# Evolution of crystalline thin films by evaporation and condensation in three
dimensions

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

The morphology of crystalline thin films evolving on flat rigid substrates by
condensation of extra film atoms or by evaporation of their own atoms in the
surrounding vapor is studied in the framework of the theory of Stress Driven
Rearrangement Instabilities (SDRI). By following the SDRI literature both the
elastic contributions due to the mismatch between the film and the substrate
lattices at their theoretical (free-standing) elastic equilibrium, and a
curvature perturbative regularization preventing the problem to be ill-posed
due to the otherwise exhibited backward parabolicity, are added in the
evolution equation. The resulting Cauchy problem under investigation consists
in an anisotropic mean-curvature type flow of the fourth order on the film
profiles, which are assumed to be parametrizable as graphs of functions
measuring the film thicknesses, coupled with a quasistatic elastic problem in
the film bulks. Periodic boundary conditions are considered. The results are
twofold: the existence of a regular solution for a finite period of time and
the stability for all times, of both Lyapunov and asymptotic type, of any
configuration given by a flat film profile and the related elastic
equilibrium. Such achievements represent both the generalization to three
dimensions of a previous result in two dimensions for a similar Cauchy
problem, and the complement of the analysis previously carried out in the
literature for the symmetric situation in which the film evolution is not
influenced by the evaporation-condensation process here considered, but it is
entirely due to the volume preserving surface-diffusion process, which is
instead here neglected. The method is based on minimizing movements, which
allow to exploit the the gradient-flow structure of the evolution equation.

###### Key words and phrases:

Thin films, evaporation, condensation, mismatch strain, 3 dimensions, gradient
flow, curvature regularization, minimizing movements, existence, regularity,
Lyapunov stability, asymptotic stability.

###### 2010 Mathematics Subject Classification:

37B25, 35K25, 35K30, 35Q74, 49J10, 74K35, 74N20

##  1\. Introduction

In this paper we study the morphological evolution of crystalline thin films
deposited on flat rigid substrates by _condensation_ of extra film atoms from
a surrounding vapor, which results in a film growth, or by _evaporation_ in
such vapor of their own film atoms, which triggers instead a film dissolution.
Besides on the phenomena at the film surface, the focus is on the elastic
properties of the film bulk material, which is subject to the so-called
_epitaxial strain_ imparted by the underlying substrate. This is due to the
fact that, on the one hand, the film and the substrate material are assumed to
adhere at their contact interface without delamination or debonding, but on
the other hand, on such interface atoms are affected by the mismatch that can
occur between the atomic lattice characterizing the film free-standing elastic
equilibrium and the atomic lattice characterizing the substrate elastic
equilibrium.

The directions of investigation are twofold: the existence of a regular
evolution for a finite period of time and the stability for all times, of both
_Lyapunov_ and _asymptotic type_ , of any film configuration consisting of a
flat profile and the related elastic equilibrium, which are established in
Theorems 3.8 and 3.9, respectively. Such results can be seen both as a
generalization to the three dimensional setting, which is the physically
relevant one for the applications, of the results previously achieved in [32],
and as an equivalent analysis to the one carried out in [14] (see also [13])
for the complementary setting in which the film evolution is not influenced by
the evaporation-condensation process here considered, but it is entirely due
to the volume preserving _surface-diffusion process_ , which is instead here
neglected. We notice that in our setting the film volume can suddenly change
directly interfering with the elastic properties of the deposited film
material.

As in [13, 14, 32] the reference framework for the modeling aspects is
provided in the literature by the theory of _Stress Driven Rearrangement
Instabilities_ (SDRI) [3, 7, 15, 19, 35] of which thin films have historically
been the SDRI foremost example, because of the interplay occurring between the
film surface and elastic energy: Among the modes of stress-relief for the film
material, the bulk deformation is energetically neutral with respect to the
surface energy, but is payed in terms of the elastic energy, while the
destabilization of the film free-boundary from the _Winterbottom optimal
shape_ [33, 34, 36] with surface roughness and interface instabilities
positively contributes to the surface energy without storage of elastic
energy. A delicate compromise between these competing stress-relief mechanisms
must therefore be achieved, of which the film morphology is the result. A
microscopic justification of the static model starting from atomic
interactions has been provided in [29], while existence and regularity results
for dimension $d=2$ are available in [4, 8, 9, 11, 12, 18, 26, 27] and have
been (partially) extended in higher dimensions for models related to the more
general setting of SDRI in [6, 28].

In regard of the evolution theory, the reference goes back to W. W. Mullins
[30, 31], who identify the equations describing the motion of a crystalline
interface $\Gamma\subset\mathbb{R}^{d}$ by the evaporation-condensation
process and by surface diffusion as the _motion by mean curvature_ , i.e.,

$V=-H\qquad\text{on $\Gamma$,}$ (1.1)

and the _motion by the Laplacian of the mean curvature_ , i.e.,

$V=\Delta_{\Gamma}H\qquad\text{on $\Gamma$,}$ (1.2)

respectively, where $V$ denotes the normal velocity on $\Gamma$, $H$ is the
mean curvature on $\Gamma$, and $\Delta_{\Gamma}$ represents the tangential
Laplacian along $\Gamma$. As described in [32] by taking into account the
film-vapor interface anisotropy $\psi$ and the elastic contribution Equations
(1.1) and (1.2) become

$V=-\mathrm{div}_{\Gamma}(D\psi(\nu))-W(Eu)\qquad\text{on $\Gamma$},$ (1.3)

and

$V=\Delta_{\Gamma}[\mathrm{div}_{\Gamma}(D\psi(\nu))+W(Eu)]\qquad\text{on
$\Gamma$},$ (1.4)

(see [2, 15, 22, 21, 23, 24] and [20, Remark 3.1, Section 8] for more
details), where $u(\cdot,t)$ is the elastic equilibrium in the region
$\Omega\subset\mathbb{R}^{d}$, $Eu$ represents the _strain_ and is the
symmetric part of the gradient $Du$, $W$ is the elastic energy density which
is defined as

$W(A):=\frac{1}{2}\mathbb{C}A:A$

for every $A\in\mathbb{R}^{2\times 2}$ and for a symmetric positive fourth-
order tensor $\mathbb{C}$, $\mathrm{div}_{\Gamma}$ is the tangential
divergence along $\Gamma\subset\partial\Omega$, and $\nu$ is the outward unit
normal of $\partial\Omega$.

We notice that (1.4) has been tackled in [16, 17] by means of fixed point
techniques to prove existence and uniqueness results. More precisely, in [16]
the authors study (1.4) for $d=2$ and obtain short time existence and
uniqueness, establishing the global-in-time existence for a specific class of
initial data, while in [17] they prove short-time existence of a smooth
solution and uniqueness for $d=3$ (with the bulk contribution given by a
forcing term, which also include the elastic setting). Furthermore, in the
absence of elasticity the strategy based on _minimizing movements_ has been
extensively used previously in the literature to treat geometric flows of the
type (1.1) and (1.2), also in order to establish global-in-time existence.
Regarding the mean curvature flow, without aiming at a comprehensive
description of the results by minimizing movements we just mention here that
in [5] the authors obtain global-in-time existence and uniqueness for the
anisotropic crystalline mean curvature flow in all dimensions for arbitrary
(possibly unbounded) initial sets and for the natural mobility, while in [25]
global-in-time existence of the flat solutions for a mean curvature flow with
volume preserving is established.

Regarding the literature results for the Equations (1.3) and (1.4) carried out
by minimizing movements in the presence of elasticity we refer to [13, 14, 32]
where the setting of _evolving graphs_ is considered under an extra
perturbative _regularization term_ added to the equations. More precisely, in
[13, 14, 32] admissible film profiles are constrained to be parametrizable as
graph $\Gamma_{h}$ of functions $h:Q\times[0,T]\rightarrow[0,+\infty)$
measuring the thickness of the film, where $Q:=(0,\ell)^{d-1}$ and $[0,T]$
relate, for given $\ell>0$ and a time $T>0$, to the spatial and the temporal
variable, respectively. Furthermore, in [13, 14, 32] a regularization term is
added on the basis of the argumentation already present in [2, 23, 24] to
overcome the fact that Equations (1.3) and (1.4) are backward parabolic in the
presence of highly anisotropic non-convex surface tension $\psi$, making the
related Cauchy problem ill-posed. By adding such regularization term Equations
(1.3) and (1.4) become

$V=-\mathrm{div}_{\Gamma}(D\psi(\nu))-W(Eu)+\varepsilon
R(k_{1},k_{2})\qquad\text{on $\Gamma_{h}$}$ (1.5)

and

$V=\Delta_{\Gamma}[\mathrm{div}_{\Gamma}(D\psi(\nu))+W(Eu)-\varepsilon
R(k_{1},k_{2})]\qquad\text{on $\Gamma_{h}$},$ (1.6)

respectively, where $\varepsilon>0$ and $R$ is a function depending on the
principal curvatures $k_{1},k_{2}$ of $\Gamma_{h}$. By defining $R$ (also on
the basis of [10]) as

$R(k_{1},k_{2}):=\left(\Delta_{\Gamma}(|H|^{p-2}H)-|H|^{p-2}H\left(k_{1}^{2}+k_{2}^{2}-\frac{1}{p}H^{2}\right)\right)$
(1.7)

(where we recall that $H:=(k_{1}+k_{2})/2$) with $p=2$ for $d=2$ in [13, 32]
and with $p>2$ for $d=3$ in [14], the backward parabolicity of the equation is
avoided as surface patterns with large curvature get penalized. As such,
minimizing movements are then used in [13, 14, 32] so that the _gradient-flow
structure_ exhibited by Equations (1.5) and (1.6) can be exploited: by
considering the functional

$\mathcal{F}(h,u_{h}):=\int_{\Omega_{h}}W(Eu)\mathrm{d}z+\int_{\Gamma_{h}}\left(\psi(\nu)+\frac{\varepsilon}{p}|H|^{p}\right)\mathrm{d}\mathcal{H}^{2},$

where $u_{h}$ denotes the elastic equilibrium in $\Omega_{h}$ (under the
proper periodic and boundary conditions) and $\mathcal{H}^{2}$ denotes the
two-dimensional Hausdorff measure, then Equations (1.5) and (1.6) formally
coincide with the gradient flow of $\mathcal{F}$ with respect to an $L^{2}$\-
and $H^{-1}$-Riemannian structure. This allowed to establish short-time
existence of a regular solution of (1.5) in [32] for $d=2$, and of (1.6) in
[13] and [14] for $d=2$ and $d=3$, respectively. It remains open the case of
(1.5) for $d=3$ that we intend here to tackle by also choosing, as in [14],
$p>2$ in (1.7).

Therefore, by recalling that the normal velocity parametrized as the graph of
the thickness functions $h$ is given by

$V=\frac{1}{\sqrt{1+|Dh|^{2}}}\frac{\partial h}{\partial t}$ (1.8)

on $\Gamma_{h}$, where $Dh$ denotes the gradient with respect to the spatial
coordinates, the _Cauchy problem_ under investigation depending on the period
of time $T>0$ is the following:

$\begin{cases}\frac{1}{\sqrt{1+|Dh|^{2}}}\frac{\partial h}{\partial
t}=-\mathrm{div}_{\Gamma}(D\psi(\nu))-W(Eu)\\\ \hskip
110.96556pt+\varepsilon\left(\Delta_{\Gamma}(|H|^{p-2}H)-|H|^{p-2}H\left(|B|^{2}-\frac{1}{p}H^{2}\right)\right)&\quad\text{in
$\mathbb{R}^{2}\times[0,T]$, }\\\ \mathrm{div}(\mathbb{C}Eu)=0&\quad\text{in
$\Omega_{h}$},\\\ \mathbb{C}Eu[\nu]=0&\quad\text{on $\Gamma_{h}$},\\\
u(x,0,t)=(e^{1}_{0}x_{1},e^{2}_{0}x_{2},0),\\\ h(\cdot,t)\text{ and
}Du(\cdot,t)\text{ are $Q$-periodic},\\\ h(\cdot,0)=h_{0},\end{cases}$ (1.9)

where $e_{0}:=(e_{0}^{1},e_{0}^{2})$ with $e_{0}^{i}>0$ is a vector which
represents the mismatch between the crystalline lattices of the film and the
substrate, $h_{0}$ is an admissible profile of the film at the initial time
$t=0$, and spatial $Q$-periodic conditions are considered. We notice that the
Cauchy problem on the period of time $T>0$ considered in [14] coincides with
(1.9) if we replace the first equation in (1.9) with (1.6) (by taking into
account (1.8)).

We can now more precisely detail the achieved results and we refer instead to
Section 3.4 for their full characterization: In Theorem 3.8 we establish the
existence of a time $T_{0}>0$ for which the Cauchy problem (1.9) admits a
solution in $[0,T_{0}]$, which is said to be _regular_ in the meaning that the
first equation in (1.9) is satisfied a.e. in $\mathbb{R}^{2}\times[0,T_{0}]$
(see Definition 3.5), while in Theorem 3.9 we differentiate the setting of a
nonconvex and of a convex surface tension $\psi$ as in [14, Section 4] and we
prove that any admissible flat configuration as from Definition 3.2 is
Lyapunov stable both in the convex and nonconvex case, and we can establish
that is asymptotic stable only in the convex case in analogy to the results of
[14].

###  1.1. Organization of the paper and description of the method

In Section 2 we fix the notation adopted throughout the paper. In Section 3 we
introduce the precise mathematical setting of the model, we introduce the
minimizing-movement scheme obtained by discretizing the time interval and by
defining the incremental family of minimum problems (3.5) at each discrete
time, we define the stability properties of the solutions of (1.9), and we
state the two main theorems of the paper, i.e., Theorems 3.8 and 3.9. In
Section 4 we show the existence of minimizers for (3.5) in Theorem 4.1 so that
a discrete-time evolution as in Definition 3.3 can be constructed, of which
then we analyze the convergence properties in Theorems 4.2 and 4.3, that then
are improved in Theorem 4.4 by selecting a specific time $T_{0}>0$. In Section
5 we prove in Theorem 3.8 the existence of a regular solution in the time
interval $[0,T_{0}]$ by passing, in view of the previously proven convergence
properties, to the discrete _Euler-Lagrange equation_ (5.1) satisfied by the
minimizers of the incremental minimum problems (3.5) (see Lemma 5.1), from
which in Theorem 5.2 we are able to deduce a crucial uniform bound on the mean
curvature. In Section 6 we prove Theorem 3.9 by following the program of [14,
Section 4] that includes the differentiation of the settings in which $\psi$
is a nonconvex and convex surface tension. Finally, in Section 7 we collect
some auxiliary results often used in the paper for the Reader’s convenience.

