$
of surface tensions given by

$\Phi:=\\{\varphi_{j},\varphi_{i,j}:\overline{\Omega}\times\mathbb{R}^{2}\to[0,\infty]\,:\,j=1,\dots,\alpha,i=0,\dots,j-1\\}.$

Furthermore, we observe that the definition of the surface tensions
$\psi_{i,j}$ and $\psi_{j,k}$ for $k\in\\{1,\dots,\alpha\\}$,
$j\in\\{1,\dots,k-1\\}$, and $i\in\\{0,\dots,j-1\\}$ are interconnected, as
they both depend on $\varphi_{j}$. We would like to point out that this is
just a choice done for simplicity (and in relation to the physical meaning of
the surface tensions of each pair of $i$th and $j$th composites for
$j=1,\dots,\alpha$ and $i=0,\dots,j-1$, i.e., $\varphi_{j}$, $\varphi_{i}$,
and $\varphi_{i,j}$, which follows from [42] and Remark 3.6). We can avoid
this interconnection, for example by working in the following settings:

  * (i)

given $\varphi_{j}:\overline{\Omega}\times\mathbb{R}^{2}\to[0,\infty]$ in the
definition of $\psi_{j-1,j}$ by replacing $\varphi_{j}$ in the definition:

    * -

of each $\psi_{j,k}$ (and of the corresponding $\varphi^{1}_{j,k}$ and
$\varphi^{2}_{j,k}$) for $k\in\\{j+1,\dots,\alpha\\}$ with a surface tension
$\varphi^{\prime}_{j,k}$ possibly different from $\varphi_{j}$,

    * -

of each $\psi_{i,j}$ (and of the corresponding $\varphi^{1}_{i,j}$ and
$\varphi^{2}_{i,j}$) for $i\in\\{0,\dots,j-2\\}$ with a surface tension
$\varphi^{\prime}_{i,j}$ possibly different from $\varphi_{j}$,

namely by enlarging the definition of $\Phi$ by considering instead

$\displaystyle\Phi^{\prime}:=\Phi\cup\\{\varphi^{\prime}_{j,k},\varphi^{\prime}_{i,j}:$
$\displaystyle\,\,\overline{\Omega}\times\mathbb{R}^{2}\to[0,\infty]\,:$
$\displaystyle j=1,\dots,\alpha,k=j+1,\dots,\alpha,i=0,\dots,j-2\\};$ (3.11)

  * (ii)

by defining some surface tensions in $\phi\in\Phi^{\prime}$ as a combination
of the other surface tensions, namely the surface tensions in
$\Phi^{\prime}\setminus\\{\phi\\}$.

Let $\alpha\in\mathbb{N}$ and let $\sigma\in\\{s,t\\}$, the _$\alpha$ -layered
total energy_
$\mathcal{F}^{\alpha}_{\sigma}:\mathcal{C}^{\alpha}\to[-\infty,\infty]$ is
defined by

$\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u):=\mathcal{S}^{\alpha}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})+\mathcal{W}({S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},}u)$

for any
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}^{\alpha}$,
where $\mathcal{W}$ stands for the elastic energy, more precisely,

$\mathcal{W}({S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},}u):=\int_{S_{h^{\alpha},K^{\alpha}}}W(x,Eu(x)-{E_{0}^{\alpha}}(x))\,dx,$

and $W$ 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 $v\in\mathrm{H}^{1}_{\mathrm{{loc}}}(\Omega)$ and ${E_{0}^{\alpha}}$
is the mismatch strain
$x\in\Omega\mapsto{E_{0}^{\alpha}}\left(x\right)\in\mathbb{M}^{2}_{\mathrm{{sym}}}$
defined as

${E_{0}^{\alpha}}:=\begin{cases}E(u_{0}^{\alpha})&\text{in }\Omega\setminus
S_{h^{\alpha-1}},\\\ E(u_{0}^{i})&\text{in }{\rm{Int}}(S_{h^{i}})\setminus
S_{h^{1-1}}\text{ for $i=1,\ldots,\alpha-1$}\\\ 0&\text{in
}{\rm{Int}}(S_{h^{0},K^{0}}),\end{cases}$

for a fixed sequence $\\{u_{0}^{i}\\}_{i=1}^{\alpha-1}\subset
H^{1}(\Omega;\mathbb{R}^{2})$.

###  3.2. Main results

We state here the main results of the paper. Let $\alpha\in\mathbb{N}$. We fix
$l,L>0$ and we consider $\Omega:=(-l,l)\times(-L,{\infty})$. For every pair of
integers $0\leq i\leq j\leq\alpha$ we consider
$\varphi^{1}_{ij}:=\min\\{\varphi_{i},\varphi_{j}+\varphi_{ij}\\}$ and
$\varphi^{2}_{ij}:=\min\\{\varphi_{j},\varphi_{i}\\}$ and we assume throughout
this manuscript that:

  * (H1)

$\varphi_{j},\varphi_{ij},\varphi^{1}_{ij},\varphi^{2}_{ij}\in
C(\overline{\Omega}\times\mathbb{R}^{2})$ are Finsler norms such that there
exists $c_{2}\geq c_{1}>0$ such that

$c_{1}\absolutevalue{\xi}\leq\varphi_{j}(x,\xi),\varphi^{1}_{ij}(x,\xi),\varphi_{ij}(x,\xi)\leq
c_{2}\absolutevalue{\xi}\quad\textrm{for every $x\in\overline{\Omega}$ and
$\xi\in\mathbb{R}^{2}$},$ (3.12)

  * (H2)

We have

$\varphi^{1}_{ij}(x,\xi)\geq\absolutevalue{\varphi_{ij}(x,\xi)-\varphi_{j}(x,\xi)}\quad\textrm{for
every $x\in\overline{\Omega}$ and $\xi\in\mathbb{R}^{2}$.}$ (3.13)

  * (H3)

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

$\mathbb{C}\left(x\right)M:M\geq 2c_{3}M:M$ (3.14)

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

We notice that under assumptions (H1)-(H3), the energy
$\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in[0,\infty]$
for every
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in{\mathcal{C}^{\alpha}}$.

We now state the main theorem of this paper.

######  Theorem 3.8 (Existence of minimizers).

Fix $\alpha\in\mathbb{N}$,
$\mathbf{m}=(m_{0},\ldots,m_{\alpha})\in\mathbb{N}^{\alpha+1}$, and
$\sigma\in\\{s,t\\}$. Let
$\\{\mathbbm{v}_{i}\\}_{i=0}^{\alpha}\subset[\mathcal{L}^{2}({\Omega})/2,\mathcal{L}^{2}({\Omega})]$
be such that $\mathbbm{v}_{i_{1}}\leq\mathbbm{v}_{i_{2}}$ for every $0\leq
i_{1}<i_{2}\leq\alpha$, and let
$\bm{\lambda}:=(\lambda_{0},\ldots,\lambda_{\alpha})\in\mathbb{R}^{\alpha+1}$
be such that $\lambda_{i}>0$ for every $i=0,\ldots,\alpha$. If (H1)-(H3) holds
true, then both the volume constrained minimum problem

$\inf_{\small\begin{aligned}
&(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}^{\alpha}_{\mathbf{m}},\\\
&\mathcal{L}^{2}({S_{h^{i},K^{i}}})=\mathbbm{v}_{i},\,i=0,\ldots,\alpha\end{aligned}}{\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)}$
(3.15)

and the unconstrained minimum problem

$\inf_{(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}^{\alpha}_{\mathbf{m}}}{\mathcal{F}^{\alpha,\bm{\lambda}}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}^{\alpha}_{\mathbf{m}}},$
(3.16)

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

$\mathcal{F}^{\alpha,\bm{\lambda}}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u):=\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)+\sum_{i=0}^{\alpha}\lambda_{i}\absolutevalue{\mathcal{L}^{2}(S_{h^{i},K^{i}})-\mathbbm{v}_{i}},$

admit a solution.

We employ the Direct Method of Calculus of Variations to prove Theorem 3.8. In
order to apply this method we prove that any energy equi-bounded sequence
$\\{(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\\}\subset\mathcal{C}^{\alpha}_{\mathbf{m}}$
satisfy the following compactness property.

######  Theorem 3.9.

Fix $\alpha\in\mathbb{N}$,
$\mathbf{m}=(m_{0},\ldots,m_{\alpha})\in\mathbb{N}^{\alpha+1}$, and
$\sigma\in\\{s,t\\}$. Let
$\\{(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})\\}_{k\in\mathbb{N}}\subset\mathcal{C}^{\alpha}_{\mathbf{m}}$
be such that

$\sup_{k\in\mathbb{N}}{\left(\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})+{\mathcal{L}^{2}\left(S_{h^{\alpha}_{k},K^{\alpha}_{k}}\right)}\right)}<\infty.$
(3.17)

Then, there exist an admissible configuration
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}^{\alpha}_{\mathbf{m}}$
of finite energy, a subsequence
$\\{(S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}},\ldots,S_{h^{0}_{k_{n}},K^{0}_{k_{n}}},u_{k_{n}})\\}_{n\in\mathbb{N}}$,
a sequence
$\\{(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}})\\}_{n\in\mathbb{N}}\subset\mathcal{C}^{\alpha}_{\mathbf{m}}$
and a sequence $\\{b_{n}\\}_{n\in\mathbb{N}}$ of piecewise rigid displacements
associated to $S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}}$ such that

$(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}}+b_{n})\xrightarrow{\tau_{\mathcal{C}^{\alpha}}}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)$

and

$\liminf_{n\rightarrow\infty}{\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}},\ldots,S_{h^{0}_{k_{n}},K^{0}_{k_{n}}},u_{k_{n}})}=\liminf_{n\rightarrow\infty}{\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}}+b_{n})}.$
(3.18)

Furthermore, we show that $\mathcal{F}^{\alpha}_{\sigma}$ is lower
semicontinuous in $\mathcal{C}^{\alpha}_{\mathbf{m}}$ with respect to the
topology $\tau_{\mathcal{C}^{\alpha}}$ for any $\alpha\in\mathbb{N}$ and
$\sigma\in\\{s,t\\}$.

######  Theorem 3.10 (Lower semicontinuity of
$\mathcal{F}^{\alpha}_{\sigma}$).

Fix $\alpha\in\mathbb{N}$,
$\mathbf{m}=(m_{0},\ldots,m_{\alpha})\in\mathbb{N}^{\alpha+1}$, and
$\sigma\in\\{s,t\\}$. Assume (H1)-(H3). If
$\\{(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})\\}_{k\in\mathbb{N}}\subset\mathcal{C}^{\alpha}_{\mathbf{m}}$
and
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}^{\alpha}_{\mathbf{m}}$
are such that

$(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})\xrightarrow{\tau_{\mathcal{C}^{\alpha}}}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u),$

then

$\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\leq\liminf_{k\rightarrow\infty}{\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})}.$
(3.19)

We conclude this section by addressing the more general setting introduced in
Remark 3.7 regarding the family of surface tensions $\Phi^{\prime}$ defined in
((i)).

