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# On the asymptotic behavior of a diffraction problem with a thin layer

P. Acampora, E. Cristoforoni

###### Abstract

We investigate the behavior of the solution to an elliptic diffraction problem
in the union of a smooth set $\Omega$ and a thin layer $\Sigma$ locally
described by $\varepsilon h$, where $h$ is a positive function defined on the
boundary $\partial\Omega$, and $\varepsilon$ is the ellipticity constant of
the differential operator in the thin layer $\Sigma$. We study the problem in
the limit for $\varepsilon$ going to zero and prove a first-order asymptotic
development by $\Gamma$-convergence of the associated energy functional.

Keywords: Diffraction, Thermal Insulation, $\Gamma$-Convergence, Robin
boundary condition

MSC 2020: 49J45, 35J25, 35B40, 80A19

##  1 Introduction

###  1.1 The general idea

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set with $C^{1,1}$
boundary, let $\nu_{0}$ denote the outer unit normal to $\partial\Omega$, and
let $h\in C^{1,1}(\partial\Omega)$. For every $\varepsilon>0$, consider the
set

$\Sigma_{\varepsilon}=\Set{\sigma+t\nu_{0}(\sigma)}{\begin{gathered}\sigma\in\partial\Omega,\\\
0<t<\varepsilon h(\sigma)\end{gathered}}.$

Figure 1: Body $\Omega$ surrounded by a thin layer
$\Sigma_{\varepsilon}$$\Omega$$\Sigma_{\varepsilon}$$\varepsilon
h(\sigma)$$\sigma$

Let $\Omega_{\varepsilon}=\overline{\Omega}\cup\Sigma_{\varepsilon}$ and let
$\nu_{\varepsilon}$ be the unit outer normal to
$\partial\Omega_{\varepsilon}$. In this paper, we study the asymptotic
behavior, as $\varepsilon$ goes to zero, of the solutions $u_{\varepsilon}$ to
the following boundary value problem:

$\begin{cases}-\Delta u_{\varepsilon}=f&\text{in }\Omega,\\\\[3.0pt]
u_{\varepsilon}^{-}=u_{\varepsilon}^{+}&\text{on }\partial\Omega,\\\\[3.0pt]
\dfrac{\partial
u_{\varepsilon}^{-}}{\partial\nu_{0}\hphantom{\scriptstyle{-}}}=\varepsilon\dfrac{\partial
u_{\varepsilon}^{+}}{\partial\nu_{0}\hphantom{\scriptstyle{+}}}&\text{on
}\partial\Omega,\\\\[6.0pt] \Delta u_{\varepsilon}=0&\text{in
}\Sigma_{\varepsilon},\\\\[3.0pt] \varepsilon\dfrac{\partial
u_{\varepsilon}}{\partial\nu_{\varepsilon}}+\beta u_{\varepsilon}=0&\text{on
}\partial\Omega_{\varepsilon},\end{cases}$ (1.1)

where $f\in L^{2}(\Omega)$ and $u_{\varepsilon}^{-}$ and $u_{\varepsilon}^{+}$
denote the traces of $u_{\varepsilon}$ on $\partial\Omega$ in $\Omega$ and in
$\Sigma_{\varepsilon}$ respectively. As shown in [11], in $\Omega$, the
functions $u_{\varepsilon}$ converges weakly in $H^{1}(\Omega)$ to $u_{0}$,
the weak solution to the boundary value problem

$\begin{cases}-\Delta{u}_{0}=f&\text{in }\Omega,\\\\[5.0pt]
\dfrac{\partial{u}_{0}}{\partial\nu_{0}}+\dfrac{\beta}{1+\beta
h}{u}_{0}=0&\text{on }\partial\Omega,\end{cases}$ (1.2)

i.e. for every $\varphi\in H^{1}(\Omega)$

$\int_{\Omega}\nabla
u_{0}\cdot\nabla\varphi\,dx+\beta\int_{\partial\Omega}\dfrac{u_{0}\varphi}{1+\beta
h}\,d\mathcal{H}^{n-1}=\int_{\Omega}f\varphi\,dx.$

Let us notice that the functions $u_{\varepsilon}$ are the minimizers, in
$H^{1}(\Omega_{\varepsilon})$, of the functionals

$\mathcal{F}_{\varepsilon}(v,h)=\displaystyle\int_{\Omega}\lvert\nabla
v\rvert^{2}\,dx+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
v\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}v^{2}\,d\mathcal{H}^{n-1}-2\int_{\Omega}fv\,dx,$

where $v\in H_{1}(\Omega_{\varepsilon})$.

Our interest in this work is focused on the behavior of solutions
$u_{\varepsilon}$ in the set $\Sigma_{\varepsilon}$. This interest is mainly
motivated by the following remark: it seems that the solutions to some
variational problems in thin sets of the type $\Sigma_{\varepsilon}$ (see for
instance [3, 11, 2]), can be approximated with functions that are linear along
the normal rays to $\partial\Omega$. We will show that, indeed, if we stretch
the set $\Sigma_{\varepsilon}$ to the set $\Sigma_{1}$, then the stretched
solutions $\tilde{u}_{\varepsilon}$ converge to a function that, in
$\Sigma_{1}$, is linear with respect to the distance from $\partial\Omega$.

In addition, we will show that this convergence property will be sufficient to
study the first-order development by $\Gamma$-convergence of the functional
$\mathcal{F}_{\varepsilon}$.

###  1.2 The main theorems

Before stating the main result, we need to fix some notations. Let

$d(x)=d(x,\Omega)=\min_{y\in\bar{\Omega}}\lvert x-y\rvert$

be the distance function from $\Omega$ and define

$\Gamma_{t}=\Set{x\in\mathbb{R}^{n}}{\,d(x)<t}\setminus\Omega.$

The regularity of $\Omega$ ensures that $\Omega$ satisfies a uniform exterior
ball condition, and in particular, there exists $d_{0}>0$ such that $d\in
C^{1,1}(\Gamma_{d_{0}})$. Moreover, for every $z\in\Gamma_{d_{0}}$ there
exists a unique $\sigma(z)\in\partial\Omega$ such that

$\lvert\sigma(z)-z\rvert=d(z),$

and we will say that $\sigma(z)$ is the _metric projection_ of $z$ over
$\Omega$; we will also denote by $\nu_{0}$ the extension of the outer unit
normal to $\partial\Omega$ such that for every $z\in\Gamma_{d_{0}}$ we have
$\nu_{0}(z)=\nu_{0}(\sigma(z))$. In the following it will be useful to have
that for every $x\in\Sigma_{\varepsilon}$ the metric projection of $x$ over
$\Omega$ exists and it will be useful to work with an extension of $h$ in the
set $\Sigma_{\varepsilon}$ which is constant along the normal directions to
$\partial\Omega$. Hence, we will assume $h$ to be a positive
$C^{1,1}(\Gamma_{d_{0}})$ function such that $\lVert h\rVert_{\infty}<d_{0}$
and $\nabla h\cdot\nu_{0}=0$. To inspect the properties of the solution
$u_{\varepsilon}$ inside $\Sigma_{\varepsilon}$, we "stretch" the solution
$u_{\varepsilon}$ via a pullback on the reference set $\Sigma_{1}$. To be more
precise, we construct a diffeomorphism

$\Psi_{\varepsilon}:\Sigma_{1}\to\Sigma_{\varepsilon}$

by rescaling the distance from $\partial\Omega$. The approach is analogous to
the dilation technique proposed in [5] for the asymptotic expansion and the
derivation of the so-called effective boundary conditions. We show the
following

######  Theorem 1.1.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open set with $C^{1,1}$
boundary, and fix a positive function $h\in C^{1,1}(\Gamma_{d_{0}})$ such that
$\nabla h\cdot\nu_{0}=0$, and let $\Psi_{\varepsilon}$ be the stretching
diffemorphism defined in subsection 3.1. Let $u_{\varepsilon}$ be the unique
weak solution to (1.1), and let $u_{0}$ be the unique weak solution to (1.2).
If we let

$\tilde{u}_{\varepsilon}(z)=\begin{cases}u_{\varepsilon}(z)&\text{if
}z\in\Omega,\\\\[5.0pt] u_{\varepsilon}(\Psi_{\varepsilon}(z))&\text{if
}z\in\Sigma_{1},\end{cases}$

then the family $\tilde{u}_{\varepsilon}$ is equibounded in
$H^{1}(\Omega_{1})$ and, up to a subsequence, converges weakly in
$H^{1}(\Omega_{1})$ to the function

$\tilde{u}_{0}(z)=\begin{cases}u_{0}(z)&\text{if }z\in\Omega,\\\\[5.0pt]
u_{0}(\sigma(z))\left(1-\dfrac{\beta d(z)}{1+\beta h(z)}\right)&\text{if
}z\in\Sigma_{1}.\end{cases}$

Notice that the limit function $\tilde{u}_{0}$ turns out to be linear with
respect to the distance $d(z)$.

The functions $u_{\varepsilon}$ are the minimizers, in
$H^{1}(\Omega_{\varepsilon})$, of the functionals

$\mathcal{F}_{\varepsilon}(v,h)=\displaystyle\int_{\Omega}\lvert\nabla
v\rvert^{2}\,dx+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
v\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}v^{2}\,d\mathcal{H}^{n-1}-2\int_{\Omega}fv\,dx,$

related to the thermal insulation of a solid, represented by the set $\Omega$,
with a thin layer of highly insulating material, represented by the set
$\Sigma_{\varepsilon}$. In this setting, $\varepsilon$ is related to both the
mean thickness and the conductivity of the insulating layer
$\Sigma_{\varepsilon}$, while the function $h$ represents the shape of the
layer. Finally, the Robin boundary condition models the case in which the heat
exchange with the environment occurs through convection, and $u_{\varepsilon}$
is the temperature of the insulated body.

Similar problems have been studied before in the context of reinforcement
problems in [6], [13], [3], and more recently in the context of thermal
insulation in [7], [11] and [2].

In particular, in [11] it was proved that the family of functionals
$\mathcal{F}_{\varepsilon}(\cdot,h)$ $\Gamma$-converges, in the strong
$L^{2}$-topology, to the functional

$\mathcal{F}_{0}(v,h)=\displaystyle\int_{\Omega}\lvert\nabla
v\rvert^{2}\,dx+\beta\int_{\partial\Omega}\dfrac{v^{2}}{1+\beta
h}\,d\mathcal{H}^{n-1}-2\int_{\Omega}fv\,dx.$

Moreover, minimizing the approximated functional $\mathcal{F}_{0}$ with
respect to both variables, the authors proved that, under an integral
constraint for the function $h$, a minimizing couple $(u_{0},h_{opt})$ exists
and that the optimal function $h_{opt}$, and hence the shape of the insulating
layer, should be related to the limit temperature $u_{0}$, while the
dependence on the geometry of $\Omega$ is only implicit. In [2] a similar
problem was studied through a first-order asymptotic development by
$\Gamma$-convergence (see subsection 2.3), the main result is in an
approximated energy functional whose minimizer $h_{opt}$ explicitly depends on
the mean curvature of $\partial\Omega$.

In the spirit of [2], we aim to study a first-order asymptotic development by
$\Gamma$-convergence of $\mathcal{F}_{\varepsilon}$. In particular, thanks to
Theorem 1.1, we are able to prove the following

######  Theorem 1.2.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open set with $C^{1,1}$
boundary, and fix a positive function $h\in C^{1,1}(\Gamma_{d_{0}})$ such that
$\nabla h\cdot\nu_{0}=0$. Then the functional

$\delta\mathcal{F}_{\varepsilon}(\cdot,h)=\dfrac{\mathcal{F}_{\varepsilon}(\cdot,h)-\mathcal{F}_{0}(u_{0},h)}{\varepsilon}$

$\Gamma$-converges, in the strong $L^{2}(\mathbb{R}^{n})$ topology, as
$\varepsilon\to 0^{+}$, to

$\mathcal{F}^{(1)}(v,h)=\begin{cases}\displaystyle\beta\int_{\partial\Omega}\frac{Hh(2+\beta
h)}{2(1+\beta h)^{2}}\,u_{0}^{2}\,d\mathcal{H}^{n-1},&\text{if
}v=u_{0},\\\\[10.0pt] +\infty&\text{if }v\neq u_{0},\end{cases}$

where $H$ denotes the mean curvature of $\partial\Omega$.

Let us look at the first-order approximation in $\varepsilon$ of the energy
$\mathcal{F}_{\varepsilon}(u_{\varepsilon},h)$, described by

$\mathcal{G}_{\varepsilon}(h)=\mathcal{F}_{0}(u_{0},h)+\varepsilon\mathcal{F}^{(1)}(u_{0},h).$

We believe that the study of $\mathcal{G}_{\varepsilon}$, either with a volume
penalization as in [9, 8, 1] or a volume constraint as in [7, 11], will
provide a deeper understanding of the optimal shape of the insulating layer.
From the previous result, we conjecture that the design of the shape of the
optimal insulating layer should not only be related to the limit temperature
$u_{0}$, but also to the mean curvature $H$. However we were not able to
provide an existence result for minimizers of the approximating energy.

The present paper is organized as follows. In section 2 we give introductory
definitions and tools about hypersurfaces and $\Gamma$-convergence. In section
3 we explicitly compute the diffeomorphism $\Psi_{\varepsilon}$ and we prove
Theorem 1.1 using some energy estimates that will be proved later in the
paper. In section 4, using the convergence of the functions
$\tilde{u}_{\varepsilon}$, we explicitly compute the first-order asymptotic
development by $\Gamma$-convergence for the functional
$\mathcal{F}_{\varepsilon}(\cdot,h)$. Finally, in section 5 we prove the
aforementioned energy estimates, using classical techniques in regularity
theory adapted to this particular case of a diffraction problem.

##  2 Notations and tools

###  2.1 Distance function and curvatures

We refer to [14, Section 14.6] for the notions in this section. Let $d$,
$\Gamma_{t}$, $\sigma$ and $\nu_{0}$ as in the introduction. For every
$x\in\Gamma_{d_{0}}$ we can write

$x=\sigma(x)+d(x)\nu_{0}(x),$

and notice that we have $\sigma,\nu_{0}\in
C^{0,1}(\Gamma_{d_{0}};\mathbb{R}^{n})$, and $\nabla d=\nu_{0}$. For
$\mathcal{H}^{n-1}$-a.e. $\sigma\in\partial\Omega$ we can consider the Hessian
matrix $D^{2}d(\sigma)$, and, since $\lvert\nabla d\rvert=1$, we have that
$\nu_{0}(\sigma)$ is an eigenvector with corresponding zero eigenvalue. Let
$\set{\tau_{1},\dots,\tau_{n-1},\nu_{0}}$ be a system of normalized
eigenvectors.