##  2\. Notation

In this section we set the main notation used throughout the paper. The space
of $m\times d$ matrices with real entries is denoted by $\mathbb{R}^{m\times
d}$, and, in case $m=d$, the subspace of symmetric matrices is denoted by
$\mathbb{R}^{d\times d}_{sym}$. Given a function
$u\colon\mathbb{R}^{d}\to\mathbb{R}^{m}$, we denote its Jacobian matrix by
$Du$, whose components are $(Du)_{ij}:=\partial_{j}u_{i}$ for $i=1,\dots,m$
and $j=1,\dots,d$, and when $u\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$, we also
denote by $Eu$ the symmetric part of the gradient, i.e.,
$Eu:=\frac{1}{2}(Du+Du^{T})$. Given a tensor field
$A\colon\mathbb{R}^{d}\to\mathbb{R}^{m\times d}$, by $\mathop{\rm
div}\nolimits A$ we mean its divergence with respect to the rows, namely
$(\mathop{\rm div}\nolimits A)_{i}:=\sum_{j=1}^{d}\partial_{j}A_{ij}$ for
$i=1,\dots,m$.

The norm of a generic Banach space $X$ is denoted by $\|\cdot\|_{X}$ and in
case $X$ is a Hilbert space, we denote by $\langle\cdot,\cdot\rangle_{X}$ its
inner product. In the case $X=\mathbb{R}^{d}$, we simplify the notation by
using the symbol $\langle\cdot,\cdot\rangle$ to denote the euclidean scalar
product, that is $\langle v,w\rangle:=\sum_{i=1}^{d}v_{i}w_{i}$ for every
$v,w\in\mathbb{R}^{d}$, while $A:B:=\sum_{i,j=1}^{d}a_{ji}b_{ij}$ denotes the
Hilbert-Schmidt product of two matrices $A,B\in\mathbb{R}^{d\times d}$. Given
two matrices $A\in\mathbb{R}^{m\times d}$ and $B\in\mathbb{R}^{n\times q}$ we
denote by $A\otimes B\in\mathbb{R}^{mn\times dq}$ the Kronecker product
between the two matrices $A$ and $B$, defined as the block matrix $A\otimes
B:=(a_{ij}B)_{i,j}$. Given two Banach spaces $X_{1}$ and $X_{2}$, the space of
linear and continuous maps from $X_{1}$ to $X_{2}$ is denoted by
$\mathscr{L}(X_{1};X_{2})$. For any $\mathbb{A}\in\mathscr{L}(X_{1};X_{2})$
and $u\in X_{1}$, we indicate the image of $u$ under $\mathbb{A}$ with
$\mathbb{A}u\in X_{2}$.

We denote the $d$-dimensional Lebesgue measure by $\mathcal{L}^{d}$ and the
$(d-1)$-dimensional Hausdorff measure by $\mathcal{H}^{d-1}$. Given a bounded
open set $\Omega$ with Lipschitz boundary, $\nu$ indicates the outer unit
normal vector of $\partial\Omega$, which is defined $\mathcal{H}^{d-1}$-a.e.
and we employ the usual definition of Lebesgue and Sobolev spaces on $\Omega$.
The values of Sobolev functions on the boundary of their set of definition are
always intended in the sense of traces. Given a set $U$, then $U^{2}:=U\times
U$ with $\times$ denoting the cartesian product. Finally, given an open
interval $(a,b)\subset\mathbb{R}$ and $p\in[1,\infty]$, we denote by
$L^{p}(a,b;X)$ the space of $L^{p}$ functions from $(a,b)$ to $X$. We use
$H^{k}(a,b;X)$ and $W^{k,p}(a,b;X)$ to denote the Sobolev space of functions
from $(a,b)$ to $X$ with $k$ weak derivatives in $L^{2}(a,b;X)$ and
$L^{p}(a,b;X)$, respectively.

##  3\. Mathematical setting and main results

In this section we introduce the model, the main definitions, and the
statements of the main results.

###  3.1. The variational model

Let us define $Q:=(0,\ell)^{2}$ with $\ell>0$. For $p>2$ we denote by
$W^{2,p}_{\\#}(Q)$ the space of all functions of $W^{2,p}(Q)$ whose
$Q$-periodic extension belong to $W^{2,p}_{loc}(\mathbb{R}^{2})$, and we
define the class of admissible profiles as

$AP:=\\{h\in W^{2,p}_{\\#}(Q):h(x)\geq 0\\},$

where we use the notation $x=(x_{1},x_{2})\in Q$.

Moreover, for $h\in W^{2,p}_{\\#}(Q)$ we refer to the sets

$\Gamma_{h}:=\\{z=(x,h(x)):x\in Q\\}\text{ and }\Omega_{h}:=\\{z=(x,y)\in
Q\times\mathbb{R}:0<y<h(x)\\}$ (3.1)

as the _film profile_ and the _region of the film with height_ $h$,
respectively, while the corresponding sets with $Q$ replaced by
$\mathbb{R}^{2}$ are denoted by $\Gamma^{\\#}_{h}$ and $\Omega^{\\#}_{h}$. We
define the _family of periodic displacements_ by

$PD:=\\{u:\Omega^{\\#}_{h}\rightarrow\mathbb{R}^{3}:u(x,y)=u(x+\ell k,y)\text{
for every }(x,y)\in\Omega^{\\#}_{h}\text{ and }k\in\mathbb{Z}^{2}\\}$

and the _the family of admissible displacements_ by

$AD_{h}:=\\{u\in L^{2}_{loc}(\Omega^{\\#}_{h},\mathbb{R}^{3})\cap
PD:u(x,0)=(e_{0}^{1}x_{1},e_{0}^{2}x_{2},0),\text{ }Eu_{|\Omega_{h}}\in
L^{2}(\Omega_{h},\mathbb{R}^{3})\\},$

where $e_{0}:=(e_{0}^{1},e_{0}^{2})$, with $e_{0}^{1},e_{0}^{2}>0$, is the
vector representing the mismatch between the free-standing equilibrium
crystalline lattice of the film and the substrate materials. Consequently, the
_family of admissible configurations_ is

$X:=\\{(h,u):h\in AP\text{ and }u\in AD_{h}\\}.$

The _elastic energy density_ $W:\mathbb{R}^{2\times
2}_{sym}\rightarrow[0,\infty)$ is defined by

$W(M):=\frac{1}{2}\mathbb{C}M:M,$

where $\mathbb{C}$ is a fourth-order tensor such that there exists $k>0$ for
which

$\mathbb{C}M:M\geq 2kM:M\quad\text{for every $M\in\mathbb{R}^{2\times
2}_{sym}$}.$

Furthermore, let $\psi:\mathbb{R}^{3}\rightarrow[0,+\infty)$ be a function of
class $C^{2}$ on $\mathbb{R}^{3}\setminus\\{0\\}$ and positively one-
homogeneous, which satisfies

$\frac{1}{c}|\xi|\leq\psi(\xi)\leq c|\xi|$ (3.2)

for every $\xi\in\mathbb{R}^{3}$ and for some positive constant $c$.

The _configuration energy functional_ $\mathcal{F}:X\rightarrow[0,\infty]$ is
given by

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

for every admissible configuration $(h,u)\in X$, where
$\mathcal{W}:X\rightarrow[0,\infty]$ and $\mathcal{S}:AP\rightarrow[0,\infty]$
represent the _elastic_ and the _surface energy_ , respectively. The elastic
energy is defined by

$\mathcal{W}(h,u):=\int_{\Omega_{h}}W(Eu)\mathrm{d}z$

while the surface energy is defined by

$\mathcal{S}(h):=\int_{\Gamma_{h}}\left(\psi(\nu)+\frac{\varepsilon}{p}|H|^{p}\right)\mathrm{d}\mathcal{H}^{2},$

where $\nu$ is the outer unit normal to $\Omega_{h}$, $\varepsilon$ is a
positive constant, and $H=\mathop{\rm div}\nolimits_{\Gamma_{h}}\nu$ denotes
the sum of the principal curvatures of $\Gamma_{h}$, i.e.,

$H=-\mathop{\rm
div}\nolimits\left(\frac{Dh}{\sqrt{1+|Dh|^{2}}}\right)\quad\text{ in $Q.$}$
(3.3)

Notice that from (3.3) it follows that

$\int_{Q}H\mathrm{d}x=0.$

######  Definition 3.1 (Elastic equilibrium).

We say that $\bar{u}\in AD_{h}$ is the _elastic equilibrium_ of $\bar{h}\in
AP$ if

$\mathcal{F}(\bar{h},\bar{u})=\min\\{\mathcal{F}(\bar{h},u):(\bar{h},u)\in
X\\}.$

Notice that the elastic equilibrium exixts for every $\bar{h}\in AP$ and it is
unique in view of the Dirichlet condition.

######  Definition 3.2 (Flat configuration).

We say that an admissible configuration $(h,u)\in X$ is a flat configuration
if $h\equiv d>0$ and $u=u_{d}$, where $u_{d}$ represents the elastic
equilibrium in $\Omega_{d}$.

###  3.2. The incremental minimum problem

Let $(h_{0},u_{0})\in X$ be such that $h_{0}>0$ and $u_{0}$ is the elastic
equilibrium of $h_{0}$, and let $\Lambda_{0}$ be a positive constant such that

$\|h_{0}\|_{C^{1}_{\\#}(Q)}<\Lambda_{0}.$ (3.4)

By considering a sequence $\tau_{N}\searrow 0$, for every $i\in\mathbb{N}$ we
define inductively $(h_{i,N},u_{i,N})$ as a solution to the following minimum
problem:

$\min\\{G_{i,N}(h,u):(h,u)\in X,\text{
}\|Dh\|_{L^{\infty}(Q)}\leq\Lambda_{0}\\}.$ (3.5)

The functional $G_{i,N}$ is given by

$G_{i,N}(h,u):=\mathcal{F}(h,u)+P_{i,N}(h),$ (3.6)

with the _penalization term_ $P_{i,N}$ defined by

$P_{i,N}(h):=\frac{1}{2\tau_{N}}\int_{\Gamma_{h_{i-1,N}}}\left(\frac{h-h_{i-1,N}}{J_{i-1,N}}\right)^{2}\mathrm{d}\mathcal{H}^{2}=\frac{1}{2\tau_{N}}\int_{Q}\frac{(h-h_{i-1,N})^{2}}{J_{i-1,N}}\mathrm{d}x,$
(3.7)

where $J_{i-1,N}:=\sqrt{1+|Dh_{i-1,N}|^{2}}$. It is proved in Theorem 4.1 that
problem (3.5) admits a minimizer.

######  Definition 3.3 (Dicrete-time evolution).

Let $(h_{0},u_{0})\in X$ be an initial configuration and for
$i,N\in\mathbb{N}$ let $(h_{i,N},u_{i,N})$ be a solution to (3.5). We refer to
the piecewise linear interpolation
$h_{N}:\mathbb{R}^{2}\times\mathbb{R}\rightarrow[0,\infty)$, given by

$h_{N}(x,t):=h_{i,N}(x)+\frac{1}{\tau_{N}}(t-(i-1)\tau_{N})(h_{i,N}(x)-h_{i-1,N}(x))$
(3.8)

for every $(x,t)\in\mathbb{R}^{2}\times[(i-1)\tau_{N},i\tau_{N}]$ and
$i\in\mathbb{N}$, as a _discrete-time evolution_ of the incremental minimum
problem (3.5). Moreover, we denote with $u_{N}(\cdot,t)$ the elastic
equilibrium corresponding to $h_{N}(\cdot,t)$.

In Theorem 3.8 we make use of a different type of interpolation of solution to
(3.5).

######  Definition 3.4.

Let $(h_{0},u_{0})\in X$ be an initial configuration. Given a solution
$(h_{i,N},u_{i,N})$ for $i,N\in\mathbb{N}$ to (3.5), we define the piecewise
constant interpolations
$\tilde{h}_{N}:\mathbb{R}^{2}\times\mathbb{R}\rightarrow[0,\infty)$ and
$\tilde{u}_{N}:\mathbb{R}^{3}\times\mathbb{R}\rightarrow\mathbb{R}^{3}$ in the
following way:

$\displaystyle\tilde{h}_{N}(x,t):=h_{i,N}(x)$ $\displaystyle\text{for
}(x,t)\in\mathbb{R}^{2}\times[(i-1)\tau_{N},i\tau_{N})\text{ and
}i\in\mathbb{N},$ (3.9) $\displaystyle\tilde{u}_{N}(x,y,t):=u_{i,N}(x,y)$
$\displaystyle\text{for
}(x,y,t)\in\Omega_{h_{i,N}}\times[(i-1)\tau_{N},i\tau_{N})\text{ and
}i\in\mathbb{N}$ (3.10)

###  3.3. Solutions of the evolution problem and properties

We now introduce the notion of the solutions to (1.9) and stability
properties, that will be proven to apply under certain conditions to the flat
configuration.

######  Definition 3.5 (Solutions of the evolution problem).

Let $(h_{0},u_{0})\in X$ be a configuration satisfying (3.4) and $T>0$. We say
that a function $h\in L^{\infty}(0,T;W^{2,p}_{\\#}(Q))\cap
H^{1}(0,T;L^{2}_{\\#}(Q))$ such that

  1. (i)

$h(\cdot,0)=h_{0}(\cdot)$ in $Q$,

  2. (ii)

$-\mathop{\rm
div}\nolimits_{\Gamma}(D\psi(\nu))-W(Eu)+\varepsilon\left(\Delta_{\Gamma}(|H|^{p-2}H)-|H|^{p-2}H\left(|B|^{2}-\frac{1}{p}H^{2}\right)\right)\in
L^{2}(0,T;L^{2}_{\\#}(Q))$,

  3. (iii)

the equation

$\frac{1}{J}\frac{\partial h}{\partial t}=-\mathop{\rm
div}\nolimits_{\Gamma}(D\psi(\nu))-W(Eu)+\varepsilon\left(\Delta_{\Gamma}(|H|^{p-2}H)-|H|^{p-2}H\left(|B|^{2}-\frac{1}{p}H^{2}\right)\right)$

is satisfied for $\mathcal{L}^{3}$-a.e. in $Q\times(0,T)$,

where $J:=\sqrt{1+|Dh|^{2}}$, $\Gamma:=\Gamma_{h(\cdot,t)}$, $u(\cdot,t)$ is
the elastic equilibrium in $\Omega_{h(\cdot,t)}$, and $W(Eu)$ denotes the
trace of $W(Eu)$ on $\Gamma_{h(\cdot,t)}$, is a _regular solution_ to (1.9) in
$[0,T]$ with initial datum $h_{0}$.

######  Definition 3.6 (Variational solutions).

We say that a solution $h$ to (1.9) (in the sense of Definition 3.5) is
variational if there exist a time step $\tau_{N}\searrow 0$ and a subsequence
$\\{h_{N_{k}}\\}$ of a discrete-time evolution $\\{h_{N}\\}$ of the minimum
incremental problem (3.5) such that

$\displaystyle h_{N_{k}}\xrightharpoonup[k\to\infty]{}h\text{ in
$H^{1}(0,T,L^{2}(Q))$},$ $\displaystyle
h_{N_{k}}\xrightarrow[k\to\infty]{}h\text{ in
$C^{0,\beta}([0,T],C^{1,\alpha}_{\\#}(Q))$}$ (3.11)

for every $\alpha\in(0,\frac{p-2}{p})$ and $\beta\in[0,\frac{(p+2)(p-2-\alpha
p)}{8p^{2}})$.