######  Remark 3.11.

We observe that Theorems 3.8–3.10 continue to hold in the more general setting
for the surface energies (3.9) and (3.10) that is described in Remark 3.7-(i)
(and so, also for the setting of Remark 3.7-(ii)) by simply adapting
hypotheses (H1) and (H2) to the more general definition of $\psi_{i,j}$ given
in Remark 3.7-(i) in terms of the family $\Phi^{\prime}$ of surface tensions
defined in ((i)). More precisely it suffices to replace $\varphi_{j}$ in (H1)
and (H2) with $\varphi^{\prime}_{i,j}\in\Phi^{\prime}$ for all
$j\in\\{2,\dots,\alpha\\}$, and $i\in\\{0,\dots,j-2\\}$. In this way, as the
application of [42, Theorem 4.2], [42, Theorem 4.3], and [42, Theorem 5.13],
respectively, in Proposition 4.1, and in Theorems 4.2 and 4.3, is preserved,
the same proof of Theorems 3.8–3.10 yields the corresponding results in the
setting of Remark 3.7.

##  4\. Single-layer films with delamination

In order to establish Theorem 3.8 we use an induction argument, and in this
section, we prove the basis of the induction. More precisely, we prove Theorem
3.8 by assuming that $\alpha=1$ and hence, in the following, we consider
$\bm{m}:=(m_{1},m_{0})\in\mathbb{N}^{2}$. We begin by observing that the
double-layered film setting of $\alpha=1$ is a particular case of the two-
phase setting considered in [42], with the only difference that the “exterior
graph condition” is assumed not only on the substrate region but also on the
film phase. The analogy comes also from the fact, that as proved below, for
energy equi-bounded admissible configurations in $\mathcal{C}^{1}$, we can
easily reduced to bounded rectangular containers
$\widetilde{\Omega}:=(-l,l)\times(-L,\widetilde{L})$ for a properly chosen
constant $\widetilde{L}>0$.

We recall that in [42] the families of admissible regions
$\mathcal{B}(\widetilde{\Omega})$ and
$\mathcal{B}_{\mathbf{m}}(\widetilde{\Omega})$ are defined as

$\displaystyle{\mathcal{B}}(\widetilde{\Omega}):=\Large\\{(A,S_{h,K}):\quad$
$\displaystyle(h,K)\in\text{AHK}(\widetilde{\Omega}),\text{ $A$ is
$\mathcal{L}^{2}$-measurable set with
$S_{h,K}\subset\overline{A}\subset\overline{\widetilde{\Omega}}$}$ such that
$\partial A\cap{\rm{Int}}(S_{h,K})=\emptyset$, $\partial A$ is
$\mathcal{H}^{1}$-rectifiable, $\displaystyle\text{$\mathcal{H}^{1}(\partial
A)+\mathcal{H}^{1}(\partial S)<\infty$}\\}$

and

$\mathcal{B}_{\mathbf{m}}(\widetilde{\Omega}):=\\{(A,S_{h,K})\in{\mathcal{B}}(\widetilde{\Omega}):\,\partial
A\text{ and }\partial S_{h,K}\text{ have at most}\\\ \text{$m_{1}$ and $m_{0}$
connected components, respectively}\\}.$

Notice that
$\mathcal{B}^{1}(\widetilde{\Omega})\subset\mathcal{B}(\widetilde{\Omega})$
and
$\mathcal{B}^{1}_{\mathbf{m}}(\widetilde{\Omega})\subset\mathcal{B}_{\mathbf{m}}(\widetilde{\Omega})$.
Furthermore, the families of admissible configurations in [42] are
$\mathcal{C}(\widetilde{\Omega})$ and
$\mathcal{C}_{\mathbf{m}}(\widetilde{\Omega})$ defined by

$\displaystyle{\mathcal{C}}(\widetilde{\Omega}):=\Large\\{(A,S_{h,K},u):\,(A,S_{h,K})\in{\mathcal{B}}(\widetilde{\Omega})\text{
and $u\in H^{1}_{\mathrm{{loc}}}({\rm{Int}}(A);\mathbb{R}^{2})$}\\}$

and

$\mathcal{C}_{\mathbf{m}}(\widetilde{\Omega}):=\\{(A,S_{h,K},u)\in{\mathcal{C}}(\widetilde{\Omega}):\,(A,S_{h,K})\in{\mathcal{B}}_{\mathbf{m}}(\widetilde{\Omega})\\},$

so that
$\mathcal{C}^{1}(\widetilde{\Omega})\subset\mathcal{C}(\widetilde{\Omega})$
and
$\mathcal{C}^{1}_{\mathbf{m}}(\widetilde{\Omega})\subset\mathcal{C}_{\mathbf{m}}(\widetilde{\Omega})$.
Therefore, since the elastic energy $\mathcal{W}$ and the surface energy
$\mathcal{S}$ considered in [42] coincide with the energies $\mathcal{W}$ and
$\mathcal{S}^{1}$ of this manuscript (by also observing that, following the
notation of [42],
$\varphi_{0}=\varphi_{\mathrm{S}},\varphi_{1}=\varphi_{\mathrm{F}},\varphi_{01}=\varphi_{\mathrm{FS}}$,
$\varphi^{1}_{01}=\varphi$ and $\varphi^{2}_{01}=\varphi^{\prime}$), we have
that

$\mathcal{S}\equiv\mathcal{S}^{1}\quad\text{and}\quad\mathcal{F}\equiv\mathcal{F}^{1},$
(4.1)

in $\mathcal{C}^{1}(\widetilde{\Omega})$ and
$\mathcal{C}^{1}_{\mathbf{m}}(\widetilde{\Omega})$. Finally, we also observe
have that the topologies $\tau_{\mathcal{B}}$ and $\tau_{\mathcal{C}}$ defined
in [42] coincide with the topologies $\tau_{\mathcal{B}^{1}}$ and
$\tau_{\mathcal{C}^{1}}$, respectively.

On the basis of these observations and by using the results for the two-phase
setting of [42], we now prove that energy-equibounded sequences in
$\mathcal{C}^{1}_{\bm{m}}$ are compact and that $\mathcal{F}^{1}$ is lower
semicontinuous with respect to the topology $\tau_{\mathcal{C}^{1}}$.

Figure 2. A single-layer film (on the substrate 0th layer $S_{h^{0},K^{0}}$)
associated to an admissible configuration
$(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)\in\mathcal{C}^{1}_{\textbf{m}}$ (see
Definition 3.1) is represented by indicating each $j$th layer with a gray
color with decreasing value with respect to the increasing order of the index
$j=0,1$ and each $j$th layer with a thinner line with respect to the
increasing order of the index $j=0,1$. Furthermore, in the $0$th layer we
distinguish between its coherent and incoherent portions by using a dashed or
a continuous line, respectively.

######  Proposition 4.1.

Let
$\\{(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}})\\}\subset\mathcal{B}^{1}_{\mathbf{m}}$
be such that

$\sup_{k\in\mathbb{N}}{\left(\mathcal{S}^{1}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}}){+\mathcal{L}^{2}(S_{h^{1}_{k},K^{1}_{k}})}\right)}<\infty.$
(4.2)

Then, there exist a not relabeled subsequence
$\\{(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}})\\}\subset\mathcal{B}^{1}_{\mathbf{m}}$
and $(S_{h^{1},K^{1}},S_{h^{0},K^{0}})\in\mathcal{B}^{1}_{\mathbf{m}}$ such
that
$(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}})\xrightarrow{\tau_{\mathcal{B}^{1}}}(S_{h^{1},K^{1}},S_{h^{0},K^{0}})$.

###### Proof.

We begin by observing that in view of [4, Theorem 3.47] from (4.2) it follows
that there exists $\widetilde{L}>0$ such that for every $k\in\mathbb{N}$,
$S_{h^{1}_{k},K^{1}_{k}}\subset(-l,l)\times(-L,\widetilde{L})=:\widetilde{\Omega}$.
Since
$(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}})\in\mathcal{B}^{1}_{\textbf{m}}$
for every $k\in\mathbb{N}$, by (4.1), where in [42] we consider
$\varphi_{0}=\varphi_{\mathrm{S}},\varphi_{1}=\varphi_{\mathrm{F}},\varphi_{01}=\varphi_{\mathrm{FS}}$,
$\varphi^{1}_{01}=\varphi$ and $\varphi^{2}_{01}=\varphi^{\prime}$, we have
that

$\sup_{k\in\mathbb{N}}{\mathcal{S}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}})}=\sup_{k\in\mathbb{N}}{\mathcal{S}^{1}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}})}<\infty.$
(4.3)

By applying [42, Theorem 4.2] with respect to the region $\widetilde{\Omega}$,
there exist a not relabeled subsequence
$\\{(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}})\\}\subset\mathcal{B}_{\textbf{m}}$
and $(S,S_{h^{0},K^{0}})\in\mathcal{B}_{\textbf{m}}$ such that

$(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}})\xrightarrow{\tau_{\mathcal{B}}}(S,S_{h^{0},K^{0}}).$
(4.4)

By definition of $\tau_{\mathcal{B}}$-convergence and by the second statement
of [42, Lemma 3.8] there exists $(h^{1},K^{1})\in\textrm{ AHK}(\Omega)$ such
that

$S=S_{h^{1},K^{1}}.$ (4.5)

In view of the definition of $h^{0}$ and $h^{1}$ that comes from [42, Lemma
3.8], we see that

$h^{0}(x_{1}):=\sup\\{\limsup_{k\to\infty}h^{0}_{k}(x_{1}^{k}):x^{k}_{1}\to
x_{1}\\}\leq\sup\\{\limsup_{k\to\infty}h^{1}_{k}(x_{1}^{k}):x^{k}_{1}\to
x_{1}\\}=:h^{1}(x_{1})$

for every $x_{1}\in[-l,l]$. Thus,
$(S_{h^{1},K^{1}},S_{h^{0},K^{0}})\in\mathcal{B}^{1}_{\mathbf{m}}$. Finally,
from (4.4) and (4.5), and by the fact that the
$\tau_{\mathcal{B}^{1}}$-convergence is similar to the
$\tau_{\mathcal{B}}$-convergence of [42] we obtain that

$(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}})\xrightarrow{\tau_{\mathcal{B}^{1}}}(S_{h^{1},K^{1}},S_{h^{0},K^{0}})$

which concludes the proof. ∎

We are in the position to prove that $\mathcal{C}^{1}_{\mathbf{m}}$ is compact
with respect to the topology $\tau_{\mathcal{C}^{1}}$.