######  Definition .

Let $\sigma\in\partial\Omega$ and let
$\set{\tau_{1}(\sigma),\dots,\tau_{n-1}(\sigma),\nu_{0}(\sigma)}$ be an
ordered system of normalized eigenvectors for $D^{2}d(\sigma)$. We define the
_principal curvatures_ $k_{1}(\sigma),\dots,k_{n-1}(\sigma)$ of
$\partial\Omega$ at the point $\sigma$ as the eigenvalues of the matrix

$D^{2}d(\sigma)$

corresponding to the eigenvectors $\tau_{1}(\sigma),\dots,\tau_{n-1}(\sigma)$.

Let $t\in(0,d_{0})$ and consider

$\gamma_{t}=\partial(\Omega\cup\Gamma_{t})=\Set{x\in\mathbb{R}^{n}}{d(x)=t}\setminus\Omega.$

Our regularity assumptions allow us to consider for every $x\in\Gamma_{d_{0}}$
the matrix $D^{2}d(x)$, which is symmetric and represents the second
fundamental form of $\gamma_{d(x)}$. Moreover, by direct computation (see for
instance [14, Lemma 14.17]) we have that $\tau_{i}(\sigma(x))$ are
eigenvectors for $D^{2}d(x)$ with eigenvalues computed in the following
definition.

######  Definition .

Let $x\in\Gamma_{d_{0}}$. For every $i=1,\dots,n-1$ we denote by

$\tau_{i}(x):=\tau_{i}(\sigma(x)),$

and we denote by $k_{1}(x),\dots,k_{n-1}(x)$ the corresponding sequence of
eigenvalues of $D^{2}d(x)$

$k_{i}(x):=\dfrac{k_{i}(\sigma(x))}{1+d(x)k_{i}(\sigma(x))},$

######  Remark .

By the properties of $D^{2}d(x)$ we can diagonalize $D\sigma(x)$ as

$D\sigma(x)\,\tau_{i}(\sigma)=\frac{1}{1+d(x)k_{i}(\sigma)}\tau_{i}(\sigma).$

###  2.2 Calculus on hypersurfaces

We refer to [15] for the notions in this section. For every
$v,w\in\mathbb{R}^{n}$ let the _tensorial product_ $v\otimes w$ be the unique
linear operator on $\mathbb{R}^{n}$ such that, for every $z\in\mathbb{R}^{n}$,

$(v\otimes w)z=(w\cdot z)v.$

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set with $C^{1}$ boundary
and let $\nu_{0}$ be the outer unit normal to its boundary. For every
$\sigma\in\partial\Omega$ we denote by

$\displaystyle
T_{\sigma}\partial\Omega=\Set{y\in\mathbb{R}^{n}}{\nu_{0}(\sigma)\cdot y=0},$

the _tangent space to $\partial\Omega$ at $\sigma$_. Let
$\tau_{1}(\sigma),\dots,\tau_{n-1}(\sigma)$ be an orthonormal basis for
$T_{\sigma}\partial\Omega$.

######  Definition  (Tangential gradient).

Let $\Omega$ be a bounded open set of class $C^{1}$, let
$U\subseteq\mathbb{R}^{n}$ be an open set containing $\partial\Omega$, and let
$\phi\in C^{0,1}(U;\mathbb{R}^{n})$. We define for $\mathcal{H}^{n-1}$-a.e.
$\sigma\in\partial\Omega$ the _tangential gradient of $\phi$_ at $\sigma$ as
the the linear map

$D^{\partial\Omega}\phi(\sigma)\colon
T_{\sigma}\partial\Omega\to\mathbb{R}^{n}$

defined as

$\begin{split}D^{\partial\Omega}\phi(\sigma)=&\sum_{h=1}^{n-1}\left(\nabla\phi(\sigma)\tau_{h}\right)\otimes\tau_{h}\\\\[5.0pt]
=&\nabla\phi(\sigma)-\left(\nabla\phi(\sigma)\nu_{0}\right)\otimes\nu_{0}.\end{split}$

Notice that the definition of tangential gradient does not depend on the
choice of the orthonormal basis $\tau_{1},\dots,\tau_{n-1}$.

######  Definition  (Tangential Jacobian).

Let $\Omega$ be a bounded open set of class $C^{1}$, let
$U\subseteq\mathbb{R}^{n}$ be an open set containing $\partial\Omega$, and let
$\phi\in C^{0,1}(U;\mathbb{R}^{n})$. We define _the tangential Jacobian of
$\phi$_ as

$J^{\partial\Omega}\phi=\sqrt{\det\big{(}(D^{\partial\Omega}\phi)^{T}(D^{\partial\Omega}\phi)\big{)}\,},\,$

where the determinant has to be intended in the space
$T_{\sigma}\partial\Omega\otimes T_{\sigma}\partial\Omega$.

######  Theorem 2.1 (Area formula on surfaces).

Let $U\subseteq\mathbb{R}^{n}$ be an open set containing $\partial\Omega$, let
$\phi\in C^{0,1}(U;\mathbb{R}^{n})$, and let $g:\mathbb{R}^{n}\to\mathbb{R}$
be a positive Borel function. We have that

$\int_{\partial\Omega}g(\phi(\sigma))\,J^{\partial\Omega}\phi\,d\mathcal{H}^{n-1}=\int_{\phi(\partial\Omega)}g(\sigma)\,d\mathcal{H}^{n-1}.$

######  Definition  (Tangential divergence).

Let $\Omega$ be a bounded open set of class $C^{1}$, let
$U\subseteq\mathbb{R}^{n}$ be an open set containing $\partial\Omega$ and let
$\phi\in C^{0,1}(U;\mathbb{R}^{n})$. We define the _tangential divergence of
$\phi$_ as

$\operatorname{div}^{\partial\Omega}\phi=\sum_{j=1}^{n-1}(D\phi\,\tau_{j})\cdot\tau_{j}$

######  Definition  (Mean Curvature).

Let $\Omega$ be a bounded open set with $C^{1,1}$ boundary and let

$\nu:\partial\Omega\to\mathbb{S}^{n-1}$

be the outer unit normal to its boundary. Let $U\subseteq\mathbb{R}^{n}$ be an
open set containing $\partial\Omega$ and let $X\in C^{0,1}(U)$ be an extension
of $\nu$. For $\mathcal{H}^{n-1}$-a.e. $\sigma\in\partial\Omega$ we define the
_mean curvature of $\partial\Omega$_ as

$H(\sigma)=\operatorname{div}^{\partial\Omega}X=\sum_{i=1}^{n-1}k_{i}(\sigma).$

######  Remark .

Let $\Omega$ be a bounded open set of class $C^{1,1}$, let
$U\subseteq\mathbb{R}^{n}$ be an open set containing $\partial\Omega$, let
$X\in C^{0,1}(U;\mathbb{R}^{n})$, and let $\phi(x)=x+tX(x)$. By direct
computations, we have that

$J^{\partial\Omega}\phi(\sigma)=1+t\operatorname{div}^{\partial\Omega}X(\sigma)+t^{2}R(t,\sigma)$

where the remainder $R$ is a bounded function. In particular, if $X$ is an
extension of $\nu$, we have

$J^{\partial\Omega}\phi(\sigma)=1+tH(\sigma)+t^{2}R(t,\sigma).$

######  Theorem 2.2 (Coarea formula).

Let $f\in C^{0,1}(\mathbb{R}^{n})$, let $g\in L^{1}(\mathbb{R}^{n})$, and let
$U\subset\mathbb{R}^{n}$ be an open set. Then

$\int_{U}g(x)\lvert\nabla
f(x)\rvert\,dx=\int_{\mathbb{R}}\int_{U\cap\Set{f=t}}g(y)\,d\mathcal{H}^{n-1}(y)\,dt.$

###  2.3 $\Gamma$-convergence

In this section, we recall some basic properties of the $\Gamma$-convergence
and the asymptotic development by $\Gamma$-convergence. We refer for instance
to [10] and [4] for the following notions.

######  Definition .

Let $X$ be a metric space and, for any $\varepsilon>0$, let us consider the
functionals
$\mathcal{F}_{\varepsilon},\mathcal{F}_{0}:X\to\mathbb{R}\cup\set{+\infty}$.
We will say that _$\mathcal{F}_{\varepsilon}$ $\Gamma$-converges, with respect
to the strong topology in $X$, as $\varepsilon\to 0^{+}$_ to $\mathcal{F}_{0}$
if for every $x\in X$ the following conditions hold:

  * •

for every sequence $\set{x_{\varepsilon}}\subset X$ converging to $x$,

$\liminf_{\varepsilon\to
0^{+}}\mathcal{F}_{\varepsilon}(x_{\varepsilon})\geq\mathcal{F}_{0}(x);$

  * •

there exists a sequence $\set{x_{\varepsilon}}\subset X$ converging to $x$
such that

$\limsup_{\varepsilon\to
0^{+}}\mathcal{F}_{\varepsilon}(x_{\varepsilon})\leq\mathcal{F}_{0}(x).$

In particular, from the definition, if $\mathcal{F}_{\varepsilon}$
$\Gamma$-converges to $\mathcal{F}_{0}$, for every $x\in X$ there exists a
recovery sequence $\set{x_{\varepsilon}}\subset X$, converging to $x$, such
that

$\lim_{\varepsilon\to
0^{+}}\mathcal{F}_{\varepsilon}(x_{\varepsilon})=\mathcal{F}_{0}(x).$

We have the following

######  Proposition .

Let $X$ be a metric space and, for any $\varepsilon>0$, let us consider the
functionals
$\mathcal{F}_{\varepsilon},\mathcal{F}_{0}:X\to\mathbb{R}\cup\set{+\infty}$
such that $\mathcal{F}_{\varepsilon}$ $\Gamma$-converges, with respect to the
strong topology in $X$ as $\varepsilon\to 0^{+}$ to $\mathcal{F}_{0}$. Let
$\set{x_{\varepsilon}}$ be a sequence in $X$ such that

$\mathcal{F}_{\varepsilon}(x_{\varepsilon})=\min_{X}\mathcal{F}_{\varepsilon}.$

If there exists $\overline{x}\in X$ such that $x_{\varepsilon}$ converges to
$\overline{x}$, then

$\mathcal{F}_{0}(\overline{x})=\min_{X}\mathcal{F}_{0}=\lim_{\varepsilon\to
0^{+}}\min_{X}\mathcal{F}_{\varepsilon}.$

Let

$m_{0}=\inf_{X}\mathcal{F}_{0},$

and, for every $x\in X$, let

$\delta\mathcal{F}_{\varepsilon}(x)=\frac{\mathcal{F}_{\varepsilon}(x)-m_{0}}{\varepsilon}$

######  Definition .

If there exists a functional $\mathcal{F}^{(1)}\colon
X\to\mathbb{R}\cup\set{+\infty}$ such that $\delta\mathcal{F}_{\varepsilon}$
$\Gamma$-converges, with respect to the strong topology in $X$, as
$\varepsilon\to 0^{+}$ to $\mathcal{F}^{(1)}$, we say that $\mathcal{F}^{(1)}$
is the _first-order asymptotic development by $\Gamma$-convergence_ for the
functional $\mathcal{F}_{\varepsilon}$.

Let

$\mathcal{U}_{0}=\Set{x\in X}{\mathcal{F}_{0}(x)=m_{0}},$

the interest in the previous definition is justified by the following

######  Remark .

Let $\set{x_{\varepsilon}}$ be a sequence in $X$ such that

$\mathcal{F}_{\varepsilon}(x_{\varepsilon})=\min_{X}\mathcal{F}_{\varepsilon},$

and assume that there exists $\overline{x}\in X$ such that $x_{\varepsilon}$
converges to $\overline{x}$; then, by subsection 2.3, we have that
$\overline{x}\in\mathcal{U}_{0}$ and

$\mathcal{F}^{(1)}(\overline{x})=\min_{X}\mathcal{F}^{(1)}=\lim_{\varepsilon\to
0^{+}}\dfrac{\mathcal{F}_{\varepsilon}(x_{\varepsilon})-m_{0}}{\varepsilon}.$

In particular, we have

$\mathcal{F}_{\varepsilon}(x_{\varepsilon})=m_{0}+\varepsilon\mathcal{F}^{(1)}(\bar{x})+o(\varepsilon).$

##  3 Stretching

In this section, we construct the diffeomorphism
$\Psi_{\varepsilon}\colon\Sigma_{1}\to\Sigma_{\varepsilon}$, and, assuming
that the family of functions
$\tilde{u}_{\varepsilon}=u_{\varepsilon}\circ\Psi_{\varepsilon}$ converges
weakly in $H^{1}(\Sigma_{1})$ to a function $\tilde{u}_{0}$, we compute the
limiting boundary value problem for the limit $\tilde{u}_{0}$.

###  3.1 Preliminary computations

Let $\Omega$ be a bounded open set with $C^{1,1}$ boundary. For every
$x\in\Gamma_{d_{0}}$, we recall we have defined

$\nu_{0}(x)=\nu_{0}(\sigma(x))$

to be the unit outer normal to $\partial\Omega$ in $\sigma(x)$, and let $h\in
C^{1,1}(\Gamma_{d_{0}})$ be a positive function such that $\nabla
h\cdot\nu_{0}=0$, so that $h$ is constant along normal radii starting from
$\partial\Omega$. We define

$\Sigma_{\varepsilon}:=\Set{x\in\mathbb{R}^{n}}{0<d(x)<\varepsilon
h(x)},\qquad\qquad\Omega_{\varepsilon}=\overline{\Omega}\cup\Sigma_{\varepsilon}.$

In particular, notice that, if $\varepsilon\lVert h\rVert_{\infty}<d_{0}$, we
can write

$\Sigma_{\varepsilon}=\Set{\sigma+t\nu_{0}(\sigma)}{\begin{gathered}\sigma\in\partial\Omega,\\\
0<t<\varepsilon h(\sigma)\end{gathered}},$ (3.1)

and the representation $x=\sigma+t\nu_{0}$ is unique and $d\in
C^{1,1}(\Sigma_{\varepsilon})$.

For simplicity’s sake, we will assume that $\lVert h\rVert_{\infty}<d_{0}$ so
that we can assume the distance function $d$ to be regular on the set
$\Sigma_{1}$.