######  Definition 3.7 (Stability of flat configurations).

We say that the flat configuration $(d,u_{d})$ is:

  * •

Lyapunov stable if, for every $\sigma>0$, there exists $\delta(\sigma)>0$ such
that, if $(h_{0},u_{0})\in X$ satisfies
$\mathcal{L}^{3}(\Omega_{h_{0}})=\mathcal{L}^{3}(\Omega_{d})$ and
$\|h_{0}-d\|_{W^{2,p}_{\\#}(Q)}\leq\delta(\sigma)$, then every variational
solution $h$ to (1.9) with initial datum $h_{0}$ (according to Definition 3.6)
exists for all times and satisfies

$\|h(\cdot,t)-d\|_{W^{2,p}_{\\#}(Q)}\leq\sigma$

for every $t>0$;

  * •

Asymptotically stable if there exists $\delta>0$ such that, if
$(h_{0},u_{0})\in X$ satisfies
$\mathcal{L}^{3}(\Omega_{h_{0}})=\mathcal{L}^{3}(\Omega_{d})$ and
$\|h_{0}-d\|_{W^{2,p}_{\\#}(Q)}\leq\delta$, then every variational solution
$h$ to (1.9) with initial datum $h_{0}$ (according to Definition 3.6) exists
for all times and satisfies

$\|h(\cdot,t)-d\|_{W^{2,p}_{\\#}(Q)}\to 0$

as $t\to+\infty$.

###  3.4. Main results

The main results of the manuscript are twofolds: we prove the existence of a
solution of (1.9) and stability properties of flat configurations.

######  Theorem 3.8 (Short time existence of a regular solution).

Let $h_{0}\in AP$ with $h_{0}>0$ and such that $h_{0}$ satisfies (3.4). There
exist $T_{0}>0$ and a regular solution $h$ to (1.9) in $[0,T]$ with initial
datum $h_{0}$ in the sense of Definition (3.5). Moreover, there exist a
nonincreasing function $g$ and a negligible set $Z_{0}$ such that

$\mathcal{F}(h(\cdot,t),u_{h}(\cdot,t))=g(t)\quad\text{for every
$t\in[0,T_{0}]\setminus Z_{0}$}$ (3.12)

and

$\mathcal{F}(h(\cdot,t),u_{h}(\cdot,t))\leq g(t)\quad\text{for every $t\in
Z_{0}$}.$ (3.13)

We notice that the solution $h$ and the time $T_{0}$ are given by Theorem 4.2
and Theorem 4.4, respectively.

In regard to stability, we prove that, under some convexity assumption on the
density $\psi$, flat configurations are both Lyapunov and asymptotically
stable. In particular, it is necessary that for every $\xi\in\mathbb{S}^{2}$
we have

$\langle D^{2}\psi(\xi)w,w\rangle>0\quad\text{for every $w\perp\xi$, $w\neq
0,$}$ (3.14)

and that the flat configuration $(d,u_{d})$ satisfies

$\partial^{2}G(d,u_{d})[\varphi]>0\quad\text{for every
$\varphi\in\tilde{H}^{1}_{\\#}(Q)\setminus\\{0\\}$},$ (3.15)

where

$G(h,u)=\int_{\Omega_{h}}W(Eu)\mathrm{d}z+\int_{\Gamma_{h}}\psi(\nu)\mathrm{d}\mathcal{H}^{2}\quad\text{and}\quad\tilde{H}^{1}_{\\#}:=\left\\{\varphi\in
H^{1}_{\\#}(Q):\int_{Q}\varphi\mathrm{d}x=0\right\\}.$

In the nonconvex case we can prove that if the boundary of the Wulff shape
$W_{\psi}$, that is defined (depending on the nonconvex density $\psi$) by

$W_{\psi}:=\\{z\in\mathbb{R}^{3}:z\cdot\nu<\psi(\nu)\text{ for every $\nu\in
S^{2}$}\\},$

contains a flat horizontal facet, then the flat configuration is always
Lyapunov stable.

######  Theorem 3.9 (Stability of the flat configuration).

Let $\psi:\mathbb{R}^{3}\to[0,+\infty)$ be a positively one-homogeneous
function satisfying (3.2). The following assertions hold true:

  1. (i)

if $\psi\in C^{2}(\mathbb{R}^{3}\setminus\\{0\\})$ and there exist
$\beta,\gamma>0$ such that

$\\{(x,y)\in\mathbb{R}^{3}:|x|\leq\beta,y=\gamma\\}\subset\partial W_{\psi},$

then, for every $d>0$ the flat configuration $(d,u_{d})$ is Lyapunov stable;

  2. (ii)

if $\psi\in C^{3}(\mathbb{R}^{3}\setminus\\{0\\})$ and (3.14) and (3.15) are
satisfied, then, $(d,u_{d})$ is Lyapunov and asymptotic stable.

##  4\. Incremental minimum problem: Existence and convergence

In this section, we construct by applying a minimizing movement scheme a
candidate to be a solution to (1.9). In Theorem 4.1 we solve the incremental
minimum problem. Then, in Theorem 4.2 and 4.3 we prove that, up to a
subsequence, the discrete-time evolution $\\{h_{N}\\}$ converges to some
function $h$ as $N\to\infty$. Finally, in Theorem 4.4 we choose $T_{0}$ small
enough to get the validity of (4.24) and that $h_{N}$ is nonnegative.

The following result allows to verify that the incremental minimum problem is
well defined, as, for each $i\in\mathbb{N}$, we can recursively find a
solution to the minimum problem (3.5).

######  Theorem 4.1.

Let $(h_{0},u_{0})\in X$ be an initial configuration such that $h_{0}$
satisfies (3.4). Then, the minimum problem (3.5) admits a solution
$(h_{i,N},u_{i,N})\in X$ for every $i\in\mathbb{N}$.

###### Proof.

We prove the theorem by induction. We fix $i\in\mathbb{N}$ such that $i>1$ and
we suppose to have a solution to (3.5) for $k=1,...,i-1$. We construct a
solution to (3.5) for $k=i$. Firstly, we begin by noticing that, in view of
the minimality of $(h_{k,N},u_{k,N})$, by (3.6) and (3.7) we have that

$\mathcal{F}(h_{k,N},u_{k,N})\leq G_{k,N}(h_{k,N},u_{k,N})\leq
G_{k,N}(h_{k-1,N},u_{k-1,N})=\mathcal{F}(h_{k-1,N},u_{k-1,N}),$ (4.1)

from which we deduce that

$0\leq\inf\\{G_{i,N}(h,u):(h,u\in X)\\}\leq
G_{i,N}(h_{i-1,N},u_{i-1,N})=\mathcal{F}(h_{i-1,N},u_{i-1,N})\leq...\leq\mathcal{F}(h_{0},u_{0}).$
(4.2)

As a consequence of (4.2), we can select a minimizing sequence
$\\{(h_{n},u_{n})\\}\in X$ such that

$\|Dh_{n}\|_{L^{\infty}(Q)}\leq\Lambda_{0}\quad\text{ and
}\quad\sup\\{G_{i,N}(h_{n},u_{n}):n\in\mathbb{N}\\}<\infty.$

By denoting with $H_{n}$ the sum of principal curvatures of $\Gamma_{h_{n}}$,
we obtain that

$\sup\\{\|H_{n}\|^{p}_{L^{p}(Q)}:n\in\mathbb{N}\\}\leq\frac{p}{\varepsilon}\sup\\{G_{i,N}(h_{n},u_{n}):n\in\mathbb{N}\\}<\infty,$

and so $\\{H_{n}\\}$ is bounded in $L^{p}(Q)$. Therefore, by Lemma 7.3 the
sequence $\\{h_{n}\\}$ is bounded in $W^{2,p}_{\\#}(Q)$. Then, up to a
subsequence, $h_{n}\rightharpoonup h$ in $W^{2,p}_{\\#}(Q)$ as $n\to\infty$,
from which we deduce that $h_{n}\rightarrow h$ in $C^{1,\alpha}_{\\#}(Q)$ for
some $\alpha\in(0,1)$ as $n\to\infty$. It follows that

$H_{n}\xrightharpoonup[n\to\infty]{L^{p}(Q)}H=-\mathop{\rm
div}\nolimits\left(\frac{Dh}{\sqrt{1+|Dh|^{2}}}\right).$

Furthermore, by lower semicontinuity

$\int_{\Gamma_{h}}\left(\psi(\nu)+\frac{\varepsilon}{p}|H|^{p}\right)\mathrm{d}\mathcal{H}^{2}\leq\liminf_{n\to\infty}\int_{\Gamma_{h_{n}}}\left(\psi(\nu_{n})+\frac{\varepsilon}{p}|H_{n}|^{p}\right)\mathrm{d}\mathcal{H}^{2},$
(4.3)

and by Fatou’s Lemma we conclude that

$P_{i,N}(h)\leq\liminf_{n\to\infty}P_{i,N}(h_{n}).$ (4.4)

Now we extend the function $u_{n}$ to $A:=Q\times(-\infty,0)$ in the following
way

$u_{n}(x,y):=(e_{0}^{1}x_{1},e_{0}^{2}x_{2},0)\varphi(y)$

for every $(x,y)\in A$, where $\varphi$ is a given cut-off function such that
$\varphi(y):=1$ in $(-1,+\infty)$ and $\varphi(y):=0$ in $(-\infty,-2)$. We
consider also the set

$A_{h}:=\\{(x,y)\in\mathbb{R}^{3}:y<h(x)\\}.$

Since

$\sup\left\\{\int_{\Omega_{h_{n}}}|Eu_{n}|^{2}\mathrm{d}z:n\in\mathbb{N}\right\\}<\infty,$

by reasoning as in [12, Proposition 2.2], we use Korn’s inequality to get the
existence of a subsequence of $\\{u_{n}\\}_{n}$, and of a function $u\in
H^{1}_{loc}(A_{h};\mathbb{R}^{3})$ with $Eu\in L^{2}(A_{h};\mathbb{R}^{3\times
3}_{sym})$, such that $u_{n}\rightharpoonup u$ in $H^{1}(D;\mathbb{R}^{3})$ as
$n\to\infty$, for every $D$ compactly contained in $A_{h}$. Then, $(h,u)\in X$
and we also have

$\int_{\Omega_{h}}W(Eu)\mathrm{d}z\leq\liminf_{n\to\infty}\int_{\Omega_{h_{n}}}W(Eu_{n})\mathrm{d}z.$
(4.5)

Thanks to (4.3), (4.4), and (4.5) we can conclude that $(h,u)$ is a minimizer
of (3.5) for $k=i$. ∎

We now prove that the discrete-time evolution $\\{h_{N}\\}$ is uniformly
bounded in $L^{\infty}(0,T;W^{2,p}_{\\#}(Q))\cap H^{1}(0,T;L^{2}(Q))$.

######  Theorem 4.2.

Let $(h_{0},u_{0})\in X$ be an initial configuration such that $h_{0}$
satisfies (3.4). For every $N,i\in\mathbb{N}$ we have

$\displaystyle\int_{0}^{+\infty}\int_{Q}\left|\frac{\partial h_{N}}{\partial
t}\right|^{2}\mathrm{d}x\mathrm{d}t\leq CF(h_{0},u_{0}),$ (4.6)
$\displaystyle\mathcal{F}(h_{i,N},u_{i,N})\leq\mathcal{F}(h_{i-1,N},u_{i-1,N})\leq\mathcal{F}(h_{0},u_{0}),$
(4.7)
$\displaystyle\sup_{t\in[0,+\infty)}\|h_{N}(\cdot,t)\|_{W^{2,p}_{\\#}(Q)}<+\infty,$
(4.8)

for some constant $C=C(\Lambda_{0})>0$. Moreover, up to a subsequence, for
every $T>0$ we have the following convergences

$\displaystyle h_{N}\xrightharpoonup[N\to\infty]{}h\text{ in
$H^{1}(0,T,L^{2}(Q))$},$ (4.9) $\displaystyle
h_{N}\xrightarrow[N\to\infty]{}h\text{ in $C^{0,\alpha}([0,T],L^{2}(Q))$ for
every $\alpha\in(0,\tfrac{1}{2})$},$ (4.10)

where the function $h$ satisfies

$h(\cdot,t)\in W^{2,p}_{\\#}(Q)\text{ and
}\mathcal{F}(h(\cdot,t),u_{h(\cdot,t)})\leq\mathcal{F}(h_{0},u_{0})\text{ for
every $t\in[0,+\infty)$}.$ (4.11)

###### Proof.

We begin by observing that (4.7) is a direct consequence of (4.1).

To prove (4.6) we notice that for each $i\in\mathbb{N}$, by (4.1) we have that

$G_{i,N}(h_{i,N},u_{i,N})\leq
G_{i,N}(h_{i-1,N},u_{i-1,N})=\mathcal{F}(h_{i-1,N},u_{i-1,N}),$

which implies

$P_{i,N}(h_{i,N})\leq\mathcal{F}(h_{i-1,N},u_{i-1,N})-\mathcal{F}(h_{i,N},u_{i,N}).$
(4.12)

Since $\|Dh_{i-1,N}\|_{L^{\infty}(Q)}<\Lambda_{0}$, by summing in (4.12) over
$i$ we get

$\sum_{i=0}^{+\infty}\tau_{N}\int_{Q}\left(\frac{h_{i,N}-h_{i-1,N}}{\tau_{N}}\right)^{2}\mathrm{d}x\leq
C(\Lambda_{0})\sum_{i=0}^{+\infty}[\mathcal{F}(h_{i-1,N},u_{i-1,N})-\mathcal{F}(h_{i,N},u_{i,N})]=C(\Lambda_{0})\mathcal{F}(h_{0},u_{0}),$
(4.13)

with $C(\Lambda_{0}):=2\sqrt{1+\Lambda_{0}^{2}}$. Furthermore, by (3.8) we
have

$\frac{\partial h_{N}}{\partial t}=\frac{h_{i,N}-h_{i-1,N}}{\tau_{N}},$

and hence, by also (4.13) we obtain (4.6).

To prove (4.8), we notice that (4.7) implies

$\sup\left\\{\int_{\Gamma_{h_{i,N}}}|H_{i,N}|^{p}\mathrm{d}\mathcal{H}^{2}:i,N\in\mathbb{N}\right\\}<\frac{p}{\varepsilon}\mathcal{F}(h_{0},u_{0}),$
(4.14)

from which, by taking into account that
$\|Dh_{i,N}\|_{L^{\infty}(Q)}<\Lambda_{0}$ and by Lemma 7.3, (4.8) follows.