[42, Theorem 4.3]

######  Theorem 4.2 (Compactness of $\mathcal{C}^{1}_{\mathbf{m}}$).

Let
$\\{(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})\\}_{k\in\mathbb{N}}\subset\mathcal{C}^{1}_{\mathbf{m}}$
be such that

$\sup_{k\in\mathbb{N}}{\left(\mathcal{F}^{1}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k}){+\mathcal{L}^{2}(S_{h^{1}_{k},K^{1}_{k}})}\right)}<\infty.$
(4.6)

Then, there exist an admissible configuration
$(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)\in\mathcal{C}^{1}_{\mathbf{m}}$ of finite
$\mathcal{F}^{1}$ energy, a subsequence
$\\{(S_{h^{1}_{k_{n}},K^{1}_{k_{n}}},S_{h^{0}_{k_{n}},K^{0}_{k_{n}}},u_{k_{n}})\\}_{n\in\mathbb{N}}$,
a sequence
$\\{(S_{h^{1}_{k_{n}},\widetilde{K}^{1}_{n}},S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}})\\}_{n\in\mathbb{N}}\subset\mathcal{C}^{1}_{\mathbf{m}}$
and a sequence $\\{b_{n}\\}_{n\in\mathbb{N}}$ of piecewise rigid displacements
associated to $S_{h^{1}_{k_{n}},\widetilde{K}^{1}_{n}}$ such that

$(S_{h^{1}_{k_{n}},\widetilde{K}^{1}_{n}},S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}}+b_{n})\xrightarrow{\tau_{\mathcal{C}^{1}}}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)$

and

$\liminf_{n\rightarrow\infty}{\mathcal{F}^{1}(S_{h^{1}_{k_{n}},K^{1}_{k_{n}}},S_{h^{0}_{k_{n}},K^{0}_{k_{n}}},u_{k_{n}})}=\liminf_{n\rightarrow\infty}{\mathcal{F}^{1}(S_{h^{1}_{k_{n}},\widetilde{K}^{1}_{n}},S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}}+b_{n})}.$
(4.7)

###### Proof.

We begin by observing that in view of [4, Theorem 3.47] from (4.2) it follows
that there exists $\widetilde{L}>0$ such that for every $k\in\mathbb{N}$,
$S_{h^{1}_{k},K^{1}_{k}}\subset(-l,l)\times(-L,\widetilde{L})=:\widetilde{\Omega}$.
In view of the observations at the beginning of the section by (4.1) and (4.6)
we have that

$\sup_{k\in\mathbb{N}}{\mathcal{F}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})}=\sup_{k\in\mathbb{N}}{\mathcal{F}^{1}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})}<\infty,$

where $\mathcal{F}:\mathcal{C}(\widetilde{\Omega})\to[0,\infty]$ is the total
energy considered in [42]. Hence, by applying [42, Theorem 4.3] with respect
to the region $\widetilde{\Omega}$, we deduce that there exist a triple
$(S,S_{h^{0},K^{0}},u)\in\mathcal{C}_{\mathbf{m}}(\widetilde{\Omega})$ of
finite $\mathcal{F}$-energy, a subsequence
$\\{(S_{h^{1}_{k_{n}},K^{1}_{k_{n}}},S_{h^{0}_{k_{n}},K^{0}_{k_{n}}},u_{k_{n}})\\}_{n\in\mathbb{N}}$,
a sequence
$\\{(\widetilde{S}_{n},S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}})\\}_{n\in\mathbb{N}}\subset\mathcal{C}_{\mathbf{m}}$
and a sequence $\\{b_{n}\\}_{n\in\mathbb{N}}$ of piecewise rigid displacements
associated to $\widetilde{S}_{n}$ such that

$(\widetilde{S}_{n},S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}}+b_{n})\xrightarrow{\tau_{\mathcal{C}}}(S,S_{h^{0},K^{0}},u)$
(4.8)

and

$\liminf_{n\rightarrow\infty}{\mathcal{F}(S_{h^{1}_{k_{n}},K^{1}_{k_{n}}},S_{h^{0}_{k_{n}},K^{0}_{k_{n}}},u_{k_{n}})}=\liminf_{n\rightarrow\infty}{\mathcal{F}(\widetilde{S}_{n},S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}}+b_{n})}.$
(4.9)

In view of the proof of [42, Theorem 4.3] we have that

$\widetilde{S}_{n}:=S_{h^{1}_{k_{n}},K^{1}_{k_{n}}}\setminus\left(\partial\widetilde{S}_{n}\setminus\partial
S_{h^{1}_{k_{n}},K^{1}_{k_{n}}}\right)$ (4.10)

In analogy to the definition of $\widetilde{K}^{0}_{n}$ in the proof of [42,
Theorem 4.3], we define

$\widetilde{K}^{1}_{n}:=K^{1}_{n}\cup\left(\partial\widetilde{S}_{n}\setminus\partial
S_{h^{1}_{k_{n}},K^{1}_{k_{n}}}\right)$ (4.11)

and we claim that $\widetilde{S}_{n}=S_{h^{1}_{k_{n}},\widetilde{K}^{1}_{n}}$.
Indeed, we have that

$\begin{split}S_{h^{1}_{k_{n}},\widetilde{K}^{1}_{n}}&:=\partial
S_{h^{1}_{k_{n}}}\cup\left(S_{h^{1}_{k_{n}}}\setminus\widetilde{K}^{1}_{n}\right)=\partial
S_{h^{1}_{k_{n}}}\cup\left(S_{h^{1}_{k_{n}}}\setminus\left(K^{1}_{n}\cup\left(\partial\widetilde{S}_{n}\setminus\partial
S_{h^{1}_{k_{n}},K^{1}_{k_{n}}}\right)\right)\right)\\\ &=\partial
S_{h^{1}_{k_{n}}}\cup\left(S_{h^{1}_{k_{n}}}\cap\left((K^{1}_{n})^{c}\cap\left(\partial\widetilde{S}_{n}\setminus\partial
S_{h^{1}_{k_{n}},K^{1}_{k_{n}}}\right)^{c}\right)\right)\\\ &=\left(\partial
S_{h^{1}_{k_{n}}}\cup\left(S_{h^{1}_{k_{n}}}\setminus{K}^{1}_{n}\right)\right)\cap\left(\partial
S_{h^{1}_{k_{n}}}\cup\left(\partial\widetilde{S}_{n}\setminus\partial
S_{h^{1}_{k_{n}},K^{1}_{k_{n}}}\right)^{c}\right)\\\
&=:S_{h^{1}_{k_{n}},K^{1}_{k_{n}}}\cap\left(\partial
S_{h^{1}_{k_{n}}}\cup(\partial\widetilde{S}_{n})^{c}\cup\partial
S_{h^{1}_{k_{n}},K^{1}_{k_{n}}}\right)=S_{h^{1}_{k_{n}},K^{1}_{k_{n}}}\setminus(\partial\widetilde{S}_{n}\setminus\partial
S_{h^{1}_{k_{n}},K^{1}_{k_{n}}})\\\ &=:\widetilde{S}_{n},\end{split}$ (4.12)

where we used (3.5) in the first and fifth equalities, (4.11) in the second
equality, De Morgan’s laws in the third, fourth and sixth equalities, and
(4.10) in the last equality, and hence
$(S_{h^{1}_{k_{n}},\widetilde{K}^{1}_{k_{n}}},S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{k_{n}}},u_{k_{n}})\in\mathcal{C}^{1}_{\textbf{m}}$.

Furthermore, by (4.6) and the non-negativeness of $\mathcal{F}$ it follows
from Proposition 4.1 that $S=S_{h^{1},K^{1}}$ for a proper pair
$(h^{1},K^{1})\in\text{AHK}(\Omega)$ and, in particular, we have that
$(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)\in\mathcal{C}^{1}_{\textbf{m}}$. Finally,
in view of the definition of $\tau_{\mathcal{C}^{1}}$-convergence, by (4.8) we
obtain that

$(S_{h^{1}_{k_{n}},\widetilde{K}^{1}_{k_{n}}},S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{k_{n}}},u_{k_{n}}+b_{n})\xrightarrow{\tau_{\mathcal{C}^{1}}}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u),$

and, by (4.1), (4.9) we obtain (4.7), which concludes the proof. ∎

Now, by applying [42, Theorem 5.14] we prove that $\mathcal{F}^{1}$ is lower
semicontinuous with respect to the $\tau_{\mathcal{C}^{1}}$-topology.

######  Theorem 4.3 (Lower semicontinuity of $\mathcal{F}^{1}$).

Assume (H1)-(H3). Let
$\\{(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})\\}_{k\in\mathbb{N}}\subset\mathcal{C}^{1}_{\mathbf{m}}$
and $(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)\in\mathcal{C}^{1}_{\mathbf{m}}$ be
such that
$(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})\xrightarrow{\tau_{\mathcal{C}^{1}}}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u).$
Then

$\mathcal{F}^{1}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)\leq\liminf_{k\rightarrow\infty}{\mathcal{F}^{1}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})}.$
(4.13)

###### Proof.

Without loss of generality, we assume that the right side of (4.13) is finite.
In view of [4, Theorem 3.47] and from the fact that and since
$(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})\xrightarrow{\tau_{\mathcal{C}^{1}}}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)$it
follows that there exists $\widetilde{L}>0$ such that for every
$k\in\mathbb{N}$,
$S_{h^{1}_{k},K^{1}_{k}}\subset(-l,l)\times(-L,\widetilde{L})=:\widetilde{\Omega}$.
Since the topology $\tau_{\mathcal{B}}$ considered in [42] coincide with the
topology $\tau_{\mathcal{B}^{1}}$, by (4.1) and by applying [42, Theorem 5.13]
applied in the regions $\Omega=\widetilde{\Omega}$, we have that

$\mathcal{S}^{1}\left(S_{h^{1},K^{1}},S_{h^{0},K^{0}}\right)\leq\liminf_{k\rightarrow\infty}{\mathcal{S}^{1}\left(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}}\right)}.$
(4.14)

Now, we are going to prove that the elastic energy is lower semicontinuous.
Indeed, let $D\subset\subset\textrm{Int}(S_{h^{1},K^{1}})$, by properties of
the signed distance convergence we have that
$D\subset\subset{\rm{Int}}{(S_{h^{1}_{k},K^{1}_{k}})}$ for $k$ large enough.
By definition of $\tau_{\mathcal{C}^{1}}$ convergence we have that
$u_{k}\rightarrow u$ a.e. in $D$. Furthermore, since $E\,u_{k}$ are bounded in
the $L^{2}(D)$ norm, we have that $E\,u_{k}\rightharpoonup E\,u$ in
$L^{2}(D)$. By convexity of $W$ we obtain that