######  Definition .

We define the _stretching diffeomorphism_ $\Psi_{\varepsilon}\in
C^{0,1}(\Gamma_{d_{0}};\Gamma_{\varepsilon d_{0}})$ as the function defined as

$\begin{split}\Psi_{\varepsilon}(z)&=\sigma(z)+\varepsilon
d(z)\nu_{0}(z)\\\\[5.0pt] &=z+(\varepsilon-1)d(z)\nu_{0}(z).\end{split}$

With this definition we have that

$\begin{split}D\Psi_{\varepsilon}(z)&=D\sigma(z)+\varepsilon\left[\nu_{0}(z)\otimes\nu_{0}(z)+d(z)D\nu_{0}(z)\right]\\\\[5.0pt]
&=\operatorname{Id}_{n}+(\varepsilon-1)\left[\nu_{0}(z)\otimes\nu_{0}(z)+d(z)D\nu_{0}(z)\right],\end{split}$

where $\operatorname{Id}_{n}$ is the identity matrix. Moreover,
$\Psi_{\varepsilon}$ is invertible with

$\Psi_{\varepsilon}^{-1}(x)=\sigma(x)+\dfrac{d(x)}{\varepsilon}\nu_{0}(x).$

By direct computations, we have the following

######  Lemma .

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open set with $C^{1,1}$
boundary, and fix a positive function $h\in C^{1,1}(\Gamma_{d_{0}})$ such that
$\nabla h\cdot\nu_{0}=0$. Let $g:\mathbb{R}^{n}\to\mathbb{R}$ be a positive
Borel function. Then

$\int_{\Sigma_{\varepsilon}}g(x)\,dx=\varepsilon\int_{\Sigma_{1}}g(\Psi_{\varepsilon}(z))J_{\varepsilon}(z)\,dz,$
(3.2)

with

$J_{\varepsilon}(z)=\prod_{i=1}^{n-1}\dfrac{1+\varepsilon
d(z)k_{i}(\sigma)}{1+d(z)k_{i}(\sigma)},$

and $\sigma=\sigma(z)$. In addition,

$\int_{\partial\Omega_{\varepsilon}}g(\xi)\,d\mathcal{H}^{n-1}(\xi)=\int_{\partial\Omega_{1}}g(\Psi_{\varepsilon}(\zeta))J^{\tau}_{\varepsilon}(\zeta)\,d\mathcal{H}^{n-1}(\zeta),$
(3.3)

where $J_{\varepsilon}^{\tau}$ is the tangential Jacobian of
$\Psi_{\varepsilon}$, and it converges uniformly, as $\varepsilon\to 0^{+}$,
to

$\begin{split}J^{\tau}_{0}(\zeta)&=\frac{1}{\sqrt{1+\lvert\nabla
h(\zeta)\rvert^{2}}}\prod_{i=1}^{n-1}\frac{1}{1+h(\zeta)k_{i}(\sigma(\zeta))}\\\\[7.0pt]
&=\frac{1}{\sqrt{1+\lvert\nabla h(\zeta)\rvert^{2}}}J_{0}(\zeta).\end{split}$

###### Proof.

Let $z\in\Gamma_{d_{0}}$, and let $\tau_{i}(z)$ be the eigenvectors of
$D^{2}d(z)=D\nu_{0}(z)$, with respective eigenvectors $k_{i}(z)$, defined in
subsection 2.1. We have that
$\set{\tau_{1}(\sigma),\dots,\tau_{n-1}(\sigma),\nu_{0}(\sigma)}$ is a basis
of eigenvectors for $D\Psi_{\varepsilon}(z)$, and, in particular,

$\begin{split}D\Psi_{\varepsilon}(z)\,\tau_{i}(z)&=\left(1+\dfrac{(\varepsilon-1)d(z)k_{i}(\sigma)}{1+d(z)k_{i}(\sigma)}\right)\tau_{i}(z)\\\\[5.0pt]
&=\dfrac{1+\varepsilon
d(z)k_{i}(\sigma)}{1+d(z)k_{i}(\sigma)}\tau_{i}(z),\end{split}$

and

$D\Psi_{\varepsilon}(z)\,\nu_{0}(z)=\varepsilon\nu_{0}(z).$

Therefore, (3.2) follows from the area formula.

On the other hand, notice that $D\Psi_{\varepsilon}$ converges uniformly, as
$\varepsilon\to 0^{+}$, to $D\sigma$, so that
$J^{\partial\Omega_{1}}\Psi_{\varepsilon}$ converges to
$J^{\partial\Omega_{1}}\sigma$. Therefore, we apply the area formula on
surfaces (Theorem 2.1) to get (3.3), and it is left to compute the tangential
Jacobian $J^{\partial\Omega_{1}}\sigma$.

Let $\zeta\in\partial\Omega_{1}$, and let $\nu_{1}$ be the unit outer normal
to $\partial\Omega_{1}$. Recalling that $\nabla d=\nu_{0}$, the definition of
$\Sigma_{1}$ ensures that

$\nu_{1}(\zeta)=\dfrac{\nu_{0}(\zeta)-\nabla h(\zeta)}{\sqrt{1+\lvert\nabla
h(\zeta)\rvert^{2}}}.$

We aim to construct an orthonormal basis for the tangent space
$T_{\zeta}\partial\Omega_{1}$ with a rotation of the tangent space
$T_{\sigma(\zeta)}\partial\Omega$. In the following, when possible, we will
drop the dependence on $\zeta$. Let us define the rotation operator

$R:\mathbb{R}^{n}\to\mathbb{R}^{n}$

as the unique linear operator having the following properties:

  1. (i)

if $w\in(\nabla h)^{\perp}\cap\nu_{0}^{\perp}$, then $Rw=w$;

  2. (ii)

$R\nu_{0}=\nu_{1}$;

  3. (iii)

$\nabla h/\lvert\nabla h\rvert$ is mapped to a unit vector laying in the plane
generated by $\nabla h$ and $\nu_{0}$, and orthogonal to $\nu_{1}$, namely

$R\frac{\nabla h}{\lvert\nabla h\rvert}=\frac{1}{\sqrt{1+\lvert\nabla
h\rvert^{2}}}\biggl{(}\frac{\nabla h}{\lvert\nabla h\rvert}+\lvert\nabla
h\rvert\nu_{0}\biggr{)}.$

Explicitly, we can write $R$ as

$\begin{split}R=&\operatorname{Id}_{n}-\nu_{0}\otimes\nu_{0}-\frac{\nabla
h}{\lvert\nabla h\rvert}\otimes\frac{\nabla h}{\lvert\nabla
h\rvert}+\nu_{1}\otimes\nu_{0}\\\\[5.0pt] &+\frac{1}{\sqrt{1+\lvert\nabla
h\rvert^{2}}}\left(\frac{\nabla h}{\lvert\nabla h\rvert}+\lvert\nabla
h\rvert\nu_{0}\right)\otimes\frac{\nabla h}{\lvert\nabla h\rvert}.\end{split}$

This operator is a rotation in $\mathbb{R}^{n}$ that maps
$T_{\sigma(\zeta)}\partial\Omega$ onto $T_{\zeta}\partial\Omega_{1}$, and, in
particular, we define an orthonormal basis $\bar{\tau}_{i}$ of
$T_{\zeta}\partial\Omega_{1}$ as follows: let
$\tau_{i}=\tau_{i}(\sigma(\zeta))$ be an orthonormal basis of
$T_{\sigma(\zeta)}\partial\Omega$, and we define

$\bar{\tau}_{s}:=R\tau_{s}=\left(\operatorname{Id}_{n}+\left(\frac{1}{\sqrt{1+\lvert\nabla
h\rvert^{2}}}-1\right)\frac{\nabla h}{\lvert\nabla h\rvert}\otimes\frac{\nabla
h}{\lvert\nabla h\rvert}\right)\tau_{s}+\frac{\nabla
h\cdot\tau_{s}}{\sqrt{1+\lvert\nabla h\rvert^{2}}}\,\nu_{0}.$

In particular, since $D\sigma\,\nu_{0}=0$, then

$(D\sigma\circ
R)\tau_{s}=D\sigma\left(\operatorname{Id}_{n}+\left(\frac{1}{\sqrt{1+\lvert\nabla
h\rvert^{2}}}-1\right)\frac{\nabla h}{\lvert\nabla h\rvert}\otimes\frac{\nabla
h}{\lvert\nabla h\rvert}\right)\tau_{s}.$

Hence, recalling the evaluation of the eigenvalues of $D\sigma$ given in
subsection 2.1, we can easily compute the determinant of the tangential
gradient

$\begin{split}J^{\partial\Omega_{1}}\sigma(\zeta)=\left(\prod_{i=1}^{n-1}(1+h(\zeta)k_{i}(\sigma(\zeta)))\sqrt{1+\lvert\nabla
h(\zeta)\rvert^{2}}\right)^{-1},\end{split}$

thus concluding the proof.

∎

###  3.2 The main result

Fix $\beta>0$. For every $0<\varepsilon<1$ let $u_{\varepsilon}\in
H^{1}(\Sigma_{\varepsilon})$ be a solution to

$\varepsilon\int_{\Sigma_{\varepsilon}}\nabla
u_{\varepsilon}\nabla\varphi\,dx+\beta\int_{\partial\Omega_{\varepsilon}}u_{\varepsilon}\varphi\,d\mathcal{H}^{n-1}=0,$
(3.4)

for every $\varphi\in H^{1}(\Sigma_{\varepsilon})$ such that $\varphi=0$ on
$\partial\Omega$. Then we have the following.

######  Proposition .

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open set with $C^{1,1}$
boundary, and fix a positive function $h\in C^{1,1}(\Gamma_{d_{0}})$ such that
$\nabla h\cdot\nu_{0}=0$. Let $u_{\varepsilon}$ be a family of weak solution
to (3.4), and let
$\tilde{u}_{\varepsilon}(z)=u_{\varepsilon}(\Psi_{\varepsilon}(z))$. If
$\tilde{u}_{\varepsilon}$ converges weakly in $H^{1}(\Sigma_{1})$ to a
function $\tilde{u}_{0}$, then $\tilde{u}_{0}$ is a solution to

$\int_{\Sigma_{1}}(\nabla\tilde{u}_{0}\cdot\nu_{0})(\nabla\varphi\cdot\nu_{0})J_{0}(z)\,dz+\beta\int_{\partial\Omega_{1}}\tilde{u}_{0}(\zeta)\varphi(\zeta)\dfrac{J_{0}(\zeta)}{\sqrt{1+\lvert\nabla
h(\zeta)\rvert^{2}}}\,d\mathcal{H}^{n-1}(\zeta)=0.$ (3.5)

for every $\varphi\in H^{1}(\Sigma_{1})$ such that $\varphi=0$ on
$\partial\Omega$.

###### Proof.

By definition, we have $\tilde{u}_{\varepsilon}\in H^{1}(\Sigma_{1})$ and

$\begin{split}\nabla
u_{\varepsilon}|_{\Psi_{\varepsilon}(z)}&=D(\Psi_{\varepsilon}^{-1})|_{\Psi_{\varepsilon}(z)}\nabla\tilde{u}_{\varepsilon}(z)\\\\[10.0pt]
&=\left(\dfrac{1}{\varepsilon}\nabla\tilde{u}_{\varepsilon}(z)\cdot\nu_{0}(z)\right)\,\nu_{0}(z)+\sum_{i=1}^{n-1}\left(\dfrac{1+d(z)k_{i}(\sigma)}{1+\varepsilon
d(z)k_{i}(\sigma)}\nabla\tilde{u}_{\varepsilon}(z)\cdot\tau_{i}(z)\right)\,\tau_{i}(z).\end{split}$
(3.6)

Let $\varphi\in H^{1}(\Sigma_{1})$ with $\varphi=0$ on $\partial\Omega$, and
let $\varphi_{\varepsilon}(x)=\varphi(\Psi_{\varepsilon}^{-1}(x))$, then
equation (3.4) yields

$\varepsilon\int_{\Sigma_{\varepsilon}}\nabla
u_{\varepsilon}\nabla\varphi_{\varepsilon}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}u_{\varepsilon}\varphi_{\varepsilon}\,d\mathcal{H}^{n-1}=0,$

from which, using subsection 3.1 and the computation (3.6), we have

$\begin{split}\int_{\Sigma_{1}}\left[(\nabla\tilde{u}_{\varepsilon}\cdot\nu_{0})(\nabla\varphi\cdot\nu_{0})+\varepsilon^{2}\sum_{i=1}^{n-1}\left(\dfrac{1+dk_{i}}{1+\varepsilon
dk_{i}}\right)^{2}(\nabla\tilde{u}_{\varepsilon}\cdot\tau_{i})(\nabla\varphi\cdot\tau_{i})\right]\prod_{i=1}^{n-1}\dfrac{1+\varepsilon
dk_{i}}{1+dk_{i}}\,dz+\\\\[15.0pt]
+\beta\int_{\partial\Omega_{1}}\tilde{u}_{\varepsilon}\varphi
J^{\tau}_{\varepsilon}\,d\mathcal{H}^{n-1}=0,\end{split}$ (3.7)

Passing to the limit in (3.7), the assertion follows. ∎

######  Remark  (Uniqueness).

For every given Dirichlet boundary condition on $\partial\Omega$, (3.5) admits
a unique solution. Indeed, let $v_{1},v_{2}\in H^{1}(\Sigma_{1})$ be two
solutions to (3.5) such that $v_{1}=v_{2}$ on $\partial\Omega$, and let
$w=v_{1}-v_{2}$. By linearity, we have that $w$ is a solution to (3.5) with
$w=0$ on $\partial\Omega$, so that we have

$\int_{\Sigma_{1}}\lvert\nabla
w\cdot\nu_{0}\rvert^{2}J_{0}\,dz+\beta\int_{\partial\Omega_{1}}w^{2}\dfrac{J_{0}}{\sqrt{1+\lvert\nabla
h\rvert^{2}}}\,d\mathcal{H}^{n-1}=0,$

and since $J_{0}>0$, then $\nabla w\cdot\nu_{0}=0$ a.e. on
$\Sigma_{\varepsilon}$. Then, for $\mathcal{L}^{n}$-a.e. $z\in\Sigma_{1}$,

$w(z)=w(\sigma(z))+\int_{0}^{d(z)}\nabla
w(\sigma(z)+t\nu_{0})\cdot\nu_{0}\,dt=0,$

that is $w=0$ and $v_{1}=v_{2}$.