Notice that (4.9) is a direct consequence of (4.6) and (4.8). It remains to
prove (4.10) and (4.11). Since $h_{N}(x,\cdot)$ is absolutely continuous on
$[0,T]$ for every $T>0$, then by Holder inequality, Fubini’s Theorem, and
(4.6), for every $t_{1},t_{2}\in[0,T]$ we obtain

$\displaystyle\|h_{N}(\cdot,t_{1})-h_{N}(\cdot,t_{2})\|_{L^{2}(Q)}$
$\displaystyle\leq\left(\int_{Q}\left(\int_{t_{1}}^{t_{2}}\frac{\partial
h_{N}}{\partial t}(x,t)\mathrm{d}t\right)^{2}\mathrm{d}x\right)^{\frac{1}{2}}$
$\displaystyle\leq\left(\int_{t_{1}}^{t_{2}}\left\|\frac{\partial
h_{N}}{\partial
t}(\cdot,t)\right\|_{L^{2}(Q)}^{2}\mathrm{d}t\right)^{\frac{1}{2}}(t_{2}-t_{1})^{\frac{1}{2}}\leq\sqrt{CF(h_{0},u_{0})}(t_{2}-t_{1})^{\frac{1}{2}}.$
(4.15)

By using (4.15) and Ascoli-Arzelà Theorem (see e.g. [1, Proposition 3.3.1]) we
get (4.10).

Finally, by (4.7), (4.9), (4.10), and the lower semicontinuity we obtain
(4.11). ∎

In the following result the convergence of $h_{N}$ to $h$ is significantly
improved, and we prove also that $\tilde{h}_{N}$ converges to $h$.

######  Theorem 4.3.

Let $(h_{0},u_{0})\in X$ be an initial configuration such that $h_{0}$
satisfies (3.4). For every $T>0$ we have the following convergences

$\displaystyle h_{N}\xrightarrow[N\to\infty]{}h\text{ in
$C^{0,\beta}([0,T],C^{1,\alpha}_{\\#}(Q))$},$ (4.16)
$\displaystyle\tilde{h}_{N}\xrightarrow[N\to\infty]{}h\text{ in
$L^{\infty}(0,T,C^{1,\alpha}_{\\#}(Q))$},$ (4.17)

for every $\alpha\in(0,\frac{p-2}{p})$ and $\beta\in[0,\frac{(p-2-\alpha
p)(p+2)}{8p^{2}})$. Moreover, $h(\cdot,t)\rightarrow h_{0}$ in
$C^{1,\alpha}_{\\#}(Q)$ as $t\to 0^{+}$.

###### Proof.

Let $t_{1},t_{2}\in[0,T]$ be such that $t_{1}<t_{2}$, then by (4.8), (4.15),
and the interpolation inequality of Lemma 7.4 we have

$\displaystyle\|Dh_{N}(\cdot,t_{2})$ $\displaystyle-
Dh_{N}(\cdot,t_{1})\|_{L^{\infty}(Q)}$ $\displaystyle\leq
C\|D^{2}h_{N}(\cdot,t_{2})-D^{2}h_{N}(\cdot,t_{1})\|_{L^{p}(Q)}^{\frac{p+2}{2p}}\|h_{N}(\cdot,t_{2})-h_{N}(\cdot,t_{1})\|_{L^{p}(Q)}^{\frac{p-2}{2p}}$
$\displaystyle\leq
C\|h_{N}(\cdot,t_{2})-h_{N}(\cdot,t_{1})\|_{L^{p}(Q)}^{\frac{p-2}{2p}}$
$\displaystyle\leq
C\left(\|D^{2}h_{N}(\cdot,t_{2})-D^{2}h_{N}(\cdot,t_{1})\|^{\frac{p-2}{2p}}_{L^{2}(Q)}\|h_{N}(\cdot,t_{2})-h_{N}(\cdot,t_{1})\|_{L^{2}(Q)}^{\frac{p+2}{2p}}\right)^{\frac{p-2}{2p}}$
$\displaystyle\leq C(t_{2}-t_{1})^{\frac{p^{2}-4}{8p^{2}}}.$ (4.18)

Moreover, thanks to Mean Value Theorem there exists $x_{0}\in Q$ such that

$h_{N}(x_{0},t_{2})-h_{N}(x_{0},t_{1})=\frac{1}{\ell^{2}}\int_{Q}(h_{N}(x,t_{2})-h_{N}(x,t_{1}))\mathrm{d}x,$

from which we deduce that for every $x\in Q$ we have

$|h_{N}(x,t_{2})-h_{N}(x,t_{1})|\leq\ell^{2}\|Dh_{N}(\cdot,t_{2})-Dh_{N}(\cdot,t_{1})\|_{L^{\infty}(Q)}+\frac{1}{\ell}\|h_{N}(\cdot,t_{2})-h_{N}(\cdot,t_{1})\|_{L^{2}(Q)}.$
(4.19)

Now, by (4.15), (4), and (4.19) we get

$\|h_{N}(\cdot,t_{2})-h_{N}(\cdot,t_{1})\|_{L^{\infty}(Q)}\leq
C\left[(t_{2}-t_{1})^{\frac{p^{2}-4}{8p^{2}}}+(t_{2}-t_{1})^{\frac{1}{2}}\right].$
(4.20)

Furthermore, notice that for every $\alpha\in(0,\frac{p-2}{p})$ we have

$[Dh_{N}(\cdot,t_{2})-Dh_{N}(\cdot,t_{1})]_{\alpha}\leq[Dh_{N}(\cdot,t_{2})-Dh_{N}(\cdot,t_{1})]^{\frac{\alpha
p}{p-2}}_{\frac{p-2}{p}}\left(2\|Dh_{N}(\cdot,t_{2})-Dh_{N}(\cdot,t_{1})\|_{L^{\infty}(Q)}\right)^{\frac{p-2-\alpha
p}{p-2}}$ (4.21)

where $[\cdot]_{\alpha}$ denotes the $\alpha$-Holder seminorm, and by Sobolev
embedding and (4.8) we obtain

$\sup\left\\{\sup_{t\in[0,T]}\|h_{N}(\cdot,t)\|_{C^{1,\frac{p-2}{p}}_{\\#}(Q)}:\text{
}N\in\mathbb{N}\right\\}<+\infty.$ (4.22)

By (4), (4.21), and (4.22) we can write

$[Dh_{N}(\cdot,t_{2})-Dh_{N}(\cdot,t_{1})]_{\alpha}\leq
C(t_{2}-t_{1})^{\frac{(p+2)(p-2-\alpha p)}{8p^{2}}}.$ (4.23)

Therefore, it follows from (4), (4.20), and (4.23), that for every
$\alpha\in(0,\frac{p-2}{p})$, $h_{N}$ is uniformly equicontinuous with respect
to the $C^{1,\alpha}(Q)$-norm topology and that

$\|h_{N}(\cdot,t_{2})-h_{N}(\cdot,t_{1})\|_{C^{1,\alpha}(Q)}\leq
C(t_{2}-t_{1})^{\frac{(p+2)(p-2-\alpha p)}{8p^{2}}}.$

In particular, by applying Ascoli-Arzelà Theorem we get both (4.16) and
(4.17). Finally, by noticing that
$\|h_{N}(\cdot,t)-h_{N}(\cdot,t_{1})\|_{C^{1,\alpha}(Q)}\to 0$ as $t\to
t_{1}$, we choose $t_{1}=0$ to conclude the proof. ∎

From now on, we assume that the initial profile $h_{0}$ is strictly positive.
Thanks to this, we can use standard elliptic estimates to establish the
convergence of $u_{N}$ and $\tilde{u}_{N}$.

######  Theorem 4.4.

Let $(h_{0},u_{0})\in X$ be an initial configuration such that $h_{0}>0$
satisfies (3.4). Then there exist $T_{0}>0$ and $C>0$ depending only on
$(h_{0},u_{0})$ such that:

  1. (i)

$h_{N},h\geq C_{0}>0$ for some positive constant $C_{0}$, and

$\sup_{t\in[0,T_{0}]}\|Dh_{N}(\cdot,t)\|_{L^{\infty}(Q)}<\Lambda_{0}$ (4.24)

for every $N\in\mathbb{N}$;

  2. (ii)

it holds

$\|Du_{i,N}\|_{C^{0,\frac{p-2}{p}}(\bar{\Omega}_{h_{i,N}};\mathbb{R}^{3\times
3}_{sym})}\leq C;$ (4.25)

  3. (iii)

for every $\alpha\in(0,\frac{p-2}{p})$ and $\beta\in[0,\frac{(p+2)(p-2-\alpha
p)}{8p^{2}})$ we have the following convergences:

$\displaystyle
E(u_{N}(\cdot,h_{N}))\xrightarrow[N\to\infty]{}E(u(\cdot,h))\text{ in
$C^{0,\beta}([0,T_{0}];C^{1,\alpha}_{\\#}(Q))$},$ (4.26) $\displaystyle
E(\tilde{u}_{N}(\cdot,\tilde{h}_{N}))\xrightarrow[N\to\infty]{}E(u(\cdot,h))\text{
in $L^{\infty}(0,T_{0};C^{1,\alpha}_{\\#}(Q))$},$ (4.27)

where $u(\cdot,t)$ is the elastic equilibrium in $\Omega_{h(\cdot,t)}$.

###### Proof.

We begin by observing that by inserting $t_{2}=t$ and $t_{1}=0$ in (4.20) we
can write

$h_{N}(x,t)\geq\min_{x\in
Q}h_{0}(x)-C\left[t^{\frac{p^{2}-4}{8p^{2}}}+t^{\frac{1}{2}}\right]\qquad\text{for
every $(x,t)\in Q\times[0,T]$}.$

Therefore, since $t\mapsto t^{\frac{p^{2}-4}{8p^{2}}}+t^{\frac{1}{2}}$ is an
increasing function, there exists $T_{1}>0$ such that $h_{N}(x,t)\geq C_{0}$
for every $(x,t)\in Q\times[0,T_{1}]$ and for some constant
$0<C_{0}<\min\\{h_{0}(x):x\in Q\\}$. As a consequence $h\geq C_{0}$. Now, we
notice that in view of (4) with $t_{2}=t$ and $t_{1}=0$ we get

$\|Dh_{N}(\cdot,t)\|_{L^{\infty}(Q)}\leq\|Dh_{N}(\cdot,t)-Dh_{0}(\cdot)\|_{L^{\infty}(Q)}+\|Dh_{0}\|_{L^{\infty}(Q)}\leq
Ct^{\frac{p^{2}-4}{8p^{2}}}+\|h_{0}\|_{C^{1}_{\\#}(Q)},$

and hence, by (3.4) we can choose $T_{2}>0$ such that

$T_{2}<\left(\frac{\Lambda_{0}-\|h_{0}\|_{C^{1}_{\\#}(Q)}}{C}\right)^{\frac{8p^{2}}{p^{2}-4}},$

from which we deduce that

$\sup_{t\in[0,T_{2}]}\|Dh_{N}(\cdot,t)\|_{L^{\infty}(Q)}<\Lambda_{0}.$

By choosing $T_{0}:=\min\\{T_{1},T_{2}\\}$ we conclude the proof of (i).

To prove (4.25), we use standard elliptic estimates to bound the norm of
$Du_{i,N}$ in $C^{0,\alpha}(\bar{\Omega}_{h_{i,N}})$ with a constant depending
only on the $C^{1,\alpha}$-norm of $h_{i,N}$ (see [18]). Then, (4.22) implies
(4.25).

Finally, by (4.16), (4.17), and Lemma 7.1, (4.26) and (4.27) follows. ∎

##  5\. Existence of evolution

In this section we prove that the candidate found in Section 4 is a solution
to (1.9) in the sense of the Definition 3.5 in the case of short time
intervals (see Theorem 3.8).

In the next lemma, thanks to the regularity given by Theorem 4.4, we derive
the Euler-Lagrange equation satisfied by the minimizer $(h_{i,N},u_{i,N})$ to
the problem (3.5).

######  Lemma 5.1.

Every minimizer $(h_{i,N},u_{i,N})$ of (3.5) satisfies the following Euler-
Lagrange equation

$\displaystyle\int_{Q}W(E(u_{i,N}(x,h_{i,N}(x))))\varphi\mathrm{d}x+\int_{Q}\langle
D\psi(-Dh_{i,N},1),(-D\varphi,0)\rangle\mathrm{d}x+\frac{\varepsilon}{p}\int_{Q}|H_{i,N}|^{p}\frac{\langle
Dh_{i,N},D\varphi\rangle}{J_{i,N}}\mathrm{d}x$
$\displaystyle-\varepsilon\int_{Q}|H_{i,N}|^{p-2}H_{i,N}\left[\Delta\varphi-\frac{\langle
D^{2}\varphi Dh_{i,N},Dh_{i,N}\rangle}{J_{i,N}^{2}}-\frac{\Delta
h_{i,N}\langle Dh_{i,N},D\varphi\rangle}{J_{i,N}^{2}}-2\frac{\langle
D^{2}h_{i,N}Dh_{i,N},D\varphi\rangle}{J_{i,N}^{2}}\right]\mathrm{d}x$
$\displaystyle-3\varepsilon\int_{Q}|H_{i,N}|^{p-2}H_{i,N}\frac{\langle
Dh_{i,N},D\varphi\rangle\langle
D^{2}h_{i,N}Dh_{i,N},Dh_{i,N}\rangle}{J_{i,N}^{4}}\mathrm{d}x+\frac{1}{\tau_{N}}\int_{Q}\frac{h_{i,N}-h_{i-1,N}}{J_{i-1,N}}\varphi\mathrm{d}x=0,$
(5.1)

for every $\varphi\in C_{\\#}^{2}(Q)$.

###### Proof.

Since $(h_{i,N},u_{i,N})$ is the minimizer of (3.5), it satisfies

$\frac{\mathrm{d}}{\mathrm{d}s}\left(\mathcal{F}(h_{i,N}(s),u_{i,N})+\frac{1}{2\tau_{N}}\int_{Q}\frac{(h_{i,N}(s)-h_{i-1,N})^{2}}{J_{i-1,N}}\mathrm{d}x\right)_{|s=0}=0$
(5.2)

for every $\varphi\in C^{2}_{\\#}(Q)$, where $h_{i,N}(s):=h_{i,N}+s\varphi$.