$\int_{D}W(x,E\,u-{E_{0}^{1}})\,dx\leq\liminf_{k\rightarrow\infty}\int_{D}W(x,E\,u_{k}-{E_{0}^{1}})\,dx\leq\liminf_{k\rightarrow+\infty}{\mathcal{W}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})}$

By taking $D\nearrow{\rm{Int}}(S_{h^{1},K^{1}})$ we conclude that

${\mathcal{W}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)}\leq\liminf_{k\rightarrow+\infty}{\mathcal{W}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})}.$
(4.15)

By (4.14) and (4.15) and thanks to the superadditivity of the _liminf_ , we
get that

$\begin{split}\mathcal{F}^{1}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)&:={\mathcal{W}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)}+\mathcal{S}^{1}\left(S_{h^{1},K^{1}},S_{h^{0},K^{0}}\right)\\\
&\leq\liminf_{k\rightarrow+\infty}{\mathcal{W}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})}+\liminf_{k\rightarrow\infty}{\mathcal{S}^{1}\left(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}}\right)}\\\
&\leq\liminf_{k\rightarrow+\infty}{\mathcal{W}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})+\mathcal{S}^{1}\left(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}}\right)}\\\
&=:\liminf_{k\rightarrow\infty}{\mathcal{F}^{1}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})},\end{split}$

which concludes the proof. ∎

Finally, we state the main result of this section. The following result is the
analogous result of Theorem 3.8, more precisely, we prove the existence of
minimizers for a volume constrained problem and for an unconstrained problem,
respectively, with respect to the admissible family of deformable film and
substrate $\mathcal{C}^{1}_{\textbf{m}}$ for every
$\textbf{m}:=(m_{0},m_{1})\in\mathbb{N}\times\mathbb{N}$.

######  Theorem 4.4 (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
$\mathbf{m}=(m_{0},m_{1})\in\mathbb{N}^{2}$, the volume constrained minimum
problem

$\inf_{(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)\in\mathcal{C}^{1}_{\mathbf{m}},\,\mathcal{L}^{2}({S_{h^{1},K^{1}}})=\mathbbm{v}_{1},\,\mathcal{L}^{2}({S_{h^{0},K^{0}}})=\mathbbm{v}_{0}}{\mathcal{F}^{1}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)}$
(4.16)

and the unconstrained minimum problem

$\inf_{(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)\in\mathcal{C}^{1}_{\textbf{m}}}{\mathcal{F}^{1,\bm{\lambda}}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)}$
(4.17)

have solution, where
$\mathcal{F}^{1,\bm{\lambda}}:\mathcal{C}^{1}_{\mathbf{m}}\rightarrow\mathbb{R}$
is defined as

$\mathcal{F}^{1,\bm{\lambda}}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u):=\mathcal{F}^{1}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)+\sum_{i=0}^{1}\lambda_{i}\absolutevalue{\mathcal{L}^{2}(S_{h^{i},K^{i}})-\mathbbm{v}_{i}}.$

for any $\bm{\lambda}=(\lambda_{0},\lambda_{1})\in\mathbb{R}^{2}$ such that
$\lambda_{0},\lambda_{1}>0$.

###### Proof.

We follow the _Direct Method of the Calculus of Variations_. Fix
$\textbf{m}:=(m_{1},m_{0})\in\mathbb{N}^{2}$. Let
$\\{(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})\\}\subset\mathcal{C}^{1}_{\mathbf{m}}$
be a minimizing sequence of $\mathcal{F}^{1}$ such that
$\mathcal{L}^{2}({S_{h^{i}_{k},K^{i}_{k}}})=\mathbbm{v}_{i}$ for $i=0,1$, and

$\sup_{k\in\mathbb{N}}{\mathcal{F}^{1}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})}<\infty.$

Since $\mathcal{L}^{2}({S_{h^{1}_{k},K^{1}_{k}}})=\mathbbm{v}_{1}$ for every
$k\in\mathbb{N}$, by Theorem 4.2, there exist a subsequence
$\\{(S_{h^{1}_{k_{l}},K^{1}_{k_{l}}},S_{h^{0}_{k_{l}},K^{0}_{k_{l}}},u_{k_{l}})\\}$,
a sequence
$\\{(S_{h^{1}_{k_{l}},\widetilde{K}^{1}_{l}},S_{h^{0}_{k_{l}},\widetilde{K}^{0}_{l}},v_{l})\\}_{n\in\mathbb{N}}\subset\mathcal{C}^{1}_{\mathbf{m}}$
and $(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)\in\mathcal{C}^{1}_{\mathbf{m}}$ such
that

$(S_{h^{1}_{k_{l}},\widetilde{K}^{1}_{l}},S_{h^{0}_{k_{l}},\widetilde{K}^{0}_{l}},v_{l})\xrightarrow{\tau_{\mathcal{C}^{1}}}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)$

as $l\to\infty$ and

$\liminf_{l\to\infty}{\mathcal{F}^{1}(S_{h^{1}_{k_{l}},\widetilde{K}^{1}_{l}},S_{h^{0}_{k_{l}},\widetilde{K}^{0}_{l}},v_{l})}=\liminf_{l\to\infty}{\mathcal{F}^{1}(S_{h^{1}_{k_{l}},K^{1}_{k_{l}}},S_{h^{0}_{k_{l}},K^{0}_{k_{l}}},u_{k_{l}})}.$
(4.18)

According to Theorem 4.3, we have that

$\mathcal{F}^{1}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)\leq\liminf_{l\to\infty}{\mathcal{F}^{1}(S_{h^{1}_{k_{l}},\widetilde{K}^{1}_{l}},S_{h^{0}_{k_{l}},\widetilde{K}^{0}_{l}},v_{l})}.$
(4.19)

We claim that $\\{(S_{h^{1},K^{1}},S_{h^{0},K^{0}})\\}$ and
$(S_{h^{1}_{k_{l}},\widetilde{K}^{1}_{l}},S_{h^{0}_{k_{l}},\widetilde{K}^{0}_{l}})$
satisfy the volume constraints of (4.16). Indeed, fix $i=0,1$, by [42, Theorem
4.3], for any $l\geq 1$,
$\mathbbm{v}_{i}=\mathcal{L}^{2}({S_{h^{i}_{k_{l}},K^{i}_{k_{l}}}})=\mathcal{L}^{2}({S_{h^{i}_{k_{l}},\widetilde{K}^{i}_{l}}})$.
Thanks to the fact that

$(S_{h^{1}_{k_{l}},\widetilde{K}^{1}_{l}},S_{h^{0}_{k_{l}},\widetilde{K}^{0}_{l}})\xrightarrow{\tau_{\mathcal{B}^{1}}}(S_{h^{1},K^{1}},S_{h^{0},K^{0}}),$

applying [35, Lemma 3.2] we infer that
$S_{h^{i}_{k_{l}},\widetilde{K}^{i}_{l}}\to S_{h^{i},K^{i}}$ in
$L^{1}(\mathbb{R}^{2})$ as $l\to\infty$, and thus
$\mathcal{L}^{2}({S_{h^{i},K^{i}}})=\mathbbm{v}_{i}$. From (4.18) and (4.19),
we deduce that

$\begin{split}&\inf_{(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)\in\mathcal{C}^{1}_{\mathbf{m}},\,\mathcal{L}^{2}({S_{h^{1},K^{1}}})=\mathbbm{v}_{1},\,\mathcal{L}^{2}({S_{h^{0},K^{0}}})=\mathbbm{v}_{0}}{\mathcal{F}^{1}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u)}\\\
&\quad=\lim_{k\to\infty}\mathcal{F}^{1}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})\geq\liminf_{l\to\infty}{\mathcal{F}^{1}(S_{h^{1}_{k_{l}},\widetilde{K}^{1}_{l}},S_{h^{0}_{k_{l}},\widetilde{K}^{0}_{l}},v_{l})}\\\
&\quad\geq\mathcal{F}^{1}(S_{h^{1},K^{1}},S_{h^{0},K^{0}},u).\end{split}$

We conclude from the previous inequality that $(A,h,K,u)$ is a minimum of
(4.16). The same arguments are used to solve the unconstrained problem (4.17)
by noticing that for a minimizing sequence
$\\{(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})\\}\subset\mathcal{C}^{1}_{\mathbf{m}}$
of $\mathcal{F}^{1,\bm{\lambda}}$ such that

$\sup_{k\in\mathbb{N}}{\mathcal{F}^{1,\bm{\lambda}}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})}<\infty$

we have that

$\mathcal{L}^{2}({S_{h^{1}_{k},K^{1}_{k}}})\leq\absolutevalue{\mathcal{L}^{2}({S_{h^{1}_{k},K^{1}_{k}}})-\mathbbm{v}_{1}}+\mathbbm{v}_{1}\leq\frac{1}{\lambda_{1}}\mathcal{F}^{1,\bm{\lambda}}(S_{h^{1}_{k},K^{1}_{k}},S_{h^{0}_{k},K^{0}_{k}},u_{k})+\mathbbm{v}_{1},$

and thus
$\sup_{k\in\mathbb{N}}\mathcal{L}^{2}({S_{h^{1}_{k},K^{1}_{k}}})<\infty$. ∎

##  5\. Multilayered films

In this section, we consider $\alpha>1$ and we denote
$\textbf{m}:=(m_{0},\ldots,m_{\alpha})\in\mathbb{N}^{\alpha+1}$. The main goal
of this section is to prove Theorem 3.8. In order to do this, first we prove
that $\mathcal{C}^{\alpha}_{{\bm{m}}}$ is compact and by induction, with
respect to $\alpha\in\mathbb{N}$ we show that $\mathcal{F}^{\alpha}$ is lower
semicontinuous with respect to the topology of $\tau_{\mathcal{C}^{\alpha}}$.
Notice that in the previous section, we proved the basis of the induction for
the lower semicontinuity property. We start by proving that
$\mathcal{B}^{\alpha}_{{\bm{m}}}$ and $\mathcal{C}^{\alpha}_{\bm{m}}$ are
compact.

######  Proposition 5.1.

Let
$\\{(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\\}\subset\mathcal{B}^{\alpha}_{\mathbf{m}}$
such that

$\sup_{k\in\mathbb{N}}{\left(\mathcal{S}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}}){+\mathcal{L}^{2}(S_{h^{\alpha}_{k},K^{\alpha}_{k}})}\right)}<\infty$
(5.1)

Then, there exist a not relabeled subsequence
$\\{(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\\}\subset\mathcal{B}^{\alpha}_{\mathbf{m}}$
and
$\\{(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})\\}\in\mathcal{B}^{\alpha}_{\mathbf{m}}$
such that
$(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\xrightarrow{\tau_{\mathcal{B}^{\alpha}}}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})$.