######  Remark  (Limit computation).

We point out that it is possible to explicitly compute the solution
$\tilde{u}_{0}$ to (3.5) in terms of its values on $\partial\Omega$ as

$\tilde{u}_{0}(z)=\tilde{u}_{0}(\sigma(z))\left(1-\frac{\beta d(z)}{1+\beta
h(z)}\right).$

We first inspect the regular case of $\Omega$ of class $C^{3}$ to obtain the
strong equation, and then we work on the general case using only the weak
equation (3.5).

Indeed, let

$A(z)=J_{0}(z)\nu_{0}(z)\otimes\nu_{0}(z),$

and notice that when $\Omega$ is smooth, then $J_{0}$ is also smooth as it can
be seen as the determinant of the smooth matrix-valued map

$z\in\Sigma_{1}\mapsto D\sigma+\nu_{0}\otimes\nu_{0}.$

Then, if $\tilde{u}_{0}$ is a regular solution to (3.5), it is a solution to

$\begin{cases}\operatorname{div}\left(A(x)\nabla\tilde{u}_{0}\right)=0&\text{in
}\Sigma_{1},\\\\[5.0pt]
\dfrac{\partial\tilde{u}_{0}}{\partial\nu_{0}}+\beta\tilde{u}_{0}=0&\text{on
}\partial\Omega_{1}.\end{cases}$ (3.8)

Moreover, we have that

$\operatorname{div}(J_{0}\nu_{0})=\nabla
J_{0}\cdot\nu_{0}+J_{0}\operatorname{Tr}(D\nu_{0}).$

In particular, we can explicitly compute the derivative of $J_{0}$ in
direction $\nu_{0}$ using the local representation

$J_{0}(z)=\prod_{i=1}^{n-1}\frac{1}{1+d(z)k_{i}(\sigma(z))},$

and recalling that $k_{i}\circ\sigma$ is constant along normal radii, so that,
for every $\zeta\in\partial\Omega_{1}$,

$\nabla
J_{0}(\zeta)\cdot\nu_{0}(\zeta)=-J_{0}(\zeta)\operatorname{Tr}(D\nu_{0}(\zeta)).$

Hence, $\operatorname{div}(J_{0}\nu_{0})=0$, and

$\operatorname{div}(A\nabla u)=J_{0}\nabla(\nabla u\cdot\nu_{0})\cdot\nu_{0}.$

Therefore, equation (3.8) reduces to

$\begin{cases}\nabla(\nabla\tilde{u}_{0}\cdot\nu_{0})\cdot\nu_{0}=0&\text{in
}\Sigma_{1},\\\\[5.0pt]
\dfrac{\partial\tilde{u}_{0}}{\partial\nu_{0}}+\beta\tilde{u}_{0}=0&\text{on
}\partial\Omega_{1}.\end{cases}$ (3.9)

The previous computation suggests that solutions to (3.5) have to be linear
with respect to the normal direction. Indeed, Since $h$ is constant along the
normal direction, it is sufficient to check that, for every $w\in
H^{1}(\Sigma_{1})$ the function

$\tilde{u}(z)=w(\sigma(z))\left(1-\dfrac{\beta d(z)}{1+\beta h(z)}\right)$

is a solution to (3.9) in $H^{1}(\Sigma_{1})$. Finally, we can show that the
previous solution to (3.9) is the solution to (3.5) also in the case in which
$\Omega$ is only $C^{1,1}$. By a change of variables and coarea formula (see
section 4) we can rewrite the integral on $\Sigma_{1}$ as

$\int_{\Sigma_{1}}g(z)J_{0}(z)\,dz=\int_{\partial\Omega}\int_{0}^{h(\sigma)}g(\sigma+t\nu_{0})\,dt\,d\mathcal{H}^{n-1}(\sigma)$

and the integral on $\partial\Omega_{1}$ as

$\int_{\partial\Omega_{1}}g(\xi)\dfrac{J_{0}(\xi)}{\sqrt{1+\lvert\nabla
h\rvert^{2}}}\,d\mathcal{H}^{n-1}(\xi)=\int_{\partial\Omega}g(\sigma+h(\sigma)\nu_{0})\,d\mathcal{H}^{n-1}(\sigma).$

Then by direct computation, we have that

$\nabla\tilde{u}\cdot\nu_{0}=-\dfrac{\beta w(\sigma(z))}{1+\beta h(z)}$

so that, for every smooth function $\varphi\in H^{1}(\Sigma_{1})$ with
$\varphi=0$ on $\partial\Omega$,

$\begin{split}\int_{\Sigma_{1}}(\nabla\tilde{u}_{0}\cdot\nu_{0})(\nabla\varphi\cdot\nu_{0})J_{0}(z)\,dz&=-\int_{\partial\Omega}\dfrac{\beta
w(\sigma}{1+\beta
h}\int_{0}^{h(\sigma)}\dfrac{d}{dt}(\varphi(\sigma+t\nu_{0}))\,dt\,d\mathcal{H}^{n-1}(\sigma)\\\\[15.0pt]
&=-\int_{\partial\Omega}\dfrac{\beta w(\sigma}{1+\beta
h}\int_{0}^{h(\sigma)}\varphi(\sigma+h(\sigma)\nu_{0})\,d\mathcal{H}^{n-1}\\\\[15.0pt]
&=-\beta\int_{\partial\Omega}\tilde{u}_{0}(\sigma+h(\sigma)\nu_{0})\varphi(\sigma+h(\sigma)\nu_{0})\,d\mathcal{H}^{n-1}\\\\[15.0pt]
&=-\beta\int_{\partial\Omega_{1}}\tilde{u}_{0}(\zeta)\varphi(\zeta)\dfrac{J_{0}(\zeta)}{\sqrt{1+\lvert\nabla
h(\zeta)\rvert^{2}}}\,d\mathcal{H}^{n-1}(\zeta)\end{split}$

and $\tilde{u}$ is in fact a solution to (3.5).

###  3.3 Proof of Theorem 1.1

Let $f\in L^{2}(\Omega)$ be a non-negative function and consider the
functional

$\mathcal{F}_{\varepsilon}(v,h)=\begin{cases}\displaystyle\int_{\Omega}\lvert\nabla
v\rvert^{2}\,dx+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
v\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}v^{2}\,d\mathcal{H}^{n-1}-2\int_{\Omega}fv\,dx&\text{if
}v\in H^{1}(\Omega_{\varepsilon}),\\\\[10.0pt] +\infty&\text{if }v\in
L^{2}(\mathbb{R}^{n})\setminus H^{1}(\Omega_{\varepsilon}),\end{cases}$

and let

$\mathcal{F}_{0}(v,h)=\begin{cases}\displaystyle\int_{\Omega}\lvert\nabla
v\rvert^{2}\,dx+\beta\int_{\partial\Omega}\dfrac{v^{2}}{1+\beta
h}\,d\mathcal{H}^{n-1}-2\int_{\Omega}fv\,dx&\text{if }v\in
H^{1}(\Omega),\\\\[10.0pt] +\infty&\text{if }v\in
L^{2}(\mathbb{R}^{n})\setminus H^{1}(\Omega),\end{cases}$

in [11] the authors prove the following

######  Theorem 3.1.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open set with $C^{1,1}$
boundary, and fix a positive Lipschitz function
$h\colon\partial\Omega\to\mathbb{R}$. Then
$\mathcal{F}_{\varepsilon}(\cdot,h)$ $\Gamma$-converges in the strong
$L^{2}(\mathbb{R}^{n})$ topology, as $\varepsilon\to 0^{+}$, to
$\mathcal{F}_{0}(\cdot,h)$.

For every $\varepsilon>0$, let $u_{\varepsilon,h}=u_{\varepsilon}\in
H^{1}(\Omega_{\varepsilon})$ be the minimizer of
$\mathcal{F}_{\varepsilon}(\cdot,h)$, then $u_{\varepsilon}$ is a solution to
the following boundary value problem

$\begin{cases}-\Delta u_{\varepsilon}=f&\text{in }\Omega,\\\\[3.0pt]
u_{\varepsilon}^{-}=u_{\varepsilon}^{+}&\text{on }\partial\Omega,\\\\[3.0pt]
\dfrac{\partial
u_{\varepsilon}^{-}}{\partial\nu_{0}\hphantom{\scriptstyle{-}}}=\varepsilon\dfrac{\partial
u_{\varepsilon}^{+}}{\partial\nu_{0}\hphantom{\scriptstyle{+}}}&\text{on
}\partial\Omega,\\\\[6.0pt] \Delta u_{\varepsilon}=0&\text{in
}\Sigma_{\varepsilon},\\\\[3.0pt] \varepsilon\dfrac{\partial
u_{\varepsilon}}{\partial\nu_{\varepsilon}}+\beta u_{\varepsilon}=0&\text{on
}\partial\Omega_{\varepsilon},\end{cases}$

where $u_{\varepsilon}^{-}$ and $u_{\varepsilon}^{+}$ denote the trace on
$u_{\varepsilon}$ from the inside $\Omega$ and from the outside of $\Omega$
respectively. By the properties of $\Gamma$-convergence, we have that

$u_{\varepsilon}\xrightharpoonup{H^{1}(\Omega)}u_{0},$ (3.10)

where $u_{0}=u_{0,h}$ is the minimizer of $\mathcal{F}_{0}(\cdot,h)$. Namely,
$u_{0}$ is the solution to

$\begin{cases}-\Delta u_{0}=f&\text{in }\Omega,\\\\[5.0pt] \dfrac{\partial
u_{0}}{\partial\nu_{0}}+\dfrac{\beta}{1+\beta h}u_{0}=0&\text{on
}\partial\Omega.\end{cases}$

We now prove the weak convergence of the family of functions
$\tilde{u}_{\varepsilon}$ defined in Theorem 1.2. The proof revolves around
some energy estimates analogous to the one proved in [6] for the solutions to
a similar boundary value problem with a transmission condition.

######  Theorem 3.2.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open set with $C^{1,1}$
boundary, and fix a positive function $h\in C^{1,1}(\Gamma_{d_{0}})$ and
$\nabla h\cdot\nu_{0}=0$. Then there exists positive constants
$\varepsilon_{0}(\Omega)$, and $C(\Omega,h,\beta,f)$ such that if

$\varepsilon\lVert h\rVert_{C^{0,1}}\leq\varepsilon_{0}$ (3.11)

and $u_{\varepsilon}$ is the weak solution to (1.1), then

$\int_{\Omega}\lvert
D^{2}u_{\varepsilon}\rvert^{2}\,dx+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert
D^{2}u_{\varepsilon}\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}\lvert\nabla^{\partial\Omega_{\varepsilon}}u_{\varepsilon}\rvert^{2}\,d\mathcal{H}^{n-1}\leq
C.$ (3.12)

######  Remark .

We want to point out that the assumption

$\nabla h(x)\cdot\nu_{0}(x)=0\qquad\forall x\in\Gamma_{d_{0}}$

is not necessary to prove Theorem 3.2, but it makes the computations easier.

For simplicity’s sake, we postpone the technicalities of the proof of Theorem
3.2 to section 5, and we directly prove Theorem 1.1. Our aim is to have
uniform $H^{1}$ estimates for $\tilde{u}_{\varepsilon}$, so we first show an
immediate consequence of Theorem 3.2. For every $x\in\Gamma_{d_{0}}$, we
denote by

$\nabla^{\partial\Omega}u_{\varepsilon}(x)=\nabla{u}_{\varepsilon}(x)-\left(\nabla{u}_{\varepsilon}(x)\cdot\nu_{0}(x)\right)\nu_{0}(x),$

and we have the following.

######  Corollary .

There exists a positive constant $C=C(\Omega,h,\beta,f)$ such that

$\int_{\Sigma_{\varepsilon}}\lvert\nabla^{\partial\Omega}u_{\varepsilon}\rvert^{2}\,dx\leq\varepsilon
C.$

###### Proof.

We start by proving that on $\partial\Omega_{\varepsilon}$

$\lvert\nabla^{\partial\Omega_{\varepsilon}}u_{\varepsilon}\rvert^{2}\geq(1-\varepsilon^{2})\lvert\nabla^{\partial\Omega}u_{\varepsilon}\rvert^{2}.$
(3.13)

Indeed, for every $\xi\in\partial\Omega_{\varepsilon}$, if $\lvert\nabla
h(\xi)\rvert=0$, then $\nu_{0}=\nu_{\varepsilon}$ and the inequality is
trivially true; on the other hand, if $\lvert\nabla h(\xi)\rvert\neq 0$ we
have that

$\lvert\nabla^{\partial\Omega_{\varepsilon}}u_{\varepsilon}\rvert^{2}=\lvert\nabla
u_{\varepsilon}\rvert^{2}-\lvert\nabla
u_{\varepsilon}\cdot\nu_{\varepsilon}\rvert^{2},$

and for every $\eta>0$

$\lvert\nabla
u_{\varepsilon}\cdot\nu_{\varepsilon}\rvert^{2}\leq\dfrac{(1+\eta)\lvert\nabla
u_{\varepsilon}\cdot\nu_{0}\rvert^{2}+\left(1+\dfrac{1}{\eta}\right)\varepsilon^{2}\lvert\nabla
u_{\varepsilon}\cdot\nabla h\rvert^{2}}{1+\lvert\nabla h\rvert^{2}}.$

Moreover $\nabla h\cdot\nu_{0}=0$, so that

$\lvert\nabla u_{\varepsilon}\cdot\nabla
h\rvert^{2}=\lvert\nabla^{\partial\Omega}u_{\varepsilon}\cdot\nabla
h\rvert^{2}\leq(\lvert\nabla u_{\varepsilon}\rvert^{2}-\lvert\nabla
u_{\varepsilon}\cdot\nu_{0}\rvert^{2})\lvert\nabla h\rvert^{2}.$

Therefore, for every $\eta>0$

$\lvert\nabla^{\partial\Omega_{\varepsilon}}u_{\varepsilon}\rvert^{2}\geq\left(1-\dfrac{\left(1+\dfrac{1}{\eta}\right)\varepsilon^{2}\lvert\nabla
h\rvert^{2}}{1+\lvert\nabla h\rvert^{2}}\right)\lvert\nabla
u_{\varepsilon}\rvert^{2}-\dfrac{1+\eta-\left(1+\dfrac{1}{\eta}\right)\varepsilon^{2}\lvert\nabla
h\rvert^{2}}{1+\lvert\nabla h\rvert^{2}}\lvert\nabla
u_{\varepsilon}\cdot\nu_{0}\rvert^{2},$

finally, letting $\eta=\lvert\nabla h\rvert^{2}$, we have

$\lvert\nabla^{\partial\Omega_{\varepsilon}}u_{\varepsilon}\rvert^{2}\geq(1-\varepsilon^{2})(\lvert\nabla
u_{\varepsilon}\rvert^{2}-\lvert\nabla
u_{\varepsilon}\cdot\nu_{0}\rvert^{2})=(1-\varepsilon^{2})\lvert\nabla^{\partial\Omega}u_{\varepsilon}\rvert^{2}.$