By considering the penalization, we have

$\frac{\mathrm{d}}{\mathrm{d}s}\left(\frac{1}{2\tau_{N}}\int_{Q}\frac{(h_{i,N}(s)-h_{i-1,N})^{2}}{J_{i-1,N}}\mathrm{d}x\right)_{|s=0}=\frac{1}{\tau_{N}}\int_{Q}\frac{(h_{i,N}-h_{i-1,N})\varphi}{J_{i-1,N}}\mathrm{d}x.$
(5.3)

For the elastic energy, we use the definition (3.1) and we get

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}\left(\int_{\Omega_{h_{i,N}+s\varphi}}W(Eu_{i,N}(x,y))\mathrm{d}x\mathrm{d}y\right)_{|s=0}=\frac{\mathrm{d}}{\mathrm{d}s}\left(\int_{Q}\int_{0}^{h_{i,N}(s)}W(Eu_{i,N}(x,y))\mathrm{d}y\mathrm{d}x\right)_{|s=0}$
$\displaystyle=\int_{Q}[W(Eu_{i,N}(x,h_{i,N}(x,s))\varphi(x)]_{|s=0}\mathrm{d}x=\int_{Q}W(Eu_{i,N}(x,h_{i,N}(x))\varphi(x)\mathrm{d}x.$
(5.4)

It remains to consider the surface energy. We begin by observing that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}\left(\int_{Q}\psi(-Dh_{i,N}(s),1)\mathrm{d}x\mathrm{d}y\right)_{|s=0}$
$\displaystyle=\int_{Q}[\langle
D\psi(-Dh_{i,N}(s),1),(-D\varphi,0)\rangle]_{|s=0}\mathrm{d}x=\int_{Q}\langle
D\psi(-Dh_{i,N},1),(-D\varphi,0)\rangle\mathrm{d}x,$ (5.5)

and that the regularization term can be treated in the following way

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}\left(\int_{\Gamma_{h_{i,N}(s)}}|H_{h_{i,N}(s)}|^{p}\mathrm{d}\mathcal{H}^{2}\right)_{|s=0}=\frac{\mathrm{d}}{\mathrm{d}s}\left(\int_{Q}f_{i,N}(s)g_{i,N}(s)\mathrm{d}x\mathrm{d}y\right)_{|s=0}$
$\displaystyle=\int_{Q}\left[\frac{\mathrm{d}f_{i,N}}{\mathrm{d}s}(0)g_{i,N}(0)+f_{i,N}(0)\frac{\mathrm{d}g_{i,N}}{\mathrm{d}s}(0)\right]\mathrm{d}x\mathrm{d}y,$
(5.6)

where

$\displaystyle g_{i,N}(s):=\sqrt{1+|Dh_{i,N}(s)|^{2}}\quad\text{ and }\quad
f_{i,N}(s):=\left|\mathop{\rm
div}\nolimits\left(\frac{Dh_{i,N}(s)}{g_{i,N}(s)}\right)\right|^{p}$

Notice that

$\displaystyle\frac{\mathrm{d}g_{i,N}}{\mathrm{d}s}(s)=\frac{1}{g_{i,N}(s)}\langle
Dh_{i,N}(s),D\varphi\rangle\Longrightarrow\frac{\mathrm{d}g_{i,N}}{\mathrm{d}s}(0)=\frac{\langle
Dh_{i,N},D\varphi\rangle}{J_{i,N}},$ (5.7)
$\displaystyle\frac{\mathrm{d}f_{i,N}}{\mathrm{d}s}(s)=p|H_{h_{i,N}(s)}|^{p-2}H_{h_{i,N}(s)}\frac{\mathrm{d}}{\mathrm{d}s}H_{h_{i,N}(s)}\Longrightarrow\frac{\mathrm{d}f_{i,N}}{\mathrm{d}s}(0)=p|H_{i,N}|^{p-2}H_{i,N}\frac{\mathrm{d}}{\mathrm{d}s}(H_{h_{i,N}(s)})_{|s=0}.$
(5.8)

By definition we have

$\displaystyle H_{h_{i,N}(s)}$ $\displaystyle=-\mathop{\rm
div}\nolimits\left(\frac{Dh_{i,N}(s)}{g_{i,N}(s)}\right)=-\sum_{k=1}^{2}\frac{\partial}{\partial
x_{k}}\left(\frac{\frac{\partial h_{i,N}(s)}{\partial
x_{k}}}{g_{i,N}(s)}\right)$
$\displaystyle=-\sum_{k=1}^{2}\left[\frac{1}{g_{i,N}(s)}\frac{\partial^{2}h_{i,N}(s)}{\partial
x_{k}^{2}}+\frac{\partial h_{i,N}(s)}{\partial x_{k}}\frac{\partial}{\partial
x_{k}}\left(\frac{1}{g_{i,N}(s)}\right)\right]$
$\displaystyle=-\sum_{k=1}^{2}\left[\frac{1}{g_{i,N}(s)}\frac{\partial^{2}h_{i,N}(s)}{\partial
x_{k}^{2}}-\frac{1}{2g_{i,N}^{3}(s)}\frac{\partial h_{i,N}(s)}{\partial
x_{k}}\sum_{m=1}^{2}\frac{\partial}{\partial x_{k}}\left(\frac{\partial
h_{i,N}(s)}{\partial x_{m}}\right)^{2}\right]$ $\displaystyle=-\frac{\Delta
h_{i,N}(s)}{g_{i,N}(s)}+\frac{\langle
D^{2}h_{i,N}(s)Dh_{i,N}(s),Dh_{i,N}(s)\rangle}{g_{i,N}^{3}(s)}:=A(s)+B(s),$

and hence, we can write

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}(H_{h_{i,N}(s)})_{|s=0}=\frac{\mathrm{d}A}{\mathrm{d}s}(0)+\frac{\mathrm{d}B}{\mathrm{d}s}(0).$
(5.9)

Now, we compute the derivative of $A$ and $B$ with respect to $s$. Since

$\displaystyle\frac{\mathrm{d}A(s)}{\mathrm{d}s}=\frac{-\Delta\varphi
g_{i,N}(s)+\frac{\Delta h_{i,N}(s)}{g_{i,N}(s)}\langle
Dh_{i,N}(s),D\varphi\rangle}{g_{i,N}^{2}(s)}=\frac{-\Delta\varphi
g_{i,N}^{2}(s)+\Delta h_{i,N}(s)\langle
Dh_{i,N}(s),D\varphi\rangle}{g_{i,N}^{3}(s)},$

we get

$\frac{\mathrm{d}A}{\mathrm{d}s}(0)=-\frac{\Delta\varphi}{J_{i,N}}+\frac{\Delta
h_{i,N}\langle Dh_{i,N},D\varphi\rangle}{J_{i,N}^{3}}.$ (5.10)

Moreover, by setting

$C(s):=\sum_{k=1}^{2}\frac{\partial h_{i,N}(s)}{\partial
x_{k}}\sum_{m=1}^{2}\frac{\partial^{2}h_{i,N}(s)}{\partial x_{k}\partial
x_{m}}\frac{\partial h_{i,N}(s)}{\partial x_{m}},$

in view of (5.7) we have

$\displaystyle\frac{\mathrm{d}B}{\mathrm{d}s}(0)$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}s}\left(\frac{C(s)}{g_{i,N}^{3}(s)}\right)_{|s=0}=\frac{\frac{\mathrm{d}C}{\mathrm{d}s}(0)g_{i,N}^{3}(0)-C(0)\frac{\mathrm{d}}{\mathrm{d}s}(g_{i,N}^{3}(s))_{|s=0}}{g_{i,N}^{6}(0)}$
$\displaystyle=\frac{1}{J_{i,N}^{3}}\frac{\mathrm{d}C}{\mathrm{d}s}(0)-\frac{3\langle
D^{2}h_{i,N}Dh_{i,N},Dh_{i,N}\rangle\langle
Dh_{i,N}(s),D\varphi\rangle}{J_{i,N}^{5}}.$ (5.11)

It remains to study the last term $C$:

$\displaystyle\frac{\mathrm{d}C(s)}{\mathrm{d}s}$
$\displaystyle=\sum_{k=1}^{2}\frac{\partial\varphi}{\partial
x_{k}}\sum_{m=1}^{2}\frac{\partial^{2}h_{i,N}(s)}{\partial x_{k}\partial
x_{m}}\frac{\partial h_{i,N}(s)}{\partial x_{m}}+\sum_{k=1}^{2}\frac{\partial
h_{i,N}(s)}{\partial
x_{k}}\sum_{m=1}^{2}\frac{\mathrm{d}}{\mathrm{d}s}\left(\frac{\partial^{2}h_{i,N}(s)}{\partial
x_{k}\partial x_{m}}\frac{\partial h_{i,N}(s)}{\partial x_{m}}\right)$
$\displaystyle=\langle
D^{2}h_{i,N}(s)Dh_{i,N}(s),D\varphi\rangle+\sum_{k=1}^{2}\frac{\partial
h_{i,N}(s)}{\partial
x_{k}}\sum_{m=1}^{2}\left(\frac{\partial^{2}\varphi}{\partial x_{k}\partial
x_{m}}\frac{\partial h_{i,N}(s)}{\partial
x_{m}}+\frac{\partial^{2}h_{i,N}(s)}{\partial x_{k}\partial
x_{m}}\frac{\partial\varphi}{\partial x_{m}}\right)$ $\displaystyle=\langle
D^{2}h_{i,N}(s)Dh_{i,N}(s),D\varphi\rangle+\langle D^{2}\varphi
Dh_{i,N}(s),Dh_{i,N}(s)\rangle+\langle
D^{2}h_{i,N}(s)D\varphi,Dh_{i,N}(s)\rangle$ $\displaystyle=2\langle
D^{2}h_{i,N}(s)Dh_{i,N}(s),D\varphi\rangle+\langle D^{2}\varphi
Dh_{i,N}(s),Dh_{i,N}(s)\rangle,$

and hence

$\frac{\mathrm{d}C}{\mathrm{d}s}(0)=2\langle
D^{2}h_{i,N}Dh_{i,N},D\varphi\rangle+\langle D^{2}\varphi
Dh_{i,N},Dh_{i,N}\rangle.$ (5.12)

By (5) and (5.12) we obtain

$\frac{\mathrm{d}B}{\mathrm{d}s}(0)=\frac{2\langle
D^{2}h_{i,N}Dh_{i,N},D\varphi\rangle+\langle D^{2}\varphi
Dh_{i,N},Dh_{i,N}\rangle}{J_{i,N}^{3}}-\frac{3\langle
D^{2}h_{i,N}Dh_{i,N},Dh_{i,N}\rangle\langle
Dh_{i,N}(s),D\varphi\rangle}{J_{i,N}^{5}}.$ (5.13)

Finally, from (5.2)-(5.10) and (5.13) the assertion follows. ∎

Now let $\widetilde{H}_{N}:\mathbb{R}^{2}\times[0,T_{0}]\rightarrow\mathbb{R}$
be the function defined by

$\widetilde{H}_{N}(x,t):=H_{i,N}(x,h_{i,N}(x))\quad\text{ for
$t\in[(i-1)\tau_{N},i\tau_{N})$},$ (5.14)

where $H_{i,N}$ is the sum of the principal curvatures of $\Gamma_{i,N}$.
Moreover, we will denote by $|B_{i,N}|^{2}$ the sum of the squares of the
principal curvatures of $\Gamma_{i,N}$, and by $J_{i,N}$ the following
quantity

$J_{i,N}:=\sqrt{1+|Dh_{i,N}|^{2}},$

######  Theorem 5.2.

Let $T_{0}$ be as in Theorem 4.4 and let $\widetilde{H}_{N}$ be given by
(5.14). Then, there exists a constant $C>0$ such that

$\int_{0}^{T_{0}}\int_{Q}|D^{2}(|\widetilde{H}_{N}|^{p-2}\widetilde{H}_{N})|^{2}\mathrm{d}x\mathrm{d}t\leq
C$ (5.15)

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

###### Proof.

We divide the proof into two steps.

Step 1 In this step we prove that for every $k\geq 1$ and
$\sigma\in(0,\frac{1}{p-1})$ we have

$|H_{i,N}|^{p-2}H_{i,N}\in W_{\\#}^{1,q}(\Gamma_{i,N})\quad\text{ and }\quad
h_{i,N}\in C_{\\#}^{2,\sigma}(Q).$

In order to do this, we begin by showing that $h_{i,N}\in W_{\\#}^{2,q}(Q)$
for every $q\geq 2$. Since $h_{i,N}$ is the solution to (3.5), thanks to Lemma
5.1 it satisfies (5.1). Now, by setting

$\displaystyle w:=|H_{i,N}|^{p-2}H_{i,N},\quad
A:=\varepsilon\left(I-\frac{Dh_{i,N}\otimes Dh_{i,N}}{J_{i,N}^{2}}\right),$
(5.16) $\displaystyle
b:=\pi(D(\psi(-Dh_{i,N},1)))-\frac{\varepsilon}{p}|H_{i,N}|^{p}Dh_{i,N}$
$\displaystyle\hskip 28.45274pt+\varepsilon w\left[-\frac{\Delta
h_{i,N}Dh_{i,N}}{J_{i,N}^{2}}-2\frac{D^{2}h_{i,N}Dh_{i,N}}{J_{i,N}^{2}}+3\frac{\langle
D^{2}h_{i,N}Dh_{i,N},Dh_{i,N}\rangle Dh_{i,N}}{J_{i,N}^{4}}\right],$
$\displaystyle
c:=-W(E(u_{i,N}(x,h_{i,N}(x))))-\frac{h_{i,N}-h_{i-1,N}}{\tau_{N}J_{i-1,N}},$

we can rephrase (5.1) as follows

$\int_{Q}wA:D^{2}\varphi\mathrm{d}x+\int_{Q}\langle
b,D\varphi\rangle\mathrm{d}x+\int_{Q}c\varphi\mathrm{d}x=0\qquad\text{for
every $\varphi\in C^{\infty}_{\\#}(Q)$}.$ (5.17)

By (4.8) and Theorem 4.4 we have that $A\in
W^{1,p}_{\\#}(Q;\mathbb{R}^{2\times 2}_{sym})$, $b\in L^{1}(Q;\mathbb{R}^{2})$
and $c\in C_{\\#}^{0,\alpha}(Q)$ for some $\alpha$, and hence, since (5.17) is
in particular satisfied for every $\varphi\in C^{\infty}_{\\#}(Q)$ with
$\int_{Q}\varphi\mathrm{d}x=0$, we can apply Lemma 7.2 to get that $w\in
L^{q}(Q)$ for $q\in(\frac{p}{p-1},2)$. By following the same argument in [14,
Theorem 3.11] we obtain $|H_{i,N}|^{p-2}H_{i,N}\in W^{1,q}_{\\#}(Q)$ for every
$q\geq 1$, then $|H_{i,N}|^{p-1}\in C^{0,\alpha}_{\\#}(Q)$ for every
$\alpha\in(0,1).$ As a consequence, $H_{i,N}\in C^{0,\sigma}_{\\#}(Q)$ for
every $\sigma\in(0,\frac{1}{p-1})$, and hence, by Schauder estimates we deduce
that $h_{i,N}\in C^{2,\sigma}_{\\#}(Q)$ for every
$\sigma\in(0,\frac{1}{p-1})$.

Step 2 In view of Lemma 7.3 and the fact that
$\|Dh_{i,N}\|_{L^{\infty}(Q)}<\Lambda_{0}$, there exists a positive constant
$C=C(q,\Lambda_{0})$ such that

$\|D^{2}h_{i,N}\|_{L^{q}}(Q)\leq C\|H_{i,N}\|_{L^{q}(Q)}.$ (5.18)

Furthermore, since $\Gamma_{i,N}$ is of class $C^{2,\sigma}$ then
$|H_{i,N}|^{p-2}H_{i,N}\in H^{2}(\Gamma_{i,N})$, and this implies that
$|H_{i,N}|^{p-2}H_{i,N}\in H^{2}(Q)$.