###### Proof.

We proceed by induction on $\alpha\in\mathbb{N}$. If $\alpha=1$ by Proposition
4.1 the assertion holds. We now prove that if the assertion holds for
$\alpha=n$, then the assertion also holds for $\alpha=n+1$. Thus, let us
assume that the assertion holds for $\alpha=n$. From the definition of
$\mathcal{S}^{n+1}_{\sigma}$ and the non-negativeness of
$\mathcal{S}^{(n+1,j)}$ for every $j=0,\ldots,n$ it follows that

$\begin{split}\mathcal{S}^{n+1}_{\sigma}(S_{h^{n+1}_{k},K^{n+1}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})&\geq\mathcal{S}^{n}_{\sigma}(S_{h^{n}_{k},K^{n}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\\\
&\qquad+\mathcal{S}^{(n+1,n)}(S_{h^{n+1}_{k},K^{n+1}_{k}},S_{h^{n}_{k},K^{n}_{k}})\end{split}$
(5.2)

for every $k\in\mathbb{N}$. By (5.1), (5.2), and the non-negativeness of
$\mathcal{S}^{(n+1,n)}$ the induction hypothesis yields that, up to extracting
a not relabeled subsequence, there exists
$(S_{h^{n},K^{n}},\ldots,S_{h^{0},K^{0}})\in\mathcal{B}^{n}_{\textbf{m}_{\textbf{n}}}$
such that
$(S_{h^{n}_{k},K^{n}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\xrightarrow{\tau_{\mathcal{B}^{n}}}(S_{h^{n},K^{n}},\ldots,S_{h^{0},K^{0}})$,
where $\textbf{m}_{\textbf{n}}:=(m_{0},\ldots,m_{n})$. Furthermore, by (5.1)
and (5.2) and the non-negativeness of $\mathcal{S}^{n}_{\sigma}$, in view of
Proposition 4.1 applied to $\mathcal{S}^{(n+1,n)}$ it follows that, up to
extracting a not relabeled subsequence, there exist a region
$S_{h^{n+1},K^{n+1}}$ with $(h^{n+1},K^{n+1})\in\text{AHK}(\Omega)$ (and
$(S_{h^{n+1},K^{n+1}},S_{h^{n},K^{n}})\in\mathcal{B}^{1}_{\textbf{m}_{1}}$ for
${\textbf{m}_{1}}:=(m_{n},m_{n+1})\in\mathbb{N}^{2}$) such that
$(S_{h^{n+1}_{k},K^{n+1}_{k}},S_{h^{n}_{k},K^{n}_{k}})\xrightarrow{\tau_{\mathcal{B}^{1}}}(S_{h^{n+1},K^{n+1}},S_{h^{n},K^{n}})$,
where we used the uniqueness of the sign-distance convergence.

It remains to prove that
$(S_{h^{n+1},K^{n+1}},\ldots,S_{h^{0},K^{0}})\in\mathcal{B}^{n+1}_{\textbf{m}}$.
Since $\partial S_{h^{i},K^{i}}$ has at most $m_{i}$ connected components for
$i=0,\ldots,n+1$, it remains only to check that $\partial
S_{h^{n+1},K^{n+1}}\cap{\rm{Int}}(S_{h^{n},K^{n}})=\emptyset$. Let us assume
by contradiction that $\partial
S_{h^{n+1},K^{n+1}}\cap{\rm{Int}}(S_{h^{n},K^{n}})\neq\emptyset$. Then, there
exists

$x\in\partial S_{h^{n+1},K^{n+1}}\cap{\rm{Int}}(S_{h^{n},K^{n}}).$ (5.3)

By properties of the signed distance convergence (see [42, Remark 3.8]) there
exists $x_{k}\in\partial S_{h^{n+1}_{k},K^{n+1}_{k}}$ such that $x_{k}\to x$,
and, since
$(S_{h^{n+1}_{k},K^{n+1}_{k}},S_{h^{n}_{k},K^{n}_{k}})\xrightarrow{\tau_{\mathcal{B}^{1}}}(S_{h^{n+1},K^{n+1}},S_{h^{n},K^{n}})$
it follows that

${\rm{sdist}}(x,\partial S_{h^{n}_{k},K^{n}_{k}})\to{\rm{sdist}}(x,\partial
S_{h^{n},K^{n}})\quad\text{as }k\to\infty.$ (5.4)

By (5.3) there exists $\varepsilon>0$ such that ${\rm{sdist}}(x,\partial
S_{h^{n},K^{n}})=-\varepsilon$, we can find $k_{0}:=k_{0}(x)$ for which
${\rm{sdist}}(x,\partial S_{h^{n}_{k_{0}},K^{n}_{k_{0}}})$ is negative. Then,
$x\in{\rm{Int}}(S_{h^{n}_{k_{0}},K^{n}_{k_{0}}})$ and so, there exists
$\delta\leq\varepsilon/2$ such that

$x_{k_{0}}\in
B_{\delta}(x)\subset{\rm{Int}}(S_{h^{n}_{k_{0}},K^{n}_{k_{0}}}),$

which is an absurd since $\partial
S_{h^{n+1}_{k_{0}},K^{n+1}_{k_{0}}}\cap{\rm{Int}}(S_{h^{n}_{k_{0}},K^{n}_{k_{0}}})=\emptyset$.
Finally, we conclude the proof by observing that there exists
$(S_{h^{n+1},K^{n+1}},\ldots,S_{h^{0},K^{0}})\in\mathcal{B}^{n+1}_{\textbf{m}}$
such that
$(S_{h^{n+1}_{k},K^{n+1}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\xrightarrow{\tau_{\mathcal{B}^{n+1}}}(S_{h^{n+1},K^{n+1}},\ldots,S_{h^{0},K^{0}})$.
∎

Now, we prove that $\mathcal{C}^{\alpha}_{{\bm{m}}}$ is compact with respect
to the topology $\tau_{\mathcal{C}^{\alpha}}$.

###### Proof of Theorem 3.9.

Denote
$R:=\sup_{k\in\mathbb{N}}\left({\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})}{+\mathcal{L}^{2}(S_{h^{\alpha}_{k},K^{\alpha}_{k}})}\right)$.
Without loss of generality (by passing, if necessary, to a not relabeled
subsequence), we assume that

$\liminf_{k\rightarrow\infty}{\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})}=\lim_{k\rightarrow\infty}{\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})}\leq
R.$ (5.5)

Since $\mathcal{W}$ is a non-negative energy, by Proposition 5.1 there exist a
subsequence
$\\{(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\\}\subset\mathcal{B}^{\alpha}_{\mathbf{m}}$
and
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})\in\mathcal{B}^{\alpha}_{\mathbf{m}}$
such that

$(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\xrightarrow{\tau_{\mathcal{B}^{\alpha}}}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}}).$

The rest of the proof is devoted to the construction of a sequence
$\\{(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}})\\}\subset\mathcal{B}^{\alpha}_{\mathbf{m}}$
to which we can apply [35, Corollary 3.8] (with
$P={\rm{Int}}{(S_{h^{\alpha},K^{\alpha}})}$ and
$P_{n}={\rm{Int}}{(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}})}$,
respectively) in order to obtain $u\in H^{1}_{{\rm
loc}}({\rm{Int}}(S_{h^{\alpha},K^{\alpha}});\mathbb{R}^{2})$ such that
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}^{\alpha}_{\mathbf{m}}$
has finite energy, and a sequence $\\{b_{n}\\}_{n\in\mathbb{N}}$ of piecewise
rigid displacements such that

$(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}}+b_{n})\xrightarrow{\tau_{\mathcal{C}^{\alpha}}}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u).$

Furthermore, we observe that also Equation (3.18) will be a consequence of
such construction and hence, the assertion of the theorem will directly
follow.

By [35, Proposition 3.6] applied to
$S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}$ and $S_{h^{\alpha},K^{\alpha}}$
there exist a not relabeled subsequence
$\\{S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}\\}$ and a sequence
$\\{\widetilde{A}_{n}\\}$ with $\mathcal{H}^{1}$-rectifiable boundary
$\partial\widetilde{A}_{n}$ of at most $m_{\alpha}$-connected components such
that

$\sup_{n\in\mathbb{N}}\mathcal{H}^{1}(\partial\widetilde{A}_{n})<\infty,$
(5.6)

that satisfy the following properties:

  * (a1)

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

  * (a2)

${\rm{sdist}}(\cdot,\partial\widetilde{A}_{n})\to{\rm{sdist}}(\cdot,\partial
S_{h^{\alpha},K^{\alpha}})$ 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
${\rm{Int}}(S_{h^{\alpha},K^{\alpha}}),$ we can find the connected components
of ${\rm{Int}}(\widetilde{A}_{n})$, which we enumerate as
$\\{E_{i}^{n}\\}_{i\in I}$, such that for any $i$ and $G\subset\subset E_{i}$
one has $G\subset\subset E_{i}^{n}$ for all $n$ large (depending only on $i$
and $G$),

  * (a4)

$\mathcal{L}^{2}(\widetilde{A}_{n})=\mathcal{L}^{2}(S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}})$.