Then, by Theorem 3.2 and (3.13), we have that

$\varepsilon\int_{\Sigma_{\varepsilon}}\lvert
D^{2}u_{\varepsilon}\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}\lvert\nabla^{\partial\Omega}u_{\varepsilon}\rvert^{2}\,d\mathcal{H}^{n-1}\leq
C.$ (3.14)

For $x\in\Sigma_{\varepsilon}$ denote by

$\xi(x)=\sigma(x)+\varepsilon h(x)\nu_{0}(x)\in\partial\Omega_{\varepsilon},$

so that for $\mathcal{L}^{n}$-a.e. $x\in\Sigma_{\varepsilon}$, we have that

$\nabla^{\partial\Omega}u_{\varepsilon}(x)=\nabla^{\partial\Omega}u_{\varepsilon}(\xi(x))-\int_{d(x)}^{\varepsilon
h(x)}\dfrac{d}{dt}\left(\nabla^{\partial\Omega}u_{\varepsilon}(\sigma(x)+t\nu_{0}(x))\right)\,dt.$

Hence,

$\lvert\nabla^{\partial\Omega}u_{\varepsilon}\rvert(x)^{2}\leq
C(h)\left(\lvert\nabla^{\partial\Omega}u_{\varepsilon}\rvert^{2}(\xi(x))+\varepsilon\int_{0}^{\varepsilon
h(x)}\lvert D^{2}u_{\varepsilon}\rvert^{2}(\sigma(x)+t\nu_{0}(x))\,dt\right),$

and integrating over $\Sigma_{\varepsilon}$, using (4.3) and (4.4), and
(3.14), we get

$\begin{split}\int_{\Sigma_{\varepsilon}}\lvert\nabla^{\partial\Omega}u_{\varepsilon}\rvert^{2}(x)\,dx\leq&\,C\int_{\partial\Omega}\int_{0}^{\varepsilon
h(\sigma)}\lvert\nabla^{\partial\Omega}u_{\varepsilon}\rvert^{2}(\sigma+\varepsilon
h(\sigma)\nu_{0})\,ds\,d\mathcal{H}^{n-1}(\sigma)+\\\\[15.0pt]
&+C\varepsilon\int_{\partial\Omega}\int_{0}^{\varepsilon
h(\sigma)}\int_{0}^{\varepsilon h(\sigma)}\lvert
D^{2}u_{\varepsilon}\rvert^{2}(\sigma(x)+t\nu_{0}(x))\,dt\,ds\,d\mathcal{H}^{n-1}(\sigma)\\\\[15.0pt]
\leq&\,\varepsilon
C\left(\int_{\partial\Omega_{\varepsilon}}\lvert\nabla^{\partial\Omega}u_{\varepsilon}\rvert^{2}\,d\mathcal{H}^{n-1}+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert
D^{2}u_{\varepsilon}\rvert^{2}\,dx\right)\\\\[15.0pt] \leq&\,\varepsilon
C.\end{split}$

∎

We are now in a position to prove the $H^{1}$ convergence of the family
$\tilde{u}_{\varepsilon}$.

###### Proof of Theorem 1.1.

We recall that we are assuming without loss of generality $\lVert
h\rVert_{\infty}<d_{0}$, so that $d\in C^{1,1}(\Sigma_{1})$. To prove the
equiboundedness in $H^{1}(\Omega_{1})$, we decompose
$\nabla\tilde{u}_{\varepsilon}$ into its normal part

$(\nabla\tilde{u}_{\varepsilon}\cdot\nu_{0})\,\nu_{0},$

and its tangential part

$\nabla^{\partial\Omega}\tilde{u}_{\varepsilon}:=\nabla\tilde{u}_{\varepsilon}-\left(\nabla\tilde{u}_{\varepsilon}\cdot\nu_{0}\right)\nu_{0}.$

Using subsection 3.1, since $J_{\varepsilon}$ and $J_{\varepsilon}^{\tau}$ are
equibounded, we can find a positive constant $C=C(\Omega,h)$ such that

$\int_{\Sigma_{1}}\lvert\nabla^{\partial\Omega}\tilde{u}_{\varepsilon}\rvert^{2}\,dz\leq\dfrac{C}{\varepsilon}\int_{\Sigma_{\varepsilon}}\lvert\nabla^{\partial\Omega}u_{\varepsilon}\rvert^{2}\,dx,$
(3.15)
$\int_{\Sigma_{1}}\lvert\nabla\tilde{u}_{\varepsilon}\cdot\nu_{0}\rvert^{2}\,dz\leq\varepsilon
C\int_{\Sigma_{\varepsilon}}\lvert\nabla u_{\varepsilon}\rvert^{2}\,dx,$
(3.16)

and

$\int_{\partial\Omega_{1}}\tilde{u}_{\varepsilon}^{2}\,d\mathcal{H}^{n-1}\leq
C\int_{\partial\Omega_{\varepsilon}}u_{\varepsilon}^{2}\,d\mathcal{H}^{n-1}.$
(3.17)

Moreover, by the weak convergence of $u_{\varepsilon}$ in $H^{1}(\Omega)$,
$u_{\varepsilon}$ are equibounded in $L^{2}(\Omega)$, while by the minimality

$\begin{split}\int_{\Omega}\lvert\nabla
u_{\varepsilon}\rvert^{2}\,dx+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
u_{\varepsilon}\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}u_{\varepsilon}^{2}\,d\mathcal{H}^{n-1}&\leq\mathcal{F}_{\varepsilon}(0)+2\int_{\Omega}fu_{\varepsilon}\,dx\\\\[7.0pt]
&\leq\int_{\Omega}(f^{2}+u_{\varepsilon}^{2})\,dx\\\\[7.0pt] &\leq
C\end{split}$ (3.18)

Joining the inequalities (3.15), (3.16), (3.17), and (3.18) with the energy
estimates in subsection 3.3, we have that for some positive constant
$C(\Omega,h,\beta)$

$\int_{\Omega_{1}}\lvert\nabla\tilde{u}_{\varepsilon}\rvert^{2}\,dz+\beta\int_{\partial\Omega_{1}}\tilde{u}_{\varepsilon}^{2}\,d\mathcal{H}^{n-1}\leq
C.$

By Poincaré’s inequality with boundary term, we have that
$\tilde{u}_{\varepsilon}$ are equibounded in $H^{1}(\Omega_{1})$. Therefore,
up to a subsequence, $\tilde{u}_{\varepsilon}$ converges weakly in
$H^{1}(\Omega_{1})$ to some function $\tilde{u}_{0}$. In particular, by the
weak convergence (3.10) inside $\Omega$, we have $\tilde{u}_{0}=u_{0}$ in
$\Omega$. On the other hand, in $\Sigma_{1}$, using subsection 3.2, we get
that $\tilde{u}_{0}$ is a solution to (3.5), so that subsection 3.2 ensures
that

$\tilde{u}_{0}(z)=\tilde{u}_{0}(\sigma(z))\left(1-\dfrac{\beta d(z)}{1+\beta
h(z)}\right).$

Finally, since $\tilde{u}_{0}\in H^{1}(\Omega_{1})$, then it cannot jump
across $\partial\Omega$, so that, since $\tilde{u}_{0}=u_{0}$ in $\Omega$, we
necessarily have that $\tilde{u}_{0}(\sigma(z))=u_{0}(\sigma(z))$ for a.e.
$z\in\Sigma_{1}$, and the theorem is proved. ∎

##  4 Asymptotic Development

In this section, we study the first-order asymptotic development by
$\Gamma$-convergence of the functional $\mathcal{F}_{\varepsilon}(\cdot,h)$.
Let us recall the notation introduced in subsection 2.1

$\gamma_{t}=\partial(\Omega\cup\Gamma_{t})=\Set{x\in\mathbb{R}^{n}}{d(x)=t}\setminus\Omega.$

######  Remark .

Using the coarea formula (Theorem 2.2) with the distance function $d$, we have
that for every $g\in L^{1}(\Omega_{\varepsilon})$

$\int_{\Sigma_{\varepsilon}}g(x)\,dx=\int_{0}^{+\infty}\int_{\gamma_{t}}g(\xi)\,\chi_{\Sigma_{\varepsilon}}\\!(\xi)\,d\mathcal{H}^{n-1}(\xi)\,dt.$

Let

$\phi_{t}\colon x\in\Gamma_{d_{0}}\mapsto x+t\nu_{0}(x)\in\Gamma_{td_{0}},$

then $\gamma_{t}=\phi_{t}(\partial\Omega)$, and by the area formula on
surfaces (Theorem 2.1)

$\begin{split}\int_{\Sigma_{\varepsilon}}g(x)\,dx&=\int_{0}^{+\infty}\int_{\partial\Omega}g(\sigma+t\nu_{0})\chi_{\Sigma_{\varepsilon}}(\sigma+t\nu_{0})J^{\tau}\phi_{t}(\sigma)\,d\mathcal{H}^{n-1}(\sigma)\,dt\\\\[10.0pt]
&=\int_{\partial\Omega}\int_{0}^{\varepsilon
h(\sigma)}g(\sigma+t\nu_{0})\prod_{i=1}^{n-1}(1+tk_{i}(\sigma))\,dt\,d\mathcal{H}^{n-1}(\sigma).\end{split}$

Similarly,

$\int_{\partial\Omega_{\varepsilon}}g(\xi)\,d\mathcal{H}^{n-1}(\xi)=\int_{\partial\Omega}g(\sigma+\varepsilon
h\nu_{0})\prod_{i=1}^{n-1}(1+\varepsilon
h(\sigma)k_{i}(\sigma))\sqrt{1+\varepsilon^{2}\lvert\nabla
h\rvert^{2}}\,d\mathcal{H}^{n-1}(\sigma).$

so that we have

$\int_{\Sigma_{\varepsilon}}g(x)\,dx=\int_{\partial\Omega}\int_{0}^{\varepsilon
h(\sigma)}g(\sigma+t\nu_{0})\left(1+tH(\sigma)+\varepsilon^{2}R_{1}(\sigma,t,\varepsilon)\right)\,dt\,d\mathcal{H}^{n-1}$
(4.1)

and

$\int_{\partial\Omega_{\varepsilon}}g(\sigma)\,d\mathcal{H}^{n-1}=\int_{\partial\Omega}g(\sigma+\varepsilon
h\nu_{0})\left(1+\varepsilon
h(\sigma)H(\sigma)+\varepsilon^{2}R_{2}(\sigma,\varepsilon)\right)\,d\mathcal{H}^{n-1},$
(4.2)

where the remainder terms $R_{1}$ and $R_{2}$ are bounded functions. In other
words, there exists $Q=Q(\Omega,h)>0$ such that $\lvert R_{1}\rvert,\lvert
R_{2}\rvert\leq Q$.

In particular, notice that there exists a positive constant $C=C(\Omega,\lVert
h\rVert_{0,1})$ such that for every $0<\varepsilon<1$

$\dfrac{1}{C}\int_{\Sigma_{\varepsilon}}g\,dx\leq\int_{\partial\Omega}\int_{0}^{\varepsilon
h(\sigma)}g(\sigma+t\nu_{0})\,dt\,d\mathcal{H}^{n-1}\leq
C\int_{\Sigma_{\varepsilon}}g\,dx,$ (4.3)

and

$\dfrac{1}{C}\int_{\partial\Omega_{\varepsilon}}g\,dx\leq\int_{\partial\Omega}g(\sigma+\varepsilon
h\nu_{0})\,d\mathcal{H}^{n-1}\leq C\int_{\partial\Omega_{\varepsilon}}g\,dx.$
(4.4)

We can now follow the lead of [2] to prove Theorem 1.2.

###### Proof of Theorem 1.2.

We start by proving the $\Gamma$-liminf inequality: without loss of
generality, we can prove the inequality for the sequence of minimizers
$u_{\varepsilon}$. Here we recall the definitions of
$\mathcal{F}_{\varepsilon}$ and $\mathcal{F}_{0}$, omitting the dependence on
$h$.

$\mathcal{F}_{\varepsilon}(u_{\varepsilon})=\int_{\Omega}\lvert\nabla
u_{\varepsilon}\rvert^{2}\,dx+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
u_{\varepsilon}\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}u_{\varepsilon}^{2}\,d\mathcal{H}^{n-1}-2\int_{\Omega}fu_{\varepsilon}\,dx,$
(4.5) $\mathcal{F}_{0}(u_{0})=\int_{\Omega}\lvert\nabla
u_{0}\rvert^{2}\,dx+\beta\int_{\partial\Omega}\dfrac{u_{0}^{2}}{1+\beta
h}\,d\mathcal{H}^{n-1}-2\int_{\Omega}fu_{0}\,dx.$ (4.6)

Moreover, notice that, by minimality of $u_{0}$,

$\mathcal{F}_{\varepsilon}(u_{\varepsilon})-\mathcal{F}_{0}(u_{0})\geq\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
u_{\varepsilon}\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}u_{\varepsilon}^{2}\,d\mathcal{H}^{n-1}-\beta\int_{\partial\Omega}\frac{u_{\varepsilon}^{2}}{1+\beta
h}\,d\mathcal{H}^{n-1}$ (4.7)

By (4.1) and (4.2) we have

$\int_{\Sigma_{\varepsilon}}\lvert\nabla
u_{\varepsilon}\rvert^{2}\,dx\geq\int_{\partial\Omega}\int_{0}^{\varepsilon
h(\sigma)}\lvert\nabla
u_{\varepsilon}(\sigma+t\nu_{0})\rvert^{2}\left(1+tH(\sigma)-\varepsilon^{2}Q\right)\,dt\,d\mathcal{H}^{n-1}$
(4.8)

and

$\dfrac{\beta}{\varepsilon}\int_{\partial\Omega_{\varepsilon}}u_{\varepsilon}^{2}\,\mathcal{H}^{n-1}\geq\dfrac{\beta}{\varepsilon}\int_{\partial\Omega}u_{\varepsilon}^{2}(\sigma+\varepsilon
h(\sigma)\nu_{0}(\sigma))\left(1+\varepsilon
h(\sigma)H(\sigma)-\varepsilon^{2}Q\right)\,d\mathcal{H}^{n-1}.$ (4.9)