It is easy to verify (see [14, Theorem 3.11]) that $h_{i,N}$ satisfies the
following Euler-Lagrange equation in intrinsic form

$\displaystyle-\varepsilon\int_{\Gamma_{i,N}}D_{\Gamma_{i,N}}($
$\displaystyle|H_{i,N}|^{p-2}H_{i,N})D_{\Gamma_{i,N}}\phi\mathrm{d}\mathcal{H}^{2}+\varepsilon\int_{\Gamma_{i,N}}|H_{i,N}|^{p-2}H_{i,N}\left(|B_{i,N}|^{2}-\frac{1}{p}H_{i,N}^{2}\right)\phi\mathrm{d}\mathcal{H}^{2}$
$\displaystyle-\int_{\Gamma_{i,N}}\left[\mathop{\rm
div}\nolimits_{\Gamma_{i,N}}(D\psi(\nu_{i,N}))+W(Eu_{i,N})\right]\phi\mathrm{d}\mathcal{H}^{2}=\int_{\Gamma_{i,N}}\frac{h_{i,N}-h_{i-1,N}}{J_{i-1,N}\tau_{N}}\phi\mathrm{d}\mathcal{H}^{2},$
(5.19)

where $\phi:=\frac{\varphi}{J_{i,N}}\circ\pi$ with $\varphi\in C^{2}_{\\#}(Q)$
such that $\int_{Q}\varphi\mathrm{d}x=0$. Now, by setting
$w:=|H_{i,N}|^{p-2}H_{i,N}$ we can rephrase (5) in the following way

$\displaystyle-\int_{Q}\langle ADw,D$
$\displaystyle\left(\frac{\varphi}{J_{i,N}}\right)\rangle
J_{i,N}\mathrm{d}x+\varepsilon\int_{Q}w\varphi\left(|B_{i,N}|^{2}-\frac{1}{p}H_{i,N}^{2}\right)\mathrm{d}x$
$\displaystyle-\int_{Q}\left[\mathop{\rm
div}\nolimits_{\Gamma_{i,N}}(D\psi(\nu_{i,N}))+W(Eu_{i,N})\right]\varphi\mathrm{d}x=\int_{Q}\frac{h_{i,N}-h_{i-1,N}}{J_{i-1,N}\tau_{N}}\varphi\mathrm{d}x,$
(5.20)

for every $\varphi\in H^{1}_{\\#}(Q)$ with $\int_{Q}\varphi\mathrm{d}x=0$,
where $A$ is defined in (5.16). We now use $\varphi=D_{k}\eta$ with $\eta\in
H^{2}_{\\#}(Q)$, and since

$\frac{D_{k}\eta}{J_{i,N}}=D_{k}\left(\frac{\eta}{J_{i,N}}\right)+\frac{\eta
D_{k}J_{i,N}}{J_{i,N}^{2}},$

we can integrate by parts to get

$\displaystyle\int_{Q}\langle ADw,$ $\displaystyle\hskip
2.0ptD\left(\frac{D_{k}\eta}{J_{i,N}}\right)\rangle J_{i,N}\mathrm{d}x$
$\displaystyle=\sum_{s,l}\int_{Q}a_{sl}D_{l}wD_{s}D_{k}\left(\frac{\eta}{J_{i,N}}\right)J_{i,N}\mathrm{d}x+\sum_{s,l}\int_{Q}a_{sl}D_{l}wD_{s}\left(\frac{\eta
D_{k}J_{i,N}}{J_{i,N}^{2}}\right)J_{i,N}\mathrm{d}x$
$\displaystyle=-\sum_{s,l}\int_{Q}D_{k}(a_{sl}J_{i,N}D_{l}w)D_{s}\left(\frac{\eta}{J_{i,N}}\right)\mathrm{d}x-\sum_{s,l}\int_{Q}D_{s}(a_{sl}J_{i,N}D_{l}w)\frac{\eta
D_{k}J_{i,N}}{J_{i,N}^{2}}\mathrm{d}x$
$\displaystyle=-\sum_{s,l}\int_{Q}D_{k}(a_{sl}J_{i,N})D_{l}wD_{s}\left(\frac{\eta}{J_{i,N}}\right)\mathrm{d}x-\sum_{s,l}\int_{Q}a_{sl}D_{k}D_{l}wD_{s}\left(\frac{\eta}{J_{i,N}}\right)J_{i,N}\mathrm{d}x$
$\displaystyle\hskip
10.0pt-\sum_{s,l}\int_{Q}D_{s}(a_{sl}J_{i,N})D_{l}w\frac{\eta
D_{k}J_{i,N}}{J_{i,N}^{2}}\mathrm{d}x-\sum_{s,l}\int_{Q}a_{sl}D_{s}D_{l}w\frac{\eta
D_{k}J_{i,N}}{J_{i,N}}\mathrm{d}x$ $\displaystyle=-\int_{Q}\langle
D_{k}(AJ_{i,N})Dw,D\left(\frac{\eta}{J_{i,N}}\right)\rangle\mathrm{d}x-\int_{Q}\langle
AD(D_{k}w),D\left(\frac{\eta}{J_{i,N}}\right)\rangle J_{i,N}\mathrm{d}x$
$\displaystyle\hskip 10.0pt-\int_{Q}\langle\mathop{\rm
div}\nolimits(AJ_{i,N}),Dw\rangle\frac{\eta
D_{k}J_{i,N}}{J_{i,N}^{2}}\mathrm{d}x-\int_{Q}A:D^{2}w\frac{\eta
D_{k}J_{i,N}}{J_{i,N}}\mathrm{d}x$ (5.21)

Hence, by (5), (5), and by a density argument, we obtain that

$\displaystyle\int_{Q}\langle AD(D_{k}w),$ $\displaystyle\hskip
2.0ptD\left(\frac{\eta}{J_{i,N}}\right)\rangle J_{i,N}\mathrm{d}x$
$\displaystyle=-\int_{Q}\langle
D_{k}(AJ_{i,N})Dw,D\left(\frac{\eta}{J_{i,N}}\right)\rangle\mathrm{d}x-\int_{Q}A:D^{2}w\frac{\eta
D_{k}J_{i,N}}{J_{i,N}}\mathrm{d}x$ $\displaystyle-\int_{Q}\langle\mathop{\rm
div}\nolimits(AJ_{i,N}),Dw\rangle\frac{\eta
D_{k}J_{i,N}}{J_{i,N}^{2}}-\varepsilon\int_{Q}wD_{k}\eta\left(|B_{i,N}|^{2}-\frac{1}{p}H_{i,N}^{2}\right)\mathrm{d}x$
$\displaystyle+\int_{Q}\left[\mathop{\rm
div}\nolimits_{\Gamma_{i,N}}(D\psi(\nu_{i,N}))+W(Eu_{i,N})\right]D_{k}\eta\mathrm{d}x+\int_{Q}\frac{1}{J_{i-1,N}}\frac{\partial
h_{N}}{\partial t}D_{k}\eta\mathrm{d}x,$

for every $\eta\in H^{1}_{\\#}(Q)$. Therefore, by choosing
$\eta=D_{k}wJ_{i,N}$ we obtain

$\displaystyle\int_{Q}\langle AD(D_{k}w),D(D_{k}w)\rangle J_{i,N}\mathrm{d}x$
$\displaystyle=-\int_{Q}\langle
D_{k}(AJ_{i,N})Dw,D(D_{k}w)\rangle\mathrm{d}x-\int_{Q}A:D^{2}wD_{k}wD_{k}J_{i,N}\mathrm{d}x$
$\displaystyle-\int_{Q}\langle\mathop{\rm
div}\nolimits(AJ_{i,N}),Dw\rangle\frac{D_{k}wD_{k}J_{i,N}}{J_{i,N}}-\varepsilon\int_{Q}wD_{k}(D_{k}wJ_{i,N})\left(|B_{i,N}|^{2}-\frac{1}{p}H_{i,N}^{2}\right)\mathrm{d}x$
$\displaystyle+\int_{Q}\left[\mathop{\rm
div}\nolimits_{\Gamma_{i,N}}(D\psi(\nu_{i,N}))+W(Eu_{i,N})\right]D_{k}(D_{k}wJ_{i,N})\mathrm{d}x+\int_{Q}\frac{1}{J_{i-1,N}}\frac{\partial
h_{N}}{\partial t}D_{k}(D_{k}wJ_{i,N})\mathrm{d}x.$ (5.22)

Now, by summing (5) for $k=1,2$, by using Young’s inequality, the ellipticity
property of matrix $A$, and the estimate of $\mathop{\rm
div}\nolimits(AJ_{i,N})$ in terms of $D^{2}h_{i,N}$, we conclude that

$\displaystyle\int_{Q}|D^{2}w|^{2}\mathrm{d}x\leq
C\int_{Q}\left(|Dw|^{2}|D^{2}h_{i,N}|^{2}+|H_{i,N}|^{2(p+1)}+|H_{i,N}|^{2(p-1)}|D^{2}h_{i,N}|^{4}+\left|\frac{\partial
h_{N}}{\partial t}\right|^{2}+1\right)\mathrm{d}x,$ (5.23)

where the constant $C$ depends only on $\Lambda_{0}$, $D^{2}\psi$, and on the
$C^{1,\alpha}$ bound of (4.25). By Young’s inequality, together to (5.18) and
Lemma 7.4, we have

$\displaystyle\int_{Q}|H_{i,N}|^{2p-2}|D^{2}h_{i,N}|^{4}\mathrm{d}x$
$\displaystyle\leq
C\int_{Q}(|H_{i,N}|^{2p+2}+|D^{2}h_{i,N}|^{2p+2})\mathrm{d}x\leq
C\int_{Q}|H_{i,N}|^{2p+2}\mathrm{d}x$
$\displaystyle=C\int_{Q}|w|^{\frac{2(p+1)}{p-1}}\mathrm{d}x\leq
C\|D^{2}w\|^{\frac{p+2}{p}}_{L^{\frac{p}{p-1}}(Q)}\|w\|^{\frac{p^{2}+p+2}{p(p-1)}}_{L^{\frac{p}{p-1}}(Q)}$
$\displaystyle\leq\frac{1}{4}\|D^{2}w\|_{L^{\frac{p}{p-1}}(Q)}^{2}+C\|w\|^{\frac{p^{2}+p+2}{(p-1)(p-2)}}_{L^{\frac{p}{p-1}}(Q)}=\frac{1}{4}\|D^{2}w\|_{L^{\frac{p}{p-1}}(Q)}^{2}+C\|H_{i,N}\|^{\frac{p^{2}+p+2}{p-2}}_{L^{p}(Q)}$
$\displaystyle\leq\frac{1}{4}\|D^{2}w\|_{L^{2}(Q)}^{2}+C,$ (5.24)

where in the last two inequalities we used (4.14) and

$L^{2}\hookrightarrow L^{\frac{p}{p-1}}$ (5.25)

since $\frac{p}{p-1}<2$, and

$\|w\|_{L^{\frac{p}{p-1}}(Q)}=\|H_{i,N}\|_{L^{p}(Q)}^{p-1}.$ (5.26)

Moreover, we have

$\displaystyle\int_{Q}|Dw|^{2}|D^{2}h_{i,N}|^{2}\mathrm{d}x$
$\displaystyle\leq\|D^{2}h\|^{2}_{L^{2(p-1)}(Q)}\|Dw\|^{2}_{L^{\frac{2(p-1)}{p-2}}(Q)}\leq
C\|w\|^{2}_{L^{\frac{2}{p-1}}(Q)}\left(\|D^{2}w\|^{\frac{p}{2(p-1)}}_{L^{2}(Q)}\|w\|^{\frac{p-2}{2(p-1)}}_{L^{2}(Q)}\right)^{2}$
$\displaystyle=C\|D^{2}w\|^{\frac{p}{p-1}}_{L^{2}(Q)}\|w\|^{\frac{p}{p-1}}_{L^{2}(Q)}\leq
C\|D^{2}w\|^{\frac{p}{p-1}}_{L^{2}(Q)}\left(\|D^{2}w\|^{\frac{p-2}{2p}}_{L^{\frac{p}{p-1}}(Q)}\|w\|^{\frac{p+2}{2p}}_{L^{\frac{p}{p-1}}(Q)}\right)^{\frac{p}{p-1}}$
$\displaystyle\leq
C\|D^{2}w\|^{\frac{3p-2}{2(p-1)}}_{L^{2}(Q)}\|w\|^{\frac{p+2}{2(p-1)}}_{L^{\frac{p}{p-1}}(Q)}\leq\frac{1}{4}\|D^{2}w\|^{2}_{L^{2}(Q)}+C\|w\|_{L^{\frac{p}{p-1}}(Q)}^{\frac{2(p+2)}{p-2}}$
$\displaystyle=\frac{1}{4}\|D^{2}w\|^{2}_{L^{2}(Q)}+C\|H_{i,N}\|^{\frac{2(p+2)(p-1)}{p-2}}_{L^{p}(Q)}\leq\frac{1}{4}\|D^{2}w\|^{2}_{L^{2}(Q)}+C,$
(5.27)

where in the first inequality we used Holder’s inequality, in the second and
third ones we used Lemma 7.4, in the fourth and the fifth ones we used (5.25)
and (5.26), and in the last one we used (4.14).

Finally, by (5.23), (5), and (5) we get

$\int_{Q}|D^{2}w|^{2}\mathrm{d}x\leq C\int_{Q}\left(1+\left|\frac{\partial
h_{N}}{\partial t}\right|^{2}\right)\mathrm{d}x,$

and by integrating with respect to time and by using (4.6) the assertion
follows. ∎

Thanks to the bound (5.15) provided by the previous theorem, in the next lemma
we obtain the convergence of powers of the sum of the squares of principal
curvatures $\widetilde{H}_{N}$ defined in (5.14).

######  Lemma 5.3.

Let $\widetilde{H}_{N}$ be the function defined in (5.14). Then

$\displaystyle|\widetilde{H}_{N}|^{p}$
$\displaystyle\xrightarrow[N\to\infty]{}|H|^{p}\quad\text{in
$L^{1}(0,T_{0};L^{1}(Q))$},$ (5.28)
$\displaystyle|\widetilde{H}_{N}|^{p-2}\widetilde{H}_{N}$
$\displaystyle\xrightarrow[N\to\infty]{}|H|^{p-2}H\quad\text{in
$L^{1}(0,T_{0};L^{2}(Q))$},$ (5.29)

where

$H:=-\mathop{\rm div}\nolimits\left(\frac{Dh}{\sqrt{1+|Dh|^{2}}}\right)$

and $h$ is the function provided by Theorem 4.4.

###### Proof.

The proof is analogous of the one in [14, Corollary 3.15] based on [14, Lemma
3.13]. ∎

Finally, we can prove the short time existence for (1.9).