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
$S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}$, see [35, Propositions 3.4 and
3.6]) it follows that

$\widetilde{A}_{n}=S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}\setminus(\partial\widetilde{A}_{n}\setminus\partial
S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}})$ (5.7)

with $\partial\widetilde{A}_{n}\setminus\partial
S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}$ given by a finite union of closed
$\mathcal{H}^{1}$-rectifiable sets connected to $\partial
S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}$. 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 $\partial
S_{k_{n}}$ such that

$\partial\widetilde{A}_{n}\setminus\partial
S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}=\bigcup_{j\in J}\Gamma_{j}.$

We define

$\widetilde{K}^{i}_{n}:=K^{i}_{k_{n}}\cup((\partial\widetilde{A}_{n}\setminus\partial
S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}})\cap S_{h^{i}_{k_{n}}})\subset
S_{h^{i}_{k_{n}}},$

for every $i=0,\ldots,\alpha$, and we observe that $\widetilde{K}^{i}_{n}$ is
closed and $\mathcal{H}^{1}$-rectifiable in view of the fact that
$\partial\widetilde{A}_{n}\setminus\partial
S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}$ is a closed set in $\Omega$ and is
$\mathcal{H}^{1}$-rectifiable, since $\partial\widetilde{A}_{n}$ is
$\mathcal{H}^{1}$-rectifiable. Therefore,
$(h^{i}_{k_{n}},\widetilde{K}^{i}_{n})\in\text{AHK}(\Omega)$ for every
$i=0,\ldots,\alpha$. Furthermore, we have that

$S_{h^{i}_{k_{n}},\widetilde{K}^{i}_{n}}\subset S_{h^{i}_{k_{n}}}\subset
S_{h^{j}_{k_{n}}}=\overline{S_{h^{j}_{k_{n}},\widetilde{K}^{j}_{n}}},$

for every $0\leq i\leq j\leq\alpha$. We claim that $\partial
S_{h^{i}_{k_{n}},\widetilde{K}^{i}_{n}}$ has at most $m_{i}$-connected
components for $i=0,\ldots,\lambda$. Indeed, let $i\in\\{0,\ldots,\alpha\\}$,
if for every $j\in J$, $S_{h^{i}_{k_{n}},K^{i}_{k_{n}}}\cap\Gamma_{j}$ is
empty there is nothing to prove, so we assume that there exists $j\in J$ such
that $S_{h^{i}_{k_{n}},K^{i}_{k_{n}}}\cap\Gamma_{j}\neq\emptyset$. On one hand
if $\Gamma_{j}\subset S_{h^{i}_{k_{n}},K^{i}_{k_{n}}}$, thanks to the facts
that $\Gamma_{j}$ is connected to $\partial
S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}$ and
$S_{h^{i}_{k_{n}},K^{i}_{k_{n}}}\subset S_{h^{\lambda}_{k_{n}}}$, we deduce
that $\Gamma_{j}$ needs to be connected to $\partial
S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}$. On the other hand, if
$\Gamma_{j}\cap(S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}\setminus
S_{h^{i}_{k_{n}}})\neq\emptyset$, then we can find $x_{1}\in\Gamma_{j}\cap
S_{h^{i}_{k_{n}},K^{i}_{k_{n}}}$ and
$x_{2}\in\Gamma_{j}\cap(S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}\setminus
S_{h_{k_{n}}})$. Since $\Gamma_{j}$ is closed and connected, by [24, 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 $\partial S_{h^{i}_{k_{n}},K^{i}_{k_{n}}}$ and we conclude that
$\Gamma_{j}$ is connected to $\partial S_{h^{i}_{k_{n}},K^{i}_{k_{n}}}$.
Finally, by repeating the same arguments of (4.12), we obtain that

$\widetilde{A}_{n}=S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{k_{n}}}$

and thus,
$(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}})\in\mathcal{B}^{\alpha}_{\textbf{m}}$.

We claim that
$(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}})\xrightarrow{\tau_{\mathcal{B}^{\alpha}}}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})$
as $n\to\infty$. In view of (5.6), (a2), by (3.7) and the previous
construction of $\widetilde{K}^{i}_{n}$,

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

for every $i=0,\ldots,\lambda$. It remains to prove that

${\rm{sdist}}(\cdot,\partial
S_{h_{k_{n}}^{i},\widetilde{K}_{n}^{i}})\to{\rm{sdist}}(\cdot,\partial
S_{h^{i},K^{i}})$ (5.8)

locally uniformly in $\mathbb{R}^{2}$ as $n\to\infty$ for every
$i=0,\ldots,\alpha$. Let us fix $i=0,\ldots,\alpha$, by properties of the
signed distance convergence, it suffices to prove that
$S_{h^{i}_{k_{n}},\widetilde{K}^{i}_{n}}\xrightarrow{\mathcal{K}}S_{h^{i}}$
and that $\Omega\setminus
S_{h^{i}_{k_{n}},\widetilde{K}^{i}_{n}}\xrightarrow{\mathcal{K}}\Omega\setminus{\rm{Int}}(S_{h^{i},K^{i}})$.
On one hand, by the $\tau_{\mathcal{B}^{\alpha}}$-convergence of
$\\{(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}})\\}$,
the fact that
$\overline{S_{h^{i}_{k_{n}},\widetilde{K}^{i}_{n}}}=S_{h^{i}_{k_{n}}}$, and
the properties of Kuratowski convergence, it follows that
$S_{h^{i}_{k_{n}},\widetilde{K}^{i}_{n}}\xrightarrow{\mathcal{K}}S_{h^{i}}$.
On the other hand, let $x\in\Omega\setminus{\rm{Int}}(S_{h^{i},K^{i}})$, since

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

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

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

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

$\begin{split}x_{n}\in\Omega\setminus{\rm{Int}}\left(S_{h^{i}_{k_{n}},\widetilde{K}^{i}_{n}}\right)&=\Omega\setminus\left({\rm{Int}}({S_{h^{i}_{k_{n}}}})\setminus\widetilde{K}^{i}_{n}\right)\\\
&=\left(\Omega\setminus{\rm{Int}}\left(S_{h^{i}_{k_{n}},K^{i}_{k_{n}}}\right)\right)\cup\left(\left(\partial
S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{k_{n}}}\setminus\partial
S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}\right)\cap
S_{h^{i}_{k_{n}}}\right),\end{split}$ (5.9)

where in the last equality we used the definition of
$\widetilde{K}^{i}_{n}:=K^{i}_{k_{n}}\cup((\partial
S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{k_{n}}}\setminus\partial
S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}})\cap S_{h^{i}_{k_{n}}})$ and the
fact that
${\rm{Int}}(S_{h^{i}_{k_{n}},K^{i}_{k_{n}}})={\rm{Int}}(S_{h^{i}_{k_{n}}})\setminus
K^{i}_{k_{n}}$. Therefore, by (5.9) we deduce that $x_{n}\in\partial
S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{k_{n}}}\setminus\partial
S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}}$ for every $n\geq n_{\epsilon}$ and
hence, $x\in\partial S_{h^{\alpha},K^{\alpha}}$ by (a2) and by [42, Remark
3.7]. We reached an absurd as it follows that
$x\in{\rm{Int}}(S_{h^{i},K^{i}})\cap\partial
S_{h^{\alpha},K^{\alpha}}=\emptyset$. This concludes the proof of (5.8) and
hence, of the claim.

By (3.12) and by conditions (a1), (a4) and (5.7), we observe that

$\lim_{n\to\infty}{\absolutevalue{\mathcal{S}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}},\ldots,S_{h^{0}_{k_{n}},K^{0}_{k_{n}}})-\mathcal{S}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}})}}=0,$
(5.10)

and

$\mathcal{W}(S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}},\ldots,S_{h^{0}_{k_{n}},K^{0}_{k_{n}}},u_{k_{n}})=\mathcal{W}(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}}).$
(5.11)

By (3.14), (5.5), (5.7), (5.11), (a3) and thanks to the fact that
$\mathcal{S}^{\alpha}_{\sigma}$ is non-negative, we obtain that

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

for every $i\in I$, for $n$ large enough and for a constant
$C:=C(u_{0}^{1},\ldots,u_{0}^{\alpha})>0$. Therefore, by a diagonal argument
and by [35, Corollary 3.8] (applied to, with the notation of [35], $P=E_{i}$
and $P_{n}=E_{i}^{n}$) up to extracting not relabelled subsequences both for
$\\{u_{{k_{n}}}\\}\subset H^{1}_{{\rm loc}}(\Omega;\mathbb{R}^{2})$ and
$\\{E_{i}^{n}\\}_{n}$ there exist $w_{i}\in
H^{1}_{\rm{loc}}(E_{i},\mathbb{R}^{2})$, 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
$S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{k_{n}}}\setminus\bigcup_{i\in
I}{E_{i}^{n}}$ 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 $u\in
H^{1}_{\rm{loc}}({\rm{Int}}(S_{h^{\alpha},K^{\alpha}});\mathbb{R}^{2})$,
${b}_{n}$ is a rigid displacement associated to
$S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{k_{n}}}$,
$u_{k_{n}}+{b}_{n}\rightarrow u$ a.e. in
${\rm{Int}}(S_{h^{\alpha},K^{\alpha}})$ and hence,
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}^{\alpha}_{\mathbf{m}}$
and
$(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}}+{b}_{n})\xrightarrow{\tau_{\mathcal{C}}}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)$.
Furthermore, as $E(u_{k_{n}}+b_{n})=Eu_{k_{n}}$, from (5.10) and (5.11) it
follows that

$\lim_{n\to\infty}{\absolutevalue{\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}},\ldots,S_{h^{0}_{k_{n}},K^{0}_{k_{n}}},u_{k_{n}})-\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}}+b_{n})}}=0,$
(5.12)

which implies (3.18) and completes the proof. ∎

In the following proof, we show by induction that
$\mathcal{F}^{\alpha}_{\sigma}$ is lower semicontinuous.

###### Proof of Theorem 3.10.

Since

$\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k}):=\mathcal{S}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})+\mathcal{W}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k}),$

and by non-negativeness of $\mathcal{S}^{\alpha}_{\sigma}$ and $\mathcal{W}$
we prove first that $\mathcal{S}^{\alpha}_{\sigma}$ is lower semicontinuous
with respect to the convergence in $\tau_{\mathcal{B}^{\alpha}}$, and then we
prove that $\mathcal{W}$ is lower semicontinuous with respect to the
convergence in $\tau_{\mathcal{C}^{\alpha}}$.

First, we assume that $\sigma=t$. To prove that $\mathcal{S}^{\alpha}_{t}$ is
lower semicontinuous we proceed by induction on $\alpha\in\mathbb{N}$. Notice
that if $\alpha=1$, by Theorem 4.3 the assertion holds. Assume now that for
$\alpha=n$ the assertion of the theorem holds. We are going to prove that the
assertion of the theorem holds if $\alpha=n+1$. By definition of the energy
$\mathcal{S}^{n+1}$ we see that

$\begin{split}\mathcal{S}^{n+1}_{t}(S_{h^{n+1}_{k},K^{n+1}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})&=\mathcal{S}^{n}_{t}(S_{h^{n}_{k},K^{n}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\\\
&\quad+\sum_{j=0}^{n}\mathcal{S}^{(n+1,j)}(S_{h^{n+1}_{k},K^{n+1}_{k}},S_{h^{j}_{k},K^{j}_{k}})\end{split}$
(5.13)

for every $k\in\mathbb{N}$. By the induction hypothesis and the fact that

$(S_{h^{n}_{k},K^{n}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\xrightarrow{\tau_{\mathcal{B}^{n}}}(S_{h^{n},K^{n}},\ldots,S_{h^{0},K^{0}})$

we have that

$\mathcal{S}^{n}_{t}(S_{h^{n},K^{n}},\ldots,S_{h^{0},K^{0}})\leq\liminf_{k\to\infty}\mathcal{S}^{n}_{t}(S_{h^{n}_{k},K^{n}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}}).$
(5.14)

Furthermore, by the definition of $\tau_{\mathcal{B}^{n+1}}$-convergence it
follows that