We choose $\varepsilon$ sufficiently small, so that for every
$\sigma\in\partial\Omega$, and $0<t<\varepsilon h(\sigma)$, we have that
$1+tH(\sigma)>0$, and, in particular, using Holder’s inequality and
integrating by parts,

$\begin{split}\int_{0}^{\varepsilon h}\lvert\nabla
u_{\varepsilon}(\sigma+t\nu_{0})\rvert^{2}\left(1+tH\right)\,dt&\geq\dfrac{1}{\varepsilon
h}\\!\left(\int_{0}^{\varepsilon h}\lvert\nabla
u_{\varepsilon}(\sigma+t\nu_{0})\rvert\sqrt{1+tH}\,dt\right)^{2}\\\\[10.0pt]
&\geq\dfrac{1}{\varepsilon h}\\!\left(\int_{0}^{\varepsilon
h}\dfrac{d}{dt}(u_{\varepsilon}(\sigma+t\nu_{0}))\sqrt{1+tH}\,dt\right)^{2}\\\\[10.0pt]
&\geq\frac{1}{\varepsilon h}\\!\left(u_{\varepsilon}(\sigma+\varepsilon
h\nu_{0})\sqrt{1+\varepsilon
hH}-\\!\left(u_{\varepsilon}(\sigma)+\\!\\!\displaystyle\int_{0}^{\varepsilon
h}\dfrac{Hu_{\varepsilon}(\sigma+t\nu_{0})}{2\sqrt{1+tH}}\,dt\right)\right)^{\\!2}\\!.\end{split}$

We then apply Young’s inequality to the absolute value of the double product,
having that for every $\lambda>0$

$\begin{split}\int_{0}^{\varepsilon h(\sigma)}\lvert\nabla
u_{\varepsilon}(\sigma+t\nu_{0})\rvert^{2}\left(1+tH(\sigma)\right)\,dt\geq&\dfrac{(1-\lambda)(1+\varepsilon
hH)u_{\varepsilon}(\sigma+\varepsilon h\nu_{0})^{2}}{\varepsilon
h}\\\\[10.0pt] &+\dfrac{1}{\varepsilon
h}\left(1-\dfrac{1}{\lambda}\right)\left(u_{\varepsilon}(\sigma)+\int_{0}^{\varepsilon
h(\sigma)}\dfrac{Hu_{\varepsilon}(\sigma+t\nu_{0})}{2\sqrt{1+tH}}\,dt\right)^{2}.\end{split}$
(4.10)

We then have, joining (4.7), (4.8), (4.10), and (4.9),

$\begin{split}\delta\mathcal{F}_{\varepsilon}(u_{\varepsilon})=&\dfrac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})-\mathcal{F}_{0}(u_{0})}{\varepsilon}\\\\[7.0pt]
\geq&\int_{\partial\Omega}\dfrac{1}{\varepsilon h(\sigma)}(1-\lambda+\beta
h)(1+\varepsilon hH)u_{\varepsilon}^{2}(\sigma+\varepsilon
h\nu_{0})\,d\mathcal{H}^{n-1}\\\\[10.0pt]
&+\int_{\partial\Omega}\dfrac{1}{\varepsilon
h}\left(\left(1-\dfrac{1}{\lambda}\right)\left(u_{\varepsilon}(\sigma)+\int_{0}^{\varepsilon
h}\dfrac{Hu_{\varepsilon}(\sigma+t\nu_{0})}{2\sqrt{1+tH}}\,dt\right)^{2}-\dfrac{\beta
h\,u_{\varepsilon}^{2}(\sigma)}{1+\beta
h}\right)\,d\mathcal{H}^{n-1}-Q\varepsilon
R(\varepsilon,u_{\varepsilon})\end{split}$ (4.11)

where, if $\varepsilon$ is small enough,

$\begin{split}R(\varepsilon,u_{\varepsilon})&=\varepsilon\int_{\partial\Omega}\int_{0}^{\varepsilon
h(\sigma)}\lvert\nabla
u_{\varepsilon}(\sigma+t\nu_{0})\rvert^{2}\,d\mathcal{H}^{n-1}+\beta\int_{\partial\Omega}u_{\varepsilon}(\sigma+\varepsilon
h(\sigma)\nu_{0}(\sigma))^{2}\,d\mathcal{H}^{n-1}\\\\[10.0pt] &\leq
C\int_{\Omega}fu_{\varepsilon}\,dx<C.\end{split}$

Letting $\lambda=\lambda(\sigma)=1+\beta h(\sigma)$ in (4.11), and using the
inequality $(a+b)^{2}\geq a^{2}+2ab$, joint with the fact that
$1-\lambda^{-1}>0$,

$\delta\mathcal{F}_{\varepsilon}(u_{\varepsilon})\geq\int_{\partial\Omega}\dfrac{\beta
Hu_{\varepsilon}(\sigma)}{\varepsilon(1+\beta h)}\int_{0}^{\varepsilon
h(\sigma)}\dfrac{u_{\varepsilon}(\sigma+t\nu_{0})}{\sqrt{1+tH}}\,dt\,d\mathcal{H}^{n-1}+O(\varepsilon).$

Moreover, for every $t\in(0,\varepsilon\lVert h\rVert_{\infty})$ we have that
$(1+tH)^{-1/2}=1+O(\varepsilon)$, so that

$\delta\mathcal{F}_{\varepsilon}(u_{\varepsilon})\geq\beta\int_{\partial\Omega}\dfrac{Hu_{\varepsilon}(\sigma)}{(1+\beta
h)}\int_{0}^{h(\sigma)}\tilde{u}_{\varepsilon}(\sigma+t\nu_{0})\,dt\,d\mathcal{H}^{n-1}+O(\varepsilon).$
(4.12)

Finally, by Theorem 1.1 we get

$\tilde{u}_{\varepsilon}\xrightarrow[]{L^{2}(\Sigma_{1})}\tilde{u}_{0},$

so that

$\int_{0}^{h(\sigma)}\tilde{u}_{\varepsilon}(\sigma+t\nu_{0})\,dt\xrightarrow[]{L^{2}(\partial\Omega)}\int_{0}^{h(\sigma)}\tilde{u}_{0}(\sigma+t\nu_{0})\,dt.$

Indeed, by (4.3),

$\int_{\partial\Omega}\left(\int_{0}^{h(\sigma)}\left(\tilde{u}_{\varepsilon}(\sigma+t\nu_{0})-\tilde{u}_{0}(\sigma+t\nu_{0})\right)\,dt\right)^{2}\,d\mathcal{H}^{n-1}\leq
C\int_{\Sigma_{1}}(\tilde{u}_{\varepsilon}-\tilde{u}_{0})^{2}\,dz.$

Therefore, passing to the limit in (4.12), and using the explicit expression
of $\tilde{u}_{0}$, we get

$\liminf_{\varepsilon\to
0^{+}}\delta\mathcal{F}_{\varepsilon}(u_{\varepsilon})\geq\beta\int_{\partial\Omega}\dfrac{hH(2+\beta
h)}{2(1+\beta h)^{2}}u_{0}^{2}(\sigma)\,d\mathcal{H}^{n-1},$

and the $\Gamma$-liminf is proved.

We now prove the $\Gamma$-limsup inequality.

Let

$\varphi_{\varepsilon}(x)=\begin{dcases}u_{0}(x)&\text{if
}x\in\Omega,\\\\[5.0pt] u_{0}(\sigma(x))\left(1-\dfrac{\beta
d(x)}{\varepsilon(1+\beta h(x))}\right)&\text{if
}x\in\Sigma_{\varepsilon},\\\\[5.0pt] 0&\text{if
}x\in\mathbb{R}^{n}\setminus\Omega_{\varepsilon},\end{dcases}$

where we recall that $\nabla h\cdot\nu_{0}=0$. We have that
$\varphi_{\varepsilon}\in H^{1}(\Omega)$ and $\varphi_{\varepsilon}$ converges
in $L^{2}(\mathbb{R}^{n})$, to $u_{0}\chi_{\Omega}$. Since
$\varphi_{\varepsilon}\equiv u_{0}$ in $\Omega$, then by definition of the
functionals (4.5), and (4.6), we can write

$\mathcal{F}_{\varepsilon}(\varphi_{\varepsilon})-\mathcal{F}_{0}(u_{0})=\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla\varphi_{\varepsilon}\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}\varphi_{\varepsilon}^{2}\,d\mathcal{H}^{n-1}-\beta\int_{\partial\Omega}\frac{u_{0}^{2}}{1+\beta
h}\,d\mathcal{H}^{n-1}.$ (4.13)

Computing the gradient of $\varphi_{\varepsilon}$, for any
$x\in\Sigma_{\varepsilon}$,

$\lvert\nabla\varphi_{\varepsilon}\rvert^{2}(x)\leq\dfrac{\beta^{2}u_{0}^{2}(\sigma(x))}{\varepsilon^{2}(1+\beta
h)^{2}}+C\left(\lvert\nabla
u_{0}(\sigma(x))\rvert^{2}+u_{0}^{2}(\sigma(x))\right),$

where $C=C(h,\beta)$. Hence, by (4.1), and noticing that
$\sigma+t\nu_{0}\in\Gamma_{d_{0}}$ implies $d(\sigma+t\nu_{0}(\sigma))=t$, we
get

$\begin{split}\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla\varphi_{\varepsilon}\rvert^{2}\,dx\leq&\dfrac{\beta^{2}}{\varepsilon}\int_{\Sigma_{\varepsilon}}\dfrac{u_{0}^{2}(\sigma)}{(1+\beta
h)^{2}}\,dx+\varepsilon C\int_{\Sigma_{\varepsilon}}\left(\lvert\nabla
u_{0}(\sigma(x))\rvert^{2}+u_{0}^{2}(\sigma(x))\right)\,dx\\\\[10.0pt]
\leq&\beta^{2}\int_{\partial\Omega}\dfrac{u_{0}^{2}(\sigma)h}{(1+\beta
h)^{2}}\left(1+\dfrac{\varepsilon
hH}{2}\right)\,d\mathcal{H}^{n-1}+O(\varepsilon^{2}).\end{split}$ (4.14)

On the other hand, for every $\sigma\in\partial\Omega$,

$\varphi_{\varepsilon}(\sigma+\varepsilon
h(\sigma)\nu_{0}(\sigma))=\frac{u_{0}(\sigma)}{1+\beta h(\sigma)},$

from which we get

$\begin{split}\beta\int_{\partial{\Omega_{\varepsilon}}}\varphi_{\varepsilon}^{2}\,d\mathcal{H}^{n-1}\leq\beta\int_{\partial\Omega}\dfrac{u_{0}^{2}(\sigma)}{(1+\beta
h)^{2}}(1+\varepsilon hH)\,d\mathcal{H}^{n-1}+O(\varepsilon^{2}).\end{split}$
(4.15)

Finally, joining (4.13), (4.14), (4.15), we have

$\begin{split}\delta\mathcal{F}_{\varepsilon}(\varphi_{\varepsilon})=\dfrac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})-\mathcal{F}_{0}}{\varepsilon}&\leq\beta\int_{\partial\Omega}\dfrac{u_{0}^{2}(\sigma)hH}{(1+\beta
h)^{2}}\left(\dfrac{\beta
h}{2}+1\right)\,d\mathcal{H}^{n-1}+O(\varepsilon)\\\\[10.0pt]
&=\beta\int_{\partial\Omega}\dfrac{hH(2+\beta h)}{2(1+\beta
h)^{2}}\,u_{0}^{2}(\sigma)\,d\mathcal{H}^{n-1}+O(\varepsilon)\end{split}$

so that

$\limsup_{\varepsilon\to
0^{+}}\,\delta\mathcal{F}_{\varepsilon}(\varphi_{\varepsilon})\leq\beta\int_{\partial\Omega}\dfrac{hH(2+\beta
h)}{2(1+\beta h)^{2}}\,u_{0}^{2}(\sigma)\,d\mathcal{H}^{n-1}$

and the $\Gamma$-limsup inequality is proved. ∎

##  5 Energy estimates

The aim of this section is to prove Theorem 3.2. The proof is mainly divided
in two steps:

  1. Step 1.

We prove some $H^{1}$ uniform estimates for the minimizer $u_{\varepsilon}$
(see subsection 5.1).

  2. Step 2.

We prove the uniform $H^{2}$ estimates in Theorem 3.2 for the minimizers
$u_{\varepsilon}$ using a local argument similar to the approach in [6]: we
focus on small neighborhoods $V_{\sigma_{0}}$ of points
$\sigma_{0}\in\partial\Omega$, and we construct a diffeomorphism
$\Phi_{\sigma_{0}}$ that flattens both $\partial\Omega$ and
$\partial\Omega_{\varepsilon}$; on the flattened set
$\Phi(V_{\sigma_{0}}\cap\Sigma_{\varepsilon})$ we are able to compute energy
estimates of the new functions
$v_{\varepsilon}=u_{\varepsilon}\circ\Phi^{-1}$.

As in the previous sections, let $u_{\varepsilon}$ be the minimizer to

$\min\Set{\mathcal{F}_{\varepsilon}(v)}{v\in H^{1}(\Omega_{\varepsilon})},$
(5.1)

where

$\mathcal{F}_{\varepsilon}(v)=\int_{\Omega}\lvert\nabla
v\rvert^{2}\,dx-2\int_{\Omega}fv\,dx+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
v\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}v^{2}\,d\mathcal{H}^{n-1},$

and $f\in L^{2}(\Omega)$. The functions $u_{\varepsilon}$ can be equivalently
seen as the unique solutions to the equations

$\int_{\Omega}\nabla
u_{\varepsilon}\cdot\nabla\varphi\,dx+\varepsilon\int_{\Sigma_{\varepsilon}}\nabla
u_{\varepsilon}\cdot\nabla\varphi\,dx+\beta\int_{\partial\Omega_{\varepsilon}}u_{\varepsilon}\varphi\,d\mathcal{H}^{n-1}=\int_{\Omega}f\varphi\,dx,$
(5.2)

for every $\varphi\in H^{1}(\Omega_{\varepsilon})$. We notice that for every
$\varepsilon>0$, by standard elliptic regularity, $u_{\varepsilon}\in
H^{2}_{\operatorname{loc}}(\Omega_{\varepsilon}\setminus\partial\Omega)$.

###  5.1 Step 1: $H^{1}$ uniform estimates

We now estimate the $H^{1}$ norm of $u_{\varepsilon}$ in terms of
$\varepsilon$.