###### Proof of Theorem 3.8.

Fix $t\in(0,T_{0})$ and let $\\{i_{k}\\}_{k}$ and $\\{N_{k}\\}_{k}$ be
sequences such that $t_{k}:=i_{k}\tau_{N_{k}}\to t$ for $k\to+\infty$. Now, by
summing (5.1) for $i=1,...,i_{k}$ we obtain

$\displaystyle\int_{0}^{t_{k}}\int_{Q}$
$\displaystyle\tilde{W}_{N_{k}}\varphi\mathrm{d}x\mathrm{d}t+\int_{0}^{t_{k}}\int_{Q}\langle
D\psi(-D\tilde{h}_{N_{k}},1),(-D\varphi,0)\rangle\mathrm{d}x\mathrm{d}t$
$\displaystyle-\varepsilon\int_{0}^{t_{k}}\int_{Q}|\widetilde{H}_{N_{k}}|^{p-2}\widetilde{H}_{N_{k}}\left[\Delta\varphi-\frac{\langle
D^{2}\varphi
D\tilde{h}_{N_{k}},D\tilde{h}_{N_{k}}\rangle}{J_{i,N}^{2}}-\frac{\Delta\tilde{h}_{N_{k}}\langle
D\tilde{h}_{N_{k}},D\varphi\rangle}{\tilde{J}_{N_{k}}^{2}}\right]\mathrm{d}x\mathrm{d}t$
$\displaystyle-\varepsilon\int_{0}^{t_{k}}\int_{Q}|\widetilde{H}_{N_{k}}|^{p-2}\widetilde{H}_{N_{k}}\left[3\frac{\langle
D\tilde{h}_{N_{k}},D\varphi\rangle\langle
D^{2}\tilde{h}_{N_{k}}D\tilde{h}_{N_{k}},D\tilde{h}_{N_{k}}\rangle}{\tilde{J}_{N_{k}}^{4}}-2\frac{\langle
D^{2}\tilde{h}_{N_{k}}D\tilde{h}_{N_{k}},D\varphi\rangle}{\tilde{J}_{N_{k}}^{2}}\right]\mathrm{d}x\mathrm{d}t$
$\displaystyle+\frac{\varepsilon}{p}\int_{0}^{t_{k}}\int_{Q}|\widetilde{H}_{N_{k}}|^{p}\frac{\langle
D\tilde{h}_{N_{k}},D\varphi\rangle}{J_{i,N}}\mathrm{d}x\mathrm{d}t+\int_{0}^{t_{k}}\int_{Q}\frac{1}{\tilde{J}_{N_{k}}(\cdot,\cdot-\tau_{N_{k}})}\frac{\partial
h_{N_{k}}}{\partial t}\varphi\mathrm{d}x\mathrm{d}t=0,$ (5.30)

where $h_{N_{k}}$, $\tilde{h}_{N_{k}}$, and $\widetilde{H}_{N_{k}}$ are
defined in (3.8), (3.9), and (5.14), and we define

$\displaystyle\tilde{W}_{N_{k}}(x,t):=W(Eu_{i,N_{k}}(x,h_{i,N_{k}}(x)))$ for
every $(x,t)\in\mathbb{R}^{2}\times[(i-1)\tau_{N_{k}},i\tau_{N_{k}})$
$\displaystyle\tilde{J}_{N_{k}}(x,t):=\sqrt{1+|D\tilde{h}_{N_{k}}|^{2}},$
$\displaystyle\text{for every $(x,t)\in\mathbb{R}^{2}\times[0,T_{0}]$}.$

Now, we claim that

$\frac{1}{\tilde{J}_{N_{k}}(\cdot,\cdot-\tau_{N_{k}})}\frac{\partial
h_{N_{k}}}{\partial t}\xrightharpoonup[k\to\infty]{}\frac{1}{J}\frac{\partial
h}{\partial t}\quad\text{in $L^{2}(0,T_{0};L^{2}(Q))$}.$ (5.31)

To prove the claim, it is enough to notice that

$\displaystyle\left|\int_{0}^{T_{0}}\int_{Q}\left(\frac{1}{\tilde{J}_{N_{k}}(\cdot,\cdot-\tau_{N_{k}})}\frac{\partial
h_{N_{k}}}{\partial t}-\frac{1}{J}\frac{\partial h}{\partial
t}\right)\eta\mathrm{d}x\mathrm{d}t\right|$
$\displaystyle\leq\left|\int_{0}^{T_{0}}\int_{Q}\left(\frac{1}{\tilde{J}_{N_{k}}(\cdot,\cdot-\tau_{N_{k}})}-\frac{1}{J}\right)\frac{\partial
h_{N_{k}}}{\partial
t}\eta\mathrm{d}x\mathrm{d}t\right|+\left|\int_{0}^{T_{0}}\int_{Q}\left(\frac{\partial
h_{N_{k}}}{\partial t}-\frac{\partial h}{\partial
t}\right)\frac{\eta}{J}\mathrm{d}x\mathrm{d}t\right|$
$\displaystyle\leq\left\|\frac{\partial h_{N_{k}}}{\partial
t}\right\|_{L^{2}(0,T_{0},L^{2}(Q))}\left\|\frac{\eta}{\tilde{J}_{N_{k}}(\cdot,\cdot-\tau_{N_{k}})}-\frac{\eta}{J}\right\|_{L^{2}(0,T_{0},L^{2}(Q))}+\left|\int_{0}^{T_{0}}\int_{Q}\left(\frac{\partial
h_{N_{k}}}{\partial t}-\frac{\partial h}{\partial
t}\right)\frac{\eta}{J}\mathrm{d}x\mathrm{d}t\right|,$

for every $\eta\in L^{2}(0,T_{0},L^{2}(Q))$, and hence, from (4.6), (4.9), and
(4.16) it follows (5.31).

Moreover, in view of Theorems 4.3 and 4.4 we get

$\displaystyle\int_{0}^{t_{k}}\int_{Q}\tilde{W}_{N_{k}}\varphi\mathrm{d}x\mathrm{d}t$
$\displaystyle\xrightarrow[k\to\infty]{}\int_{0}^{t}\int_{Q}W(E(u(x,h(x,s),s))\varphi\mathrm{d}x\mathrm{d}s,$
(5.32) $\displaystyle\int_{0}^{t_{k}}\int_{Q}\langle
D\psi(-D\tilde{h}_{N_{k}},1),(-D\varphi,0)\rangle\mathrm{d}x\mathrm{d}t$
$\displaystyle\xrightarrow[k\to\infty]{}\int_{0}^{t}\int_{Q}\langle
D\psi(-Dh,1),(-D\varphi,0)\rangle\mathrm{d}x\mathrm{d}s,$ (5.33)

and by using again Theorem 4.3, (5.28), and (5.29), we obtain

$\displaystyle\int_{0}^{t_{k}}\int_{Q}|\widetilde{H}_{N_{k}}|^{p}\frac{\langle
D\tilde{h}_{N_{k}},D\varphi\rangle}{J_{i,N}}\mathrm{d}x\mathrm{d}t\xrightarrow[k\to\infty]{}\int_{0}^{t}\int_{Q}|H|^{p}\frac{\langle
Dh,D\varphi\rangle}{J}\mathrm{d}x\mathrm{d}s,$ (5.34)
$\displaystyle\int_{0}^{t_{k}}\int_{Q}|\widetilde{H}_{N_{k}}|^{p-2}\widetilde{H}_{N_{k}}\left[\Delta\varphi-\frac{\langle
D^{2}\varphi
D\tilde{h}_{N_{k}},D\tilde{h}_{N_{k}}\rangle}{J_{i,N}^{2}}\right]\mathrm{d}x\mathrm{d}t$
$\displaystyle\hskip
142.26378pt\xrightarrow[k\to\infty]{}\int_{0}^{t}\int_{Q}|H|^{p-2}H\left[\Delta\varphi-\frac{\langle
D^{2}\varphi Dh,Dh\rangle}{J^{2}}\right]\mathrm{d}x\mathrm{d}s.$ (5.35)

In order to establish the convergence of the other terms, we firstly observe
that by (4.8) and (4.16) we have
$\Delta\tilde{h}_{N_{k}}(\cdot,t)\rightharpoonup\Delta h(\cdot,t)$ for every
$t\in(0,T_{0})$. Moreover, by (5.29) we have for a.e. $t\in(0,T_{0})$ we have
that
$(|\widetilde{H}_{N_{k}}|^{p-2}\widetilde{H}_{N_{k}})(\cdot,t)\rightarrow(|H|^{p-2}H)(\cdot,t)$
in $L^{2}(Q)$, and hence

$\int_{Q}|\widetilde{H}_{N_{k}}|^{p-2}\widetilde{H}_{N_{k}}\frac{\Delta\tilde{h}_{N_{k}}\langle
D\tilde{h}_{N_{k}},D\varphi\rangle}{\tilde{J}_{N_{k}}^{2}}\mathrm{d}x\xrightarrow[k\to\infty]{}\int_{Q}|H|^{p-2}H\frac{\Delta
h\langle Dh,D\varphi\rangle}{J^{2}}\mathrm{d}x.$ (5.36)

Since by (3.8), (4.8), and (5.29) we have

$\displaystyle\left|\int_{Q}|\widetilde{H}_{N_{k}}|^{p-2}\widetilde{H}_{N_{k}}\frac{\Delta\tilde{h}_{N_{k}}\langle
D\tilde{h}_{N_{k}},D\varphi\rangle}{\tilde{J}_{N_{k}}^{2}}\mathrm{d}x\right|$
$\displaystyle\leq
C\|\Delta\tilde{h}_{N_{k}}\|_{L^{2}(Q)}\||\widetilde{H}_{N_{k}}|^{p-2}\widetilde{H}_{N_{k}}\|_{L^{2}(Q)}$
$\displaystyle\leq
C\||\widetilde{H}_{N_{k}}|^{p-2}\widetilde{H}_{N_{k}}\|_{L^{2}(Q)}\xrightarrow[k\to\infty]{L^{1}(0,T_{0})}\||H|^{p-2}H\|_{L^{2}(Q)},$
(5.37)

the generalized Lebesgue dominated convergence theorem, together with (5) and
(5.36), implies that

$\int_{0}^{t_{k}}\int_{Q}|\widetilde{H}_{N_{k}}|^{p-2}\widetilde{H}_{N_{k}}\frac{\Delta\tilde{h}_{N_{k}}\langle
D\tilde{h}_{N_{k}},D\varphi\rangle}{\tilde{J}_{N_{k}}^{2}}\mathrm{d}x\mathrm{d}t\xrightarrow[k\to\infty]{}\int_{0}^{t}\int_{Q}|H|^{p-2}H\frac{\Delta
h\langle Dh,D\varphi\rangle}{J^{2}}\mathrm{d}x\mathrm{d}s.$ (5.38)

For the remaining terms, by arguing in analogously we obtain

$\int_{0}^{t_{k}}\int_{Q}|\widetilde{H}_{N_{k}}|^{p-2}\widetilde{H}_{N_{k}}\frac{\langle
D^{2}\tilde{h}_{N_{k}}D\tilde{h}_{N_{k}},D\varphi\rangle}{\tilde{J}_{N_{k}}^{2}}\mathrm{d}x\mathrm{d}t\xrightarrow[k\to\infty]{}\int_{0}^{t}\int_{Q}|H|^{p-2}H\frac{\langle
D^{2}hDh,D\varphi\rangle}{J^{2}}\mathrm{d}x\mathrm{d}s,$ (5.39)

and

$\displaystyle\int_{0}^{t_{k}}\int_{Q}|\widetilde{H}_{N_{k}}|^{p-2}\widetilde{H}_{N_{k}}$
$\displaystyle\frac{\langle D\tilde{h}_{N_{k}},D\varphi\rangle\langle
D^{2}\tilde{h}_{N_{k}}D\tilde{h}_{N_{k}},D\tilde{h}_{N_{k}}\rangle}{\tilde{J}_{N_{k}}^{4}}\mathrm{d}x\mathrm{d}t$
$\displaystyle\hskip
85.35826pt\xrightarrow[k\to\infty]{}\int_{0}^{t}\int_{Q}|H|^{p-2}H\frac{\langle
Dh,D\varphi\rangle\langle D^{2}hDh,Dh\rangle}{J^{4}}\mathrm{d}x\mathrm{d}s.$
(5.40)

Finally, by (5.31), (5.32), (5.33), (5.34), (5.35), (5.38), (5.39), and (5),
we pass to the limit in (5), obtaining that the limiting function $h$
satisfies

$\displaystyle\int_{0}^{t}\int_{Q}\frac{1}{J}\frac{\partial h}{\partial
t}\varphi\mathrm{d}x\mathrm{d}s=-\int_{0}^{t}\int_{Q}W(E(u(x,h(x,s),s))\varphi\mathrm{d}x\mathrm{d}s-\int_{0}^{t}\int_{Q}\langle
D\psi(-Dh,1),(-D\varphi,0)\rangle\mathrm{d}x\mathrm{d}s$
$\displaystyle+\varepsilon\int_{0}^{t}\int_{Q}|H|^{p-2}H\left[\Delta\varphi-\frac{\langle
D^{2}\varphi Dh,Dh\rangle}{J^{2}}-\frac{\Delta h\langle
Dh,D\varphi\rangle}{J^{2}}-2\frac{\langle
D^{2}hDh,D\varphi\rangle}{J^{2}}\right]\mathrm{d}x\mathrm{d}s$
$\displaystyle+\varepsilon\int_{0}^{t}\int_{Q}\left[3|H|^{p-2}H\frac{\langle
Dh,D\varphi\rangle\langle
D^{2}hDh,Dh\rangle}{J^{4}}-\frac{1}{p}|H|^{p}\frac{\langle
Dh,D\varphi\rangle}{J}\right]\mathrm{d}x\mathrm{d}s.$ (5.41)

Now, by letting $\varphi$ vary in a countable dense subset of $H^{1}_{\\#}(Q)$
and by differentiating (5) with respect to t, we obtain

$\displaystyle-\int_{Q}W(E(u(x,h(x,t),t))\varphi\mathrm{d}x-\int_{Q}\langle
D\psi(-Dh,1),(-D\varphi,0)\rangle\mathrm{d}x-\frac{\varepsilon}{p}\int_{Q}|H|^{p}\frac{\langle
Dh,D\varphi\rangle}{J}\mathrm{d}x$
$\displaystyle+\varepsilon\int_{Q}|H|^{p-2}H\left[\Delta\varphi-\frac{\langle
D^{2}\varphi Dh,Dh\rangle}{J^{2}}-\frac{\Delta h\langle
Dh,D\varphi\rangle}{J^{2}}-2\frac{\langle
D^{2}hDh,D\varphi\rangle}{J^{2}}\right]\mathrm{d}x$
$\displaystyle+3\varepsilon\int_{Q}|H|^{p-2}H\frac{\langle
Dh,D\varphi\rangle\langle
D^{2}hDh,Dh\rangle}{J^{4}}\mathrm{d}x=\int_{Q}\frac{1}{J}\frac{\partial
h}{\partial t}\varphi\mathrm{d}x.$ (5.42)

Since by (5.15) $|H|^{p-2}H\in L^{2}(0,T_{0};H^{2}_{\\#}(Q))$, by arguing as
in Theorem 5.2 we have that (5) is equivalent to

$\displaystyle-\varepsilon\int_{\Gamma_{h}}D_{\Gamma_{h}}(|H|^{p-2}H)D_{\Gamma_{h}}\phi\mathrm{d}\mathcal{H}^{2}$
$\displaystyle+\varepsilon\int_{\Gamma_{h}}|H|^{p-2}H\left(|B|^{2}-\frac{1}{p}H^{2}\right)\phi\mathrm{d}\mathcal{H}^{2}$
$\displaystyle-\int_{\Gamma_{h}}\left[\mathop{\rm
div}\nolimits_{\Gamma_{h}}(D\psi(\nu))+W(Eu)\right]\phi\mathrm{d}\mathcal{H}^{2}=\int_{\Gamma_{h}}V\phi\mathrm{d}\mathcal{H}^{2}$

for every $t\in(0,T_{0})$, with $\phi:=\frac{\varphi}{J}$.