$(S_{h^{n+1}_{k},K^{n+1}_{k}},S_{h^{j}_{k},K^{j}_{k}})\xrightarrow{\tau_{\mathcal{B}^{1}}}(S_{h^{n+1},K^{n+1}},S_{h^{j}_{k},K^{j}_{k}})$
(5.15)

for every $j=0,\ldots,n$. Thus, by (5.15) and by the basis of the induction
(see the proof Theorem 4.3) we deduce that

$\mathcal{S}^{(n+1,j)}(S_{h^{n+1},K^{n+1}},S_{h^{j}_{k},K^{j}_{k}})\leq\liminf_{k\to\infty}\mathcal{S}^{(n+1,j)}(S_{h^{n+1}_{k},K^{n+1}_{k}},S_{h^{j}_{k},K^{j}_{k}})$
(5.16)

for every $j=0,\ldots,n$. It follows from the superadditivity of the _liminf_
that

$\begin{split}\sum_{j=0}^{n}\mathcal{S}^{(n+1,j)}(S_{h^{n+1},K^{n+1}},S_{h^{j}_{k},K^{j}_{k}})&\leq\sum_{j=0}^{n}\liminf_{k\to\infty}\mathcal{S}^{(n+1,j)}(S_{h^{n+1}_{k},K^{n+1}_{k}},S_{h^{j}_{k},K^{j}_{k}})\\\
&\leq\liminf_{k\to\infty}\sum_{j=0}^{n}\mathcal{S}^{(n+1,j)}(S_{h^{n+1}_{k},K^{n+1}_{k}},S_{h^{j}_{k},K^{j}_{k}}),\end{split}$
(5.17)

where in the first inequality we used (5.16). Therefore, we have that

$\begin{split}\mathcal{S}^{n+1}_{t}(S_{h^{n+1},K^{n+1}},\ldots,S_{h^{0},K^{0}})&=\mathcal{S}^{n}_{t}(S_{h^{n},K^{n}},\ldots,S_{h^{0},K^{0}})\\\
&\quad+\sum_{j=0}^{n}\mathcal{S}^{(n+1,j)}(S_{h^{n+1},K^{n+1}},S_{h^{j},K^{j}})\\\
&\leq\liminf_{k\to\infty}\mathcal{S}^{n}_{t}(S_{h^{n}_{k},K^{n}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\\\
&\quad+\liminf_{k\to\infty}\sum_{j=0}^{n}\mathcal{S}^{(n+1,j)}(S_{h^{n+1}_{k},K^{n+1}_{k}},S_{h^{j}_{k},K^{j}_{k}})\\\
&\leq\liminf_{k\to\infty}\left(\mathcal{S}^{n}_{t}(S_{h^{n}_{k},K^{n}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\right.\\\
&\quad\left.+\sum_{j=0}^{n}\mathcal{S}^{(n+1,j)}(S_{h^{n+1}_{k},K^{n+1}_{k}},S_{h^{j}_{k},K^{j}_{k}})\right)\\\
&=\liminf_{k\to\infty}\mathcal{S}^{n+1}_{t}(S_{h^{n+1}_{k},K^{n+1}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}}),\end{split}$
(5.18)

where in the first and the second equality we used (5.13), in the first
inequality we used (5.14) and (5.17), and in the second inequality we used the
superadditivity of the _liminf_ , and thus, $\mathcal{S}^{\alpha}$ is lower
semicontinuous with respect to the topology $\tau_{\mathcal{B}^{\alpha}}$.

For the case with $\sigma=s$, we proceed also by induction and, on the basis
of the induction hypothesis with $\sigma=s$, in order to obtain the lower
semicontinuous property for $\mathcal{S}^{n+1}_{s}$, we proceed in the same
way, with the only difference that the subscript $t$ is replaced by $s$,
instead of (5.13) we use

$\begin{split}\mathcal{S}^{n+1}_{s}(S_{h^{n+1}_{k},K^{n+1}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})&=\mathcal{S}^{n}_{s}(S_{h^{n}_{k},K^{n}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\\\
&\quad+\mathcal{S}^{(n,n+1)}(S_{h^{n+1}_{k},K^{n+1}_{k}},S_{h^{n}_{k},K^{n}_{k}}),\end{split}$

and that, because of the definition of $\mathcal{S}^{n+1}_{s}$, it is enough
to use (5.16) only with $j=n$.

With respect to the elastic energy it is enough to repeat the same arguments
of the proof of lower semicontinuity of $\mathcal{W}$ in the proof of Theorem
4.3, from which we obtain that

$\mathcal{W}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u)\leq\liminf_{k\to\infty}\mathcal{W}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k}).$
(5.19)

Finally, we conclude the proof by observing that by the superadditivity of the
_liminf_ it follows that

$\begin{split}\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)&:=\mathcal{S}^{\alpha}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}})+\mathcal{W}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u)\\\
&\leq\liminf_{k\to\infty}\mathcal{S}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})\\\
&\quad\quad+\liminf_{k\to\infty}\mathcal{W}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})\\\
&\leq\liminf_{k\to\infty}\left(\mathcal{S}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}})+\mathcal{W}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})\right)\\\
&=:\liminf_{k\to\infty}\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k}),\end{split}$

where in the first inequality we used the lower semicontinuity of
$\mathcal{S}^{\alpha}$ and $\mathcal{W}$. ∎

Finally, we are now in a position to prove the main result of this paper.

###### Proof of Theorem 3.8.

We follow the _Direct Method of the Calculus of Variations_. Fix
$\mathbf{m}=(m_{0},\ldots,m_{\alpha})\in\mathbb{N}^{\alpha+1}$ and let
$\\{(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})\\}\subset\mathcal{C}^{\alpha}_{\mathbf{m}}$
be a minimizing sequence of $\mathcal{F}^{\alpha}_{\sigma}$ such that
$\mathcal{L}^{2}({S_{h^{i}_{k},K^{i}_{k}}})=\mathbbm{v}_{i}$ for
$i=0,\ldots,\alpha$, and

$\sup_{k\in\mathbb{N}}{\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})}<\infty.$

Since
$\mathcal{L}^{2}({S_{h^{\alpha}_{k},K^{\alpha}_{k}}})=\mathbbm{v}_{\alpha}$,
by Theorem 3.9 there exist a subsequence
$\\{(S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}},\ldots,S_{h^{0}_{k_{n}},K^{0}_{k_{n}}},u_{k_{n}})\\}$,
a sequence
$\\{(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},v_{n})\\}_{n\in\mathbb{N}}\subset\mathcal{C}^{\alpha}_{\mathbf{m}}$
and
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}^{\alpha}_{{\sigma},\mathbf{m}}$
such that

$(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},v_{n})\xrightarrow{\tau_{\mathcal{C}^{\alpha}}}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)$

as $n\to\infty$ and

$\liminf_{n\to\infty}{\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},v_{n})}=\liminf_{n\to\infty}{\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k_{n}},K^{\alpha}_{k_{n}}},\ldots,S_{h^{0}_{k_{n}},K^{0}_{k_{n}}},u_{k_{n}})}.$
(5.20)

According to Theorem 3.10, we have that

$\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\leq\liminf_{n\to\infty}{\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},v_{n})}.$
(5.21)

We claim that for every $i=0,\ldots,\alpha$,
$S_{h^{i}_{k_{n}},\widetilde{K}^{i}_{n}}$ and $S_{h^{i},\widetilde{K}^{i}}$
satisfy the volume constraints of (3.15). Indeed, by Theorem 3.9, for any
$n\in\mathbb{N}$,
$\mathbbm{v}_{i}=\mathcal{L}^{2}({S_{h^{i}_{k_{n}},K^{i}_{k_{n}}}})=\mathcal{L}^{2}({S_{h^{i}_{k_{n}},\widetilde{K}^{i}_{n}}})$
for every $i=0,\ldots,\alpha$. Fix $i=0,\ldots,\alpha$. By definition of
$\tau_{\mathcal{B}^{\alpha}}$-convergence and by applying [35, Lemma 3.2] we
infer that $S_{h^{i}_{k_{n}},\widetilde{K}^{i}_{n}}\to S_{h^{i},K^{i}}$ in
$L^{1}(\mathbb{R}^{2})$ as $n\to\infty$, and thus
$\mathcal{L}^{2}({S_{h^{i},K^{i}}})=\mathbbm{v}_{i}$. From (5.20) and (5.21),
we deduce that

$\begin{split}&\inf_{\small\begin{split}(&S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)\in\mathcal{C}^{\alpha}_{\mathbf{m}},\\\
&\mathcal{L}^{2}({S_{h^{i},K^{i}}})=\mathbbm{v}_{i},\,i=0,\ldots,\alpha\end{split}}{\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)}\\\
&\qquad=\lim_{n\to\infty}{\mathcal{F}^{\alpha}_{{\sigma}}(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}})}\\\
&\qquad\geq\liminf_{n\to\infty}{\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha}_{k_{n}},\widetilde{K}^{\alpha}_{n}},\ldots,S_{h^{0}_{k_{n}},\widetilde{K}^{0}_{n}},u_{k_{n}})}\geq\mathcal{F}^{\alpha}_{\sigma}(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u).\end{split}$

We conclude from the previous inequality that
$(S_{h^{\alpha},K^{\alpha}},\ldots,S_{h^{0},K^{0}},u)$ is a minimum of (3.15).

The same strategy is used to solve the unconstrained problem (3.16) thanks to
the extra observation that for any minimizing sequence
$\\{(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})\\}\subset\mathcal{C}^{\alpha}_{\mathbf{m}}$
of $\mathcal{F}^{\alpha,\bm{\lambda}}_{\sigma}$ such that

$\sup_{k\in\mathbb{N}}{\mathcal{F}^{\alpha,\bm{\lambda}}_{\sigma}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})}<\infty$

we have that

$\mathcal{L}^{2}(S_{h^{\alpha}_{k},K^{\alpha}_{k}})\leq\absolutevalue{\mathcal{L}^{2}(S_{h^{\alpha}_{k},K^{\alpha}_{k}})-\mathbbm{v}_{\alpha}}+\mathbbm{v}_{\alpha}\leq\frac{1}{\lambda_{\alpha}}\mathcal{F}^{\alpha,\bm{\lambda}}(S_{h^{\alpha}_{k},K^{\alpha}_{k}},\ldots,S_{h^{0}_{k},K^{0}_{k}},u_{k})+\mathbbm{v}_{\alpha}$

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

This concludes the proof. ∎

## Acknowledgments

The authors acknowledge the support received from the Austrian Science Fund
(FWF) projects P 29681 and TAI 293, from the Vienna Science and Technology
Fund (WWTF) together with the City of Vienna and Berndorf Privatstiftung
through Project MA16-005, and from BMBWF through the OeAD-WTZ project HR
08/2020. R. Llerena thanks for the support obtained by a French public grant
as part of the “Investissement d’avenir” project ANR-11-LABX-0056-LMH, LabEx
LMH. P. Piovano is member of the Italian “Gruppo Nazionale per l’Analisi
Matematica, la Probabilità e le loro Applicazioni” (GNAMPA) and has received
funding from the GNAMPA-INdAM 2022 project CUP: E55F22000270001 and 2023
Project CUP: E53C22001930001. P. Piovano also acknowledges the support
obtained by the Italian Ministry of University and Research (MUR) through the
PRIN Project “Partial differential equations and related geometric-functional
inequalities”. Finally, P. Piovano is also grateful for the support received
as _Visiting Professor and Excellence Chair_ and from the _Visitor in the
Theoretical Visiting Sciences Program_ (TSVP) at the Okinawa Institute of
Science and Technology (OIST), Japan.