######  Lemma .

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open set with $C^{1,1}$
boundary, and fix a positive function $h\in C^{0,1}(\Gamma_{d_{0}})$ such that
$\nabla h\cdot\nu_{0}=0$. Then there exist positive constants
$\varepsilon_{0}(\Omega)$, and $C(\Omega,\lVert h\rVert_{C^{0,1}},\beta,f)$
such that if

$\varepsilon\lVert h\rVert_{\infty}\leq\varepsilon_{0},$ (5.3)

then for every $v\in H^{1}(\Omega_{\varepsilon})$

$\int_{\Omega_{\varepsilon}}v^{2}\,dx\leq C\left[\int_{\Omega}\lvert\nabla
v\rvert^{2}\,dx+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
v\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}v^{2}\,d\mathcal{H}^{n-1}\right].$
(5.4)

###### Proof.

Up to using a density argument, it is enough to prove the assertion for every
smooth function in $H^{1}(\Omega_{\varepsilon})$. Let $v\in
C^{1}(\overline{\Omega_{\varepsilon}})$. For every $x\in\Sigma_{\varepsilon}$
we recall that we can represent

$x=\sigma(x)+d(x)\nu_{0}(x),$

and we define

$\xi(x):=\sigma(x)+\varepsilon h(x)\,\nu_{0}(x).$

This construction allows us to write for every $x\in\Sigma_{\varepsilon}$

$v(x)=v(\xi(x))-\int_{d(x)}^{\varepsilon
h(x)}\dfrac{\partial}{\partial\nu_{0}}v(\sigma(x)+t\nu_{0}(x))\,dt.$

Hence, by Young and Hölder inequalities,

$v^{2}(x)\leq 2v^{2}(\xi(x))+2\varepsilon\lVert
h\rVert_{\infty}\int_{0}^{\varepsilon h(x)}\lvert\nabla
v\rvert^{2}(\sigma(x)+t\nu_{0}(x))\,dt.$ (5.5)

Integrating over $\Sigma_{\varepsilon}$, using (4.3) and (4.4), and recalling
that $\xi(x)=\xi(\sigma(x))$, we can find a positive constant
$C=C(\Omega,\lVert h\rVert_{C^{0,1}})$ such that

$\begin{split}\int_{\Sigma_{\varepsilon}}v^{2}\,dx\leq&C\int_{\partial\Omega}\int_{0}^{\varepsilon
h(\sigma)}v^{2}(\xi(\sigma))\,ds\,d\mathcal{H}^{n-1}(\sigma)\\\\[10.0pt]
&+\varepsilon C\int_{\partial\Omega}\int_{0}^{\varepsilon
h(\sigma)}\int_{0}^{\varepsilon h(\sigma)}\lvert\nabla
v\rvert^{2}(\sigma+t\nu_{0})\,dt\,ds\,d\mathcal{H}^{n-1}(\sigma)\\\\[10.0pt]
\leq&\varepsilon
C\left(\int_{\partial\Omega_{\varepsilon}}v^{2}\,d\mathcal{H}^{n-1}+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
v\rvert^{2}\,dx\right).\end{split}$ (5.6)

Similarly, we can integrate (5.5) over $\partial\Omega$ and have

$\int_{\partial\Omega}v^{2}\,d\mathcal{H}^{n-1}\leq
C\left(\int_{\partial\Omega_{\varepsilon}}v^{2}\,d\mathcal{H}^{n-1}+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
v\rvert^{2}\,dx\right).$ (5.7)

From the Poincaré inequality with trace term in $\Omega$ and the Bossel-Daners
inequality, we have

$\int_{\Omega}v^{2}\,dx\leq
C_{p}(\lvert\Omega\rvert)\left(\int_{\Omega}\lvert\nabla
v\rvert^{2}\,dx+\int_{\partial\Omega}v^{2}\,d\mathcal{H}^{n-1}\right),$ (5.8)

so that joining (5.6) and (5.8), and using (5.7) we have the assertion.

∎

######  Corollary .

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open set with $C^{1,1}$
boundary, and fix a positive function $h\in C^{0,1}(\Gamma_{d_{0}})$ such that
$\nabla h\cdot\nu_{0}=0$. Then there exists a positive constant
$C=C(\Omega,h,\beta,f)$ such that if $u_{\varepsilon}\in
H^{1}(\Omega_{\varepsilon})$ is the minimizer to (5.1), then

$\int_{\Omega}\lvert\nabla
u_{\varepsilon}\rvert^{2}\,dx+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
u_{\varepsilon}\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}u_{\varepsilon}^{2}\,d\mathcal{H}^{n-1}\leq
C,$ (5.9)

and

$\int_{\Omega_{\varepsilon}}u_{\varepsilon}^{2}\,dx\leq C.$ (5.10)

###### Proof.

For every $\eta>0$, we can write

$\begin{split}\int_{\Omega}\lvert\nabla
u_{\varepsilon}\rvert^{2}\,dx+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
u_{\varepsilon}\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}u_{\varepsilon}^{2}\,d\mathcal{H}^{n-1}&\leq\mathcal{F}_{\varepsilon}(0)+2\int_{\Omega}fu_{\varepsilon}\,dx\\\\[15.0pt]
&\leq\eta\int_{\Omega}f^{2}\,dx+\dfrac{1}{\eta}\int_{\Omega}u_{\varepsilon}^{2}\,dx.\end{split}$

Using (5.4), for a suitable choice of $\eta$, we get the result. ∎

###  5.2 Flattening the boundaries

Here we aim to construct a flattening diffeomorphism that locally transforms
$\partial\Omega$ and $\partial\Omega_{\varepsilon}$ in subsets of parallel
planes. To do so, we have to represent locally $\partial\Omega$ and
$\partial\Omega_{\varepsilon}$.

######  Lemma  (Uniform local representation of $\partial\Omega$ and
$\partial\Omega_{\varepsilon}$.).

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open set with $C^{1,1}$
boundary, fix a positive function $h\in C^{0,1}(\Gamma_{d_{0}})$ such that
$\nabla h\cdot\nu_{0}=0$, and let $\sigma_{0}\in\partial\Omega$. There exists
$\varepsilon_{0}=\varepsilon_{0}(\Omega,\sigma_{0})$ such that, if

$\varepsilon\lVert h\rVert_{C^{0,1}}<\varepsilon_{0},$

then there exist an open set $V$ containing $\sigma_{0}$ and
$\sigma_{0}+\varepsilon h(\sigma_{0})\nu_{0}(\sigma_{0})$, and there exist
functions $g,k_{\varepsilon}\colon\mathbb{R}^{n-1}\to\mathbb{R}$ such that, up
to a rototranslation,

$\begin{split}\Omega\cap V&=\Set{(x^{\prime},x_{n})}{x_{n}\leq
g(x^{\prime})}\cap V,\\\ \Omega_{\varepsilon}\cap
V&=\Set{(x^{\prime},x_{n})}{x_{n}\leq g(x^{\prime})+\varepsilon
k_{\varepsilon}(x^{\prime})}\cap V,\end{split}$

and

$\begin{split}\partial\Omega\cap
V&=\Set{(x^{\prime},x_{n})}{x_{n}=g(x^{\prime})}\cap V,\\\
\partial\Omega_{\varepsilon}\cap
V&=\Set{(x^{\prime},x_{n})}{x_{n}=g(x^{\prime})+\varepsilon
k_{\varepsilon}(x^{\prime})}\cap V,\end{split}$

###### Proof.

Without loss of generality we can assume that
$\nu_{0}(\sigma_{0})=\mathbf{e}_{n}$ and $\sigma_{0}=0$. We already know by
definition that $\Omega$ can be represented locally near $0$, that is, there
exist a neighborhood $U$ of $\sigma_{0}$ and a function $g\in
C^{1,1}(\mathbb{R}^{n-1})$ with $g(0)=0$ and $\nabla g(0)=0$, such that

$\Omega\cap U=\Set{(x^{\prime},x_{n})\in\mathbb{R}^{n}}{x_{n}\leq
g(x^{\prime})}\cap U.$

For every $r\in(0,1)$ let

$B_{r}^{\prime}=\Set{x^{\prime}\in\mathbb{R}^{n-1}}{\lvert
x^{\prime}\rvert\leq r},$

and $V_{r}=B_{r}^{\prime}\times[-2r,2r]$. For every $x^{\prime}\in
B^{\prime}_{r}$ let

$F_{x^{\prime}}:t\in(g(x^{\prime}),2r]\longmapsto d(x^{\prime},t)-\varepsilon
h(x^{\prime},t),$

where we recall that $d(x)$ denotes the distance from $\Omega$. Let us also
recall that

$\Sigma_{\varepsilon}=\Omega_{\varepsilon}\setminus\overline{\Omega}=\Set{x\in\mathbb{R}^{n}}{0<d(x)<\varepsilon
h(x)}.$

The definition of $F_{x^{\prime}}$ gives us the possibility to characterize
the property $(x^{\prime},t)\in\Omega_{\varepsilon}$ in the equivalent way
$F_{x^{\prime}}(t)<0$. As an immediate consequence,
$F_{x^{\prime}}(g(x^{\prime}))<0$ for every $x^{\prime}\in B_{r}^{\prime}$.

The idea of the proof can be divided in two main steps: first we show that for
a right choice of $\varepsilon_{0}$ the set $\partial\Omega_{\varepsilon}$
cannot touch the upper base of the cylinder $V_{r}$ by proving
$F_{x^{\prime}}(2r)>0$ for every $x^{\prime}\in B_{r}^{\prime}$; next, we show
that for a right choice of $\varepsilon_{0}$ we have that
$\partial\Omega_{\varepsilon}$ can be represented as a graph over the whole
$B_{r}^{\prime}$ by showing that $F_{x^{\prime}}$ is strictly increasing in
$t$ for every $x^{\prime}\in B_{r}^{\prime}$

Let us choose $r=r(\sigma_{0},\Omega)$ small enough so that we have
$V_{r}\subseteq U$ and, thanks to the continuity of $\nu_{0}$ in
$\Gamma_{d_{0}}$

$\lvert\nu_{0}(x)-\mathbf{e}_{n}\rvert<\dfrac{1}{2},$ (5.11)

for every $x\in V_{r}\cap\Gamma_{d_{0}}$. Note in addition that up to choosing
a smaller $r$ we can assume the vertical distance between $\partial\Omega$ and
the upper base of $V_{r}$ being greater than $r$, namely

$2r-\max_{x^{\prime}\in B_{r}^{\prime}}g(x^{\prime})>r.$ (5.12)

Indeed, since $\nabla g$ is continuous and $\nabla g(0)=0$, then for small
enough $r$ we also have, for every $x^{\prime}\in B_{r}^{\prime}$,

$\lvert\nabla g(x^{\prime})\rvert<1,$

which joint with the fact that $g(0)=0$, ensures (5.12).

Let $x^{\prime}\in B^{\prime}_{r}$, fix $t\in(g(x^{\prime}),2r]$, and let
$x=(x^{\prime},t)$. We claim that under the assumption (5.11), we have

$d(x)>\frac{t-g(x^{\prime})}{2}.$ (5.13)

Indeed, let $\bar{x}$ denote the point at distance
$d(\bar{x})=t-g(x^{\prime})$ whose projection onto $\Omega$ is
$(x^{\prime},g(x^{\prime}))$, namely

$\overline{x}=(x^{\prime},g(x^{\prime}))+(t-g(x^{\prime}))\nu_{0}(x^{\prime},g(x^{\prime})).$

We have

$\lvert
x-\overline{x}\rvert=(t-g(x^{\prime}))\,\lvert\mathbf{e}_{n}-\nu_{0}(x^{\prime},g(x^{\prime}))\rvert<\dfrac{t-g(x^{\prime})}{2},$

and, using the fact that the distance $d$ is Lipschitzian with constant 1,

$\begin{split}d(x)&\geq d(\overline{x})-\lvert
d(x)-d(\overline{x})\rvert\\\\[5.0pt] &\geq d(\overline{x})-\lvert
x-\overline{x}\rvert>\dfrac{t-g(x^{\prime})}{2}.\end{split}$

We now join (5.13), (5.12), and $\varepsilon\lVert
h\rVert_{\infty}<\varepsilon_{0}$ to get that for $\varepsilon_{0}<r/2$

$F_{x^{\prime}}(2r)>\dfrac{2r-g(x^{\prime})}{2}-\varepsilon_{0}>0,$ (5.14)

which concludes the first step.

We now prove that $F_{x^{\prime}}(t)$ is monotone increasing in $t$. Indeed,
by the assumption (5.11), we have that

$\nu_{0}(x)\cdot\mathbf{e}_{n}>\frac{1}{2},$

and

$\lvert\nabla h(x)\cdot\mathbf{e}_{n}\rvert=\lvert\nabla
h(x)\cdot(\mathbf{e}_{n}-\nu_{0}(x))\rvert\leq\frac{\lVert\nabla
h\rVert_{\infty}}{2},$

so that, since $r<1$ and $\varepsilon_{0}<r/2$, we have
$\varepsilon\lVert\nabla h\rVert_{\infty}<1/2$, which ensures

$\dfrac{d}{dt}F_{x^{\prime}}(t)=\nu_{0}(x^{\prime},t)\cdot\mathbf{e}_{n}-\varepsilon\nabla
h(x^{\prime},t)\cdot\mathbf{e}_{n}>\frac{1}{4}.$ (5.15)

Joining (5.14) and (5.15), we get that for every $x^{\prime}\in
B_{r}^{\prime}$ there exists a unique $t(x^{\prime})\in(g(x^{\prime}),2r)$
such that $(x^{\prime},t)\in\Omega_{\varepsilon}\cap V_{r}$ if and only if
$t\leq t(x^{\prime})$, thus concluding the proof. ∎

Thanks to subsection 5.2, we can now represent both $\partial\Omega$ and
$\partial\Omega_{\varepsilon}$ as graphs, uniformly in $\varepsilon$, and
flatten the boundaries of $\Omega$ and $\Omega_{\varepsilon}$. We define the
invertible map

$\Phi_{\sigma_{0}}\colon(x^{\prime},x_{n})\in
V\mapsto\left(x^{\prime},\dfrac{x_{n}-g(x^{\prime})}{k_{\varepsilon}(x^{\prime})}\right)\in\tilde{V},$

where $\tilde{V}=\Phi_{\sigma_{0}}(V)$. For simplicity’s sake, when possible,
we will drop the explicit dependence on the point
$\sigma_{0}\in\partial\Omega$. Notice that the map $\Phi$ indeed flattens the
boundaries of $\Omega$ and $\Omega_{\varepsilon}$, in the sense that

$\Phi(\partial\Omega\cap V)=\set{y_{n}=0}\cap\tilde{V},$

and

$\Phi(\partial\Omega_{\varepsilon}\cap
V)=\set{y_{n}=\varepsilon}\cap\tilde{V}.$

For every $\delta>0$ we define the cube

$\tilde{Q}_{\delta}=\set{y\in\mathbb{R}^{n}}{\lvert
y_{i}\rvert\leq\delta,\,\text{for every }1\leq i\leq n}.$

Up to choosing a smaller neighborhood $V$ of $\sigma_{0}$, we may assume that

$\tilde{V}=\Phi(V)=\tilde{Q}_{\delta_{0}}$

for some $\delta_{0}=\delta_{0}(\sigma_{0},\Omega)>0$. For every
$0<\delta<\delta_{0}$ we define

$Q_{\delta}=\Phi^{-1}(\tilde{Q}_{\delta}).$

######  Remark  (Estimates for $k_{\varepsilon}$, $\Phi$, and $\Phi^{-1}$).

We claim that there exists a positive constant $C=C(\Omega,\lVert
h\rVert_{C^{1,1}})$ such that

$\min k_{\varepsilon}\geq\frac{1}{C},\qquad\qquad\lVert
k_{\varepsilon}\rVert_{C^{1,1}}+\lVert\Phi\rVert_{C^{1,1}}+\lVert\Phi^{-1}\rVert_{C^{1,1}}\leq
C.$ (5.16)

Notice that, since $\Omega\subset\Omega_{\varepsilon}$ for every
$\varepsilon>0$, then $k_{\varepsilon}$ is non negative. Moreover, for every
$x=(x^{\prime},x_{n})\in\partial\Omega_{\varepsilon}\cap V$

$\varepsilon k_{\varepsilon}(x^{\prime})\geq
d(x,\partial\Omega)\geq\varepsilon\min_{\Gamma_{d_{0}}}h,$

so that $k_{\varepsilon}$ is strictly bounded from below uniformly in
$\varepsilon$. Similarly, $k_{\varepsilon}$ is bounded in $C^{1,1}$ norm
uniformly in $\varepsilon$, indeed for every
$x\in\partial\Omega_{\varepsilon}\cap V$, we can write
$x=(x^{\prime},g(x^{\prime})+\varepsilon k_{\varepsilon}(x^{\prime}))$, and we
have that

$d(x,\partial\Omega_{\varepsilon})=0.$

Differentiating with respect to $x^{\prime}$ we get

$\nu_{\varepsilon}^{\prime}+\nu_{\varepsilon}\cdot\mathbf{e}_{n}(\nabla
g+\varepsilon\nabla k_{\varepsilon})=0,$ (5.17)

where $\nu_{\varepsilon}^{\prime}\in\mathbb{R}^{n-1}$ is the vector whose
components are the first $n-1$ components of $\nu_{\varepsilon}$. Using the
fact that

$\nu_{\varepsilon}=\dfrac{\nu_{0}-\varepsilon\nabla
h}{\sqrt{1+\varepsilon^{2}\lvert\nabla h\rvert^{2}}},$

and that in $V$

$\nu_{0}=\left(-\dfrac{\nabla g}{\sqrt{1+\lvert\nabla
g\rvert^{2}}},\dfrac{1}{\sqrt{1+\lvert\nabla g\rvert^{2}}}\right),$

we can rewrite equation (5.17) as

$\nabla k_{\varepsilon}=\dfrac{(\nabla h)^{\prime}+(\nabla
h\cdot\mathbf{e}_{n})\nabla g}{(\nu_{0}-\varepsilon\nabla
h)\cdot\mathbf{e}_{n}}.$ (5.18)

Finally, the choice of $\varepsilon_{0}$ in subsection 5.2 (see (5.15))
ensures us that

$(\nu_{0}-\varepsilon\nabla h)\cdot\mathbf{e}_{n}>\dfrac{1}{4},$ (5.19)

so that from equation (5.18) and (5.19), (5.16) follows.

######  Remark  (The equation in the flattened set).

Let $u_{\varepsilon}$ be the solution to (5.2), fix
$\sigma_{0}\in\partial\Omega$, and let

$v(y)=u_{\varepsilon}(\Phi^{-1}(y)),$

then

$v\in H^{1}\bigl{(}\set{y_{n}<\varepsilon}\cap\tilde{V}\bigr{)}\cap
H^{2}_{\operatorname{loc}}\Bigl{(}\bigl{(}\set{y_{n}<\varepsilon}\cap\tilde{V}\bigr{)}\setminus\set{y_{n}=0}\Bigr{)}$

and, for all $\varphi\in H^{1}_{0}(\tilde{V})$, $v$ solves the equation

$\displaystyle\int_{\\{y_{n}<\varepsilon\\}\cap\tilde{V}}\varepsilon(y_{n})A_{\varepsilon}\nabla
v\cdot\nabla\varphi\,dy+\beta\int_{\\{y_{n}=\varepsilon\\}\cap\tilde{V}}v\varphi
J_{\varepsilon}\,d\mathcal{H}^{n-1}=\int_{\\{y_{n}<0\\}\cap\tilde{V}}\tilde{f}_{\varepsilon}\varphi\,dy,$
(5.20)

where

$\varepsilon(y_{n})=\begin{cases}\varepsilon&\text{if }y_{n}>0,\\\ 1&\text{if
}y_{n}\leq 0,\end{cases}$
$A_{\varepsilon}(y)=k_{\varepsilon}(y^{\prime})(D(\Phi^{-1})(y))^{-1}(D(\Phi^{-1})(y))^{-T},\qquad\qquad\quad\tilde{f}_{\varepsilon}(y)=f(\Phi^{-1}(y))\,k_{\varepsilon}(y^{\prime}),$

and

$J_{\varepsilon}(y)=\sqrt{1+\lvert\nabla g(y^{\prime})+\varepsilon\nabla
k_{\varepsilon}(y^{\prime})\rvert^{2}}.$

Notice that $A_{\varepsilon}$ is elliptic and bounded, uniformly in $y$ and
$\varepsilon$. Moreover, using (5.9) in subsection 5.1, we also get that there
exists a positive constant $C=C(\Omega,h,\beta,f,\sigma_{0})$ such that

$\int_{\set{y_{n}<\varepsilon}\cap\tilde{V}}\varepsilon(y_{n})\lvert\nabla
v\rvert^{2}\,dy+\beta\int_{\\{y_{n}=\varepsilon\\}\cap\tilde{V}}v^{2}\,d\mathcal{H}^{n-1}\leq
C.$ (5.21)

###  5.3 Step 2: $H^{2}$ uniform estimates

Since we aim to prove $H^{2}$ estimates with a local approach, we define the
energy quantities $I_{\delta}$ and $\tilde{I}_{\delta}$ as follows: given a
function

$\varphi\in H^{1}\bigl{(}\set{y_{n}<\varepsilon}\cap\tilde{V}\bigr{)}\cap
H^{2}\Bigl{(}\bigl{(}\set{y_{n}<\varepsilon}\cap\tilde{V}\bigr{)}\setminus\set{y_{n}=0}\Bigr{)}$

and $0<\delta<\delta_{0}$, we denote by

$\tilde{I}_{\delta,\sigma_{0}}(\varphi)=\int_{\set{y_{n}<\varepsilon}\cap\tilde{Q}_{\delta}}\varepsilon(y_{n})\lvert
D^{2}\varphi\rvert^{2}\,dy+\beta\int_{\\{y_{n}=\varepsilon\\}\cap\tilde{Q}_{\delta}}\lvert\nabla_{n-1}\varphi\rvert^{2}\,d\mathcal{H}^{n-1},$
(5.22)

where

$\nabla_{n-1}\varphi=\nabla\varphi-\dfrac{\partial\varphi}{\partial
y_{n}}\mathbf{e}_{n},$

and, as in subsection 5.2,

$\varepsilon(t)=\begin{cases}\varepsilon&\text{if }t>0,\\\ 1&\text{if }t\leq
0.\end{cases}$

Analogously in $\Omega_{\varepsilon}$, for every

$\varphi\in H^{1}\bigl{(}\Omega_{\varepsilon}\cap V\bigr{)}\cap
H^{2}\Bigr{(}\bigr{(}\Omega_{\varepsilon}\cap
V\bigl{)}\setminus\partial\Omega\Bigr{)},$

we let

$I_{\delta,\sigma_{0}}(\varphi)=\int_{\Omega_{\varepsilon}\cap{Q}_{\delta}}\varepsilon(d(x))\lvert
D^{2}\varphi\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}\cap{Q}_{\delta}}\lvert\nabla^{\partial\Omega_{\varepsilon}}\varphi\rvert^{2}\,d\mathcal{H}^{n-1},$
(5.23)

where we recall that

$\nabla^{\partial\Omega_{\varepsilon}}\varphi=\nabla\varphi-\left(\nabla\varphi\cdot\nu_{\varepsilon}\right)\nu_{\varepsilon}.$

When possible, we will drop the dependence on $\sigma_{0}$. Uniform bounds for
${I}_{\delta}$ can be read as uniform bounds for $\tilde{I}_{\delta}$ and
viceversa. Indeed, we have the following

######  Lemma .

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open set with $C^{1,1}$
boundary, fix a positive function $h\in C^{1,1}(\Gamma_{d_{0}})$ such that
$\varepsilon\lVert h\rVert_{C^{0,1}}\leq\varepsilon_{0}$ and $\nabla
h\cdot\nu_{0}=0$. Then, for every $\sigma_{0}\in\partial\Omega$, there exists
a positive constant $C=C(\Omega,\lVert h\rVert_{C^{1,1}},\sigma_{0})$ such
that if

$u\in H^{1}(Q_{\delta})\cap H^{2}(Q_{\delta}\setminus\partial\Omega),$

and

$v(y)=u(\Phi^{-1}(y)),$

then, for every $0<\delta<\delta_{0}$,

${I}_{\delta}(u)\leq
C\left(\tilde{I}_{\delta}(v)+\int_{{\\{y_{n}<\varepsilon\\}}\cap\tilde{Q}_{\delta}}\varepsilon(y_{n})\lvert\nabla
v\rvert^{2}\,dy\right),$ (5.24)

and

$\tilde{I}_{\delta}(v)\leq
C\left({I}_{\delta}(u)+\int_{\Omega_{\varepsilon}\cap
Q_{\delta}}\varepsilon(d(x))\lvert\nabla u\rvert^{2}\,dx\right).$ (5.25)

###### Proof.

We start by evaluating the trace term in the definition of
$\tilde{I}_{\delta}$. By means of the change of variables $y=\Phi(x)$, we have

$\begin{split}\int_{{\\{y_{n}=\varepsilon\\}}\cap\tilde{Q}_{\delta}}\lvert\nabla_{n-1}v\rvert^{2}\,d\mathcal{H}^{n-1}(y)&=\sum_{i=1}^{n-1}\int_{\partial\Omega_{\varepsilon}\cap
V}(\nabla u\cdot
w_{i})^{2}\,J^{\partial\Omega_{\varepsilon}}\Phi\,d\mathcal{H}^{n-1}(x),\end{split}$
(5.26)

where

$w_{i}=\mathbf{e}_{i}+(\partial_{i}g+\varepsilon\partial_{i}k_{\varepsilon})\,\mathbf{e}_{n}.$

The vectors $w_{i}$ (that, in general, could be non-orthogonal) form a basis
for the tangent plane $T_{\sigma_{0}}\partial\Omega_{\varepsilon}$. In
particular, we have that there exists a positive constant
$C=C(\Omega,h,\sigma_{0})$ such that

$\lvert\nabla^{\partial\Omega_{\varepsilon}}u\rvert^{2}\leq
C\sum_{i=1}^{n-1}(\nabla u\cdot w_{i})^{2}\leq
C^{2}\lvert\nabla^{\partial\Omega_{\varepsilon}}u\rvert^{2}.$ (5.27)

Therefore, using the uniform bounds (5.16) we get that for some positive
constant $C=C(\Omega,h)$

$\frac{1}{C}\leq
J^{\partial\Omega_{\varepsilon}}\Phi=\frac{1}{\sqrt{1+\lvert\nabla
g+\varepsilon\nabla k_{\varepsilon}\rvert^{2}}}\leq C,$ (5.28)

and joining (5.26), (5.27), and (5.28), we get

$\int_{\partial\Omega_{\varepsilon}\cap{Q}_{\delta}}\lvert\nabla^{\partial\Omega_{\varepsilon}}\varphi\rvert^{2}\,d\mathcal{H}^{n-1}\leq
C\int_{{\\{y_{n}=\varepsilon\\}}\cap\tilde{Q}_{\delta}}\lvert\nabla_{n-1}v\rvert^{2}\,d\mathcal{H}^{n-1}\leq
C^{2}\int_{\partial\Omega_{\varepsilon}\cap{Q}_{\delta}}\lvert\nabla^{\partial\Omega_{\varepsilon}}\varphi\rvert^{2}\,d\mathcal{H}^{n-1}.$
(5.29)

For what concerns the second order term, we evaluate for a.e.
$y\in\\{y_{n}<\varepsilon\\}\cap\tilde{V}$,

$\big{(}D^{2}v(y)\big{)}_{ij}=\sum_{k=1}^{n}\frac{\partial^{2}(\Phi^{-1})_{k}}{\partial
y_{i}\,\partial y_{j}}\frac{\partial u}{\partial
x_{k}}+\sum_{k,l=1}^{n}\frac{\partial(\Phi^{-1})_{k}}{\partial
y_{i}}\frac{\partial^{2}u}{\partial x_{k}\,\partial
x_{l}}\frac{\partial(\Phi^{-1})_{l}}{\partial y_{j}},$

so that, for some positive constant $C=C(n,\lVert\Phi^{-1}\rVert_{C^{1,1}})$,

$\lvert D^{2}v\big{(}\Phi(x)\big{)}\rvert^{2}\leq C\left(\lvert\nabla
u(x)\rvert^{2}+\lvert D^{2}u(x)\rvert^{2}\right).$ (5.30)

In a similar way we have that, for a.e. $x\in\Omega_{\varepsilon}\cap V$, and
for some positive constant $C=C(n,\lVert\Phi\rVert_{C^{1,1}})$,

$\lvert D^{2}u\big{(}\Phi^{-1}(y)\big{)}\rvert^{2}\leq C\left(\lvert\nabla
v(y)\rvert^{2}+\lvert