Now, we want to show that the energy decreases during the evolution. Thanks to
(4.7) we can say that the function
$t\mapsto\mathcal{F}(\tilde{h}_{N}(\cdot,t),\tilde{u}_{N}(\cdot,t))$ is
nonincreasing, where $\tilde{h}_{N}(\cdot,t)$ and $\tilde{u}_{N}(\cdot,t)$ are
defined in (3.9) and (3.10). Moreover, by using (5.28) we have, up to a
subsequence, that for a.e. $t\in(0,T_{0})$ it holds
$\widetilde{H}_{N}(\cdot,t)\rightarrow H(\cdot,t)$ in $L^{p}(Q)$. Thanks to
this, we can use (4.17) and (4.27) to get
$\mathcal{F}(\tilde{h}_{N}(\cdot,t),\tilde{u}_{N}(\cdot,t))\to\mathcal{F}(h(\cdot,t),u(\cdot,t))$
as $N\to\infty$ for all such $t$, which implies (3.12). Now, to prove (3.13)
we consider $t\in Z_{0}$ and $t_{N}\to t+$ with $t_{N}\notin Z_{0}$ as
$N\to\infty$ for every $N\in\mathbb{N}$. Since $h(\cdot,t_{N})\rightharpoonup
h(\cdot,t)$ in $W^{2,p}_{\\#}(Q)$ by (4.8), by using the lower semicontinuity
of $\mathcal{F}$ we obtain

$\mathcal{F}(h(\cdot,t),u(\cdot,t))\leq\liminf_{N\to\infty}\mathcal{F}(h(\cdot,t_{N}),u(\cdot,t_{N}))=\lim_{N\to\infty}g(t_{N})=g(t+),$

which concludes the proof. ∎

##  6\. Stability of flat configurations

In this section, inspired by [14, Section 4], we prove the Lyapunov stability
and the asymptotic stability of any admissible flat configuration. We prove
Lyapunov stability both in convex and nonconvex case, and the asymptotic
stability only in convex case. We begin with the following proposition:

######  Proposition 6.1.

Let $h$ be a variational solution to (1.9) in the sense of Definition 3.6.
Then, there exist a nonincreasing function $g$ and a negligible set $Z_{0}$
such that

$\mathcal{F}(h(\cdot,t),u_{h}(\cdot,t))=g(t)\quad\text{for every
$t\in[0,T_{0}]\setminus Z_{0}$}$

and

$\mathcal{F}(h(\cdot,t),u_{h}(\cdot,t))\leq g(t)\quad\text{for every $t\in
Z_{0}$}.$

###### Proof.

Since $h$ is a variational solution to (1.9) in the sense of Definition 3.6,
there exists a dicrete-time evolution $\\{h_{N}\\}$ such that (3.11) holds.
Thanks to this, we can repeat the arguments in Theorems 5.2, in Lemma 5.3, and
in the final part of the proof of Theorem 3.8 to conclude the proof. ∎

Now, we show that, if a variational solution exists for every times, then
there exists a sequence of times $\\{t_{n}\\}\subset(0,+\infty)$ such that
$t_{n}\to+\infty$ and $h(\cdot,t_{n})$ converges to a critical profile, which
will be used to prove (ii) of Theorem 3.9.

######  Proposition 6.2.

Let $h_{0}\in W^{2,p}_{\\#}(Q)$ be an initial datum such that there exists a
global-in-time variational solution $h$ (in the sense of Definition (3.6)).
Then, there exist a sequence $\\{t_{n}\\}_{n}\subset(0,+\infty)\setminus
Z_{0}$ and a critical profile $\bar{h}$ of $\mathcal{F}$ such that
$t_{n}\to+\infty$ and $h(\cdot,t_{n})\to\bar{h}$ in $W^{2,p}_{\\#}(Q)$.

###### Proof.

By (4.6) we have that

$\int_{0}^{+\infty}\left\|\frac{\partial h}{\partial
t}\right\|^{2}_{L^{2}(Q)}\mathrm{d}t\leq C\mathcal{F}(h_{0},u_{0}),$

and hence, there exists $\\{t_{n}\\}_{n}\subset(0,+\infty)\setminus Z_{0}$
with $t_{n}\to+\infty$ such that

$\left\|\frac{\partial h}{\partial
t}(\cdot,t_{n})\right\|_{L^{2}(Q)}\xrightarrow[n\to+\infty]{}0.$

Since $h\in L^{\infty}(0,+\infty;W^{2,p}_{\\#}(Q))\cap
H^{1}(0,+\infty;L^{2}(Q))$, by setting $h_{n}:=h(\cdot,t_{n})$ there exists
$\bar{h}\in W^{2,p}_{\\#}(Q)$ such that, up to extracting a non-relabelled
subsequence, we obtain that

$h_{n}\xrightharpoonup[n\to+\infty]{}\bar{h}\quad\text{in
$W^{2,p}_{\\#}(Q)$}.$

Furthermore, from Lemma 7.1 it follows that

$u_{h_{n}}(\cdot,h_{n}(\cdot))\xrightarrow[n\to+\infty]{}u_{\bar{h}}(\cdot,\bar{h}(\cdot))\quad\text{in
$C^{1,\alpha}_{\\#}(Q)$},$

where $u_{h_{n}}$ and $u_{\bar{h}}$ are the elastic equilibria with respect to
$h_{n}$ and $\bar{h}$, respectively. We now notice that the equation satisfied
by $h_{n}$ as in (5.1) is

$\displaystyle\int_{Q}W(E(u_{h^{n}}(x,h^{n}(x))))\varphi\mathrm{d}x+\int_{Q}\langle
D\psi(-Dh^{n},1),(-D\varphi,0)\rangle\mathrm{d}x+\frac{\varepsilon}{p}\int_{Q}|H_{n}|^{p}\frac{\langle
Dh^{n},D\varphi\rangle}{J_{n}}\mathrm{d}x$
$\displaystyle-\varepsilon\int_{Q}|H_{n}|^{p-2}H_{n}\left[\Delta\varphi-\frac{\langle
D^{2}\varphi Dh^{n},Dh^{n}\rangle}{J_{n}^{2}}-\frac{\Delta h^{n}\langle
Dh^{n},D\varphi\rangle}{J_{n}^{2}}-2\frac{\langle
D^{2}h^{n}Dh^{n},D\varphi\rangle}{J_{n}^{2}}\right]\mathrm{d}x$
$\displaystyle-3\varepsilon\int_{Q}|H_{n}|^{p-2}H_{n}\frac{\langle
Dh^{n},D\varphi\rangle\langle
D^{2}h^{n}Dh^{n},Dh^{n}\rangle}{J_{n}^{4}}\mathrm{d}x+\int_{Q}\frac{1}{J_{n}}\frac{\partial
h}{\partial t}(\cdot,t_{n})\varphi\mathrm{d}x=0$ (6.1)

for every $\varphi\in C^{2}_{\\#}(Q)$, with $J_{n}=\sqrt{1+|Dh_{n}|^{2}}$ and
$H_{n}=-\mathop{\rm div}\nolimits\left(\frac{Dh_{n}}{J_{n}}\right)$.

By repeating the same arguments of Theorem 5.2 we obtain that

$\int_{Q}|D^{2}(|H_{n}|^{p-2}H_{n})|^{2}\mathrm{d}x\leq\int_{Q}\left(1+\left|\frac{\partial
h}{\partial t}(\cdot,t_{n})\right|^{2}\right)\mathrm{d}x,$

and hence, by arguing as in the proof of Lemma 5.3 we have

$\displaystyle|H_{n}|^{p-2}H_{n}$
$\displaystyle\xrightharpoonup[n\to+\infty]{}|\bar{H}|^{p-2}\bar{H}$
$\displaystyle\text{in $H^{2}_{\\#}(Q)$},$ $\displaystyle|H_{n}|^{p-2}H_{n}$
$\displaystyle\xrightarrow[n\to+\infty]{}|\bar{H}|^{p-2}\bar{H}$
$\displaystyle\text{in $H^{1}_{\\#}(Q)$},$ $\displaystyle|H_{n}|^{p}$
$\displaystyle\xrightarrow[n\to+\infty]{}|\bar{H}|^{p}$ $\displaystyle\text{in
$L^{1}(Q)$},$

where

$\bar{H}=-\mathop{\rm
div}\nolimits\left(\frac{D\bar{h}}{\sqrt{1+|D\bar{h}|^{2}}}\right).$

Finally, in view of such convergences, by passing to the limit in (6) in the
same way as done in the proof of Theorem 3.8, we prove that $\bar{h}$ is a
critical profile. ∎

The proof of Theorem 3.9 directly follows from the arguments already used in
[14, Section 4], as the functional $\mathcal{F}$ is the same used in [14].

###### Proof of Theorem 3.9.

The proof of assertion ((i)) is exactly the same as the proof of [14, Theorem
4.7]. The proof of assertion ((ii)) instead follows by replacing [14,
Proposition 4.11] with Proposition 6.2 in the proof of [14, Theorem 4.14]. ∎

##  7\. Appendix

For the proof of the following lemmas see [14].

######  Lemma 7.1.

Let $M<0$ and $c_{0}>0$. Let $h_{1},h_{2}\in C^{1,\alpha}_{\\#}(Q)$ for some
$\alpha\in(0,1)$, with $\|h_{i}\|_{C^{1,\alpha}_{\\#}(Q)}\leq M$ and
$h_{i}\geq c_{0}$ for $i=1,2$, and let $u_{1}$ and $u_{2}$ be the
corresponding elastic equilibria of $h_{1}$ and of $h_{2}$, respectively. Then

$\|E(u_{1}(\cdot,h_{1}(\cdot)))-E(u_{2}(\cdot,h_{2}(\cdot)))\|_{C^{1,\alpha}_{\\#}(Q)}\leq
C\|h_{1}-h_{2}\|_{C^{1,\alpha}_{\\#}(Q)},$

for some constant $C>0$ depending only on $M$, $c_{0}$, and $\alpha$.

######  Lemma 7.2.

Let $p>2$, $u\in L^{\frac{p}{p-1}}(Q)$ such that

$\int_{Q}uA:D^{2}\varphi\mathrm{d}x+\int_{Q}\langle
b,D\varphi\rangle\mathrm{d}x+\int_{Q}c\varphi\mathrm{d}x=0$

for every $\varphi\in C^{\infty}_{\\#}(Q)$ with
$\int_{Q}\varphi\mathrm{d}x=0$, where $A\in
W^{1,p}_{\\#}(Q;\mathbb{R}^{2\times 2}_{sym})$ satisfies standard uniform
elliptic condition (7.2), $b\in L^{1}(Q;\mathbb{R}^{2})$ and $c\in L^{1}(Q)$.
Then $u\in L^{q}(Q)$ for every $q\in(1,2)$. Moreover, if $b,u\mathop{\rm
div}\nolimits A\in L^{r}(Q;\mathbb{R}^{2})$ and $c\in L^{r}(Q)$ for some
$r>1$, then $u\in W^{1,r}_{\\#}(Q)$.

In the next lemma we denote by $Lu$ an elliptic operator of the form

$Lu:=\sum_{i,j}a_{ij}(x)D_{ij}u+\sum_{i}b_{i}(x)D_{i}u$ (7.1)

where all the coefficients are $Q$-periodic functions, the $a_{ij}$ are
continuous, and the $b_{i}$ are bounded. Moreover, there exist
$\lambda,\Lambda>0$ such that

$\Lambda|\xi|^{2}\geq\sum_{i,j}a_{ij}\xi_{i}\xi_{j}\geq\lambda|\xi|^{2}$ (7.2)

for every $\xi\in\mathbb{R}^{2}$, $\sum_{i}|b_{i}|\leq\Lambda$.

######  Lemma 7.3.

Let $p\geq 2$. Then, there exists $C>0$ such that for all $u\in
W^{2,p}_{\\#}(Q)$ we have

$\|D^{2}u\|_{L^{p}(Q)}\leq C\|Lu\|_{L^{p}(Q)},$

where $L$ is the differential operator defined in (7.1). The constant $C$
depends only on $p$, $\lambda$, $\Lambda$, and the moduli of continuity of the
coefficients $a_{ij}$.

######  Lemma 7.4.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set satisfying the cone
condition. Let $s$, $j$ , and $m$ be integers such that $0\leq s\leq j\leq m$.
Let $1\leq p\leq q<\infty$ if $(m-j)p\geq n$, and let $1\leq p\leq
q\leq\infty$ if $(m-j)p>n$. Then, there exists $C>0$ such that

$\|D^{j}f\|_{L^{q}(Q)}\leq
C\left(\|D^{m}f\|_{L^{p}(Q)}^{\theta}\|D^{s}f\|_{L^{p}(Q)}^{1-\theta}+\|D^{s}f\|_{L^{p}(Q)}\right)$
(7.3)

for every $f\in W^{m,p}(\Omega)$, where

$\theta:=\frac{1}{m-s}\left(\frac{n}{p}-\frac{n}{q}+j-s\right).$

Moreover, if $\Omega$ is a cube, $f\in W^{m,p}_{\\#}(Q)$ and, if either $f$
vanishes at the boundary or $\int_{\Omega}f\mathrm{d}x=0$, then (7.3) holds in
the stronger form

$\|D^{j}f\|_{L^{q}(Q)}\leq
C\|D^{m}f\|_{L^{p}(Q)}^{\theta}\|D^{s}f\|_{L^{p}(Q)}^{1-\theta}.$

######  Acknowledgements.

The authors acknowledge the financial support received from the Austrian
Science Fund (FWF) through the project TAI 293. Furthermore, the authors are
members of the Italian “Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni” (GNAMPA-INdAM) and have received funding
from the GNAMPA 2022 project CUP: E55F22000270001 and GNAMPA 2023 project CUP:
E53C22001930001. Moreover, P. Piovano thanks for the funding received from the
Austrian Science Fund (FWF) through the project P 29681 and from BMBWF through
the OeAD-WTZ project HR 08/2020, and is grateful for the support received as
_Visitor in the Theoretical Visiting Sciences Program_ (TSVP) at the Okinawa
Institute of Science and Technology (OIST), Japan. Finally, P. Piovano
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”.

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