## References

  * [1] F. J. Almgren, _Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints_ , Mem. Amer. Math. Soc. 4-165 (1976), viii+199.
  * [2] L. Ambrosio and A. Braides, _Functionals defined on partitions in sets of finite perimeter I: integral_ , J. Math. Pures Appl. 69 (1990), 285–305.
  * [3] by same author, _Functionals defined on partitions in sets of finite perimeter II: semicontinuity_ , J. Math. Pures Appl. 69 (1990), 307–333.
  * [4] L. Ambrosio, N. Fusco, and D. Pallara, _Functions of bounded variation and free discontinuity problems_ , Oxford University Press, New York, 2000.
  * [5] E.-Y. Andrei and A.-H. MacDonald, _Graphene bilayers with a twist_ , Nat. Mater 19 (2020), no. 12, 1265–1275.
  * [6] R. Asaro and W. Tiller, _Interface morphology development during stress corrosion cracking: Part I. Via surface diffusion_ , Metall. Trans. 3 (1972), 1789–1796.
  * [7] M. Bonacini and R. Cristoferi, _Area Quasi-minimizing Partitions with a Graphical Constraint: Relaxation and Two-Dimensional Partial Regularity_ , J Nonlinear Sci 32 (2022), no. 6, 93.
  * [8] D. Bucur, I. Fragalà, and A. Giacomini, _The Multiphase Mumford-Shah problem_ , SIAM J. Imaging Sci. 12 (2019), 1561–1583.
  * [9] by same author, _Multiphase free discontinuity problems: Monotonicity formula and regularity results_ , Ann. Inst. Henri Poincare (C) Anal. Non Lineaire 38 (2021), 1553–1582.
  * [10] D. Caraballo, _The triangle inequalities and lower semi-continuity of surface energy of partitions_ , Proc. R. Soc. Edinb. A: Math. 139 (2009), no. 3, 449–457.
  * [11] by same author, _BV-Ellipticity and Lower Semicontinuity of Surface Energy of Caccioppoli Partitions of $\mathbb{R}^{n}$_, J Geom Anal 23 (2013), 202–220.
  * [12] by same author, _Existence of surface energy minimizing partitions of $\mathbb{R}^{n}$ satisfying volume constraints_, Trans. Amer. Math. Soc. 369 (2016), 1517–1546.
  * [13] P. Cermelli and M.E. Gurtin, _The dynamics of solid-solid phase transitions 2. Incoherent interfaces_ , Arch. Rational Mech. Anal. 127 (1994), no. 1, 41–99.
  * [14] A. Chambolle and E. Bonnetier, _Computing the equilibrium configuration of epitaxially strained crystalline films_ , SIAM J. Appl. Math. 62 (2002), 1093–1121.
  * [15] A. Chambolle and V. Crismale, _Existence of strong solutions to the Dirichlet problem for the Griffith energy_ , Calc. Var. Partial Differential Equations 58 (2019), 136.
  * [16] B.G. Chirranjeevi, T.A. Abinandanan, and M.P. Gururajan, _A phase field study of morphological instabilities in multilayer thin films_ , Acta Mater. 57 (2009), no. 4, 1060–1067.
  * [17] M. Conti, S. Terracini, and G. Verzini, _A variational problem for the spatial segregation of reaction-diffusion systems_ , Indiana Univ. Math. J. 54-3 (2005), 779–815.
  * [18] G. Dal Maso, J. Morel, and S. Solimini, _A variational method in image segmentation: existence and approximation results_ , Acta Math. 168 (1992), 89–151.
  * [19] A Danescu, _The Asaro–Tiller–Grinfeld instability revisited_ , Int. J. Solids Struct. 38 (2001), 4671–4684.
  * [20] E. Davoli and P. Piovano, _Analytical validation of the Young–Dupré law for epitaxially-strained thin films_ , Math. Models Methods Appl. Sci 29 (2019), 2183–2223.
  * [21] by same author, _Derivation of a heteroepitaxial thin-film model_ , Interfaces Free Bound. 22 (2020), 1–26.
  * [22] G. De Philippis and F. Maggi, _Regularity of Free Boundaries in Anisotropic Capillarity Problems and the Validity of Young’s Law_ , Arch. Ration. Mech. Anal. 216 (2015), 473–568.
  * [23] L.C. Evans and R.F. Gariepy, _Measure Theory and Fine Properties of Functions_ , Taylor & Francis, New York, 1991.
  * [24] K. Falconer, _The geometry of fractal sets_ , no. 85, Cambridge University Press, Cambridge, 1986.
  * [25] I. Fonseca, N. Fusco, G. Leoni, and V. Millot, _Material voids in elastic solids with anisotropic surface energies_ , J. Math. Pures Appl. 96 (2011), 591–639.
  * [26] I. Fonseca, N. Fusco, G. Leoni, and M. Morini, _Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results_ , Arch. Ration. Mech. Anal. 186 (2007), 477–537.
  * [27] I. Fonseca and G. Leoni, _Modern methods in the Calculus of Variations: $L^{p}$-spaces_, Springer, New York, 2007.
  * [28] G.A. Francfort and J.-J. Marigo, _Revisiting brittle fracture as an energy minimization problem_ , Mech. Phys. Solids 46 (1998), 1319–1342.
  * [29] E. Fried and M. Gurtin, _A unified treatment of evolving interfaces accounting for small deformations and atomic transport with emphasis on grain-boundaries and epitaxy_ , Adv. Appl. Mech. 40 (2004), 1–177.
  * [30] M. Friedrich, M. Perugini, and F. Solombrino, _Lower semicontinuity for functionals defined on piecewise rigid functions and on GSBD_ , J. Funct. Anal. 280 (2021), 108929.
  * [31] A. Giacomini, _A generalization of Gołab’s theorem and applications to fracture mechanics_ , Math. Models Methods Appl. Sci. 12 (2002), 1245–1267.
  * [32] M.A Grinfeld, _The stress driven instability in elastic crystals: Mathematical models and physical manifestations_ , J. Nonlinear Sci. 3 (1993), 35–83.
  * [33] M.E. Gurtin, _The dynamics of solid-solid phase transitions 1. Coherent interfaces_ , Arch. Rational Mech. Anal. 123 (1993), no. 4, 305–335.
  * [34] Z.-F. Huang and R. C. Desai, _Stress-driven instability in growing multilayer films_ , Phys. Rev. B 67 (2003), 075416.
  * [35] Sh. Kholmatov and P. Piovano, _A Unified Model for Stress-Driven Rearrangement Instabilities_ , Arch. Rational Mech. Anal. (2020), 415–488.
  * [36] by same author, _Existence of minimizers for the SDRI model in 2d: wetting and dewetting regime with mismatch strain_ , Adv. Calc. (2023).
  * [37] by same author, _Existence of minimizers for the SDRI model in $\mathbb{R}^{n}$: Wetting and dewetting regimes with mismatch strain_, arXiv: Analysis of PDEs (2023).
  * [38] L. Kreutz and P. Piovano, _Microscopic Validation of a Variational Model of Epitaxially Strained Crystalline Films_ , SIAM J. Appl. Math. 53 (2021), 453–490.
  * [39] F. K. LeGoues, P. M. Mooney, and J. Tersoff, _Measurement of the activation barrier to nucleation of dislocations in thin films_ , Phys. Rev. Lett. 71 (1993), 396–399.
  * [40] A.A. León Baldelli, J.-F. Babadjian, B. Bourdin, D. Henao, and C. Maurini, _A variational model for fracture and debonding of thin films under in-plane loadings_ , J. Mech. Phys. Solids 70 (2014), 320–348.
  * [41] G. Leoni, _A first course in Sobolev spaces_ , Graduate studies in mathematics; 105, American Math. Soc., Providence, 2009.
  * [42] R. Llerena and P. Piovano, _Existence of minimizers for a two-phase free boundary problem with coherent and incoherent interfaces_ , Submitted (2023).
  * [43] F. Maggi, _Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory_ , Cambridge University Press, 2012\.
  * [44] P. Mattila, _Geometry of sets and measures in Euclidean spaces: fractals and rectifiability_ , Cambridge University Press, Cambridge, 1999.
  * [45] F. Morgan, _Lowersemicontinuity of energy clusters_ , Proc. R. Soc. Edinb. A: Math. 127 (1997), 819–822.
  * [46] D. Mumford and J. Shah, _Optimal approximations by piecewise smooth functions and associated variational problems_ , Comm. Pure Appl. Math. 42-5 (1989), 577–685.
  * [47] J. Prentice, _Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures_ , J. Phys. D: Appl. Phys. 33 (2000), no. 24, 3139.
  * [48] B. J. Spencer, _Asymptotic derivation of the glued-wetting-layer model and contact-angle condition for stranski-krastanow islands_ , Phys. Rev. B 59 (1999), 2011–2017.
  * [49] B. J. Spencer and J. Tersoff, _Equilibrium shapes and properties of epitaxially strained islands_ , Phys. Rev. Lett. 79 (1997), 4858–4861.
  * [50] N. Sridhar, J. M. Rickman, and D. J. Srolovitz, _Multilayer film stability_ , J. Appl. Phys. 82 (1997), no. 10, 4852–4859.
  * [51] D.J Srolovitz, _On the stability of surfaces of stressed solids_ , Acta Metal. 37 (1989), 621–625.
  * [52] D.J. Srolovitz and M.P. Anderson, _A criterion for compressive failure of a continuous, protective surface film_ , Acta Metall. 32 (1984), no. 7, 1089–1092.
  * [53] C. Teichert, M. G. Lagally, L. J. Peticolas, J. C. Bean, and J. Tersoff, _Stress-induced self-organization of nanoscale structures in sige/si multilayer films_ , Phys. Rev. B 53 (1996), 16334–16337.
  * [54] J. Tersoff, C. Teichert, and M. G. Lagally, _Self-organization in growth of quantum dot superlattices_ , Phys. Rev. Lett. 76 (1996), 1675–1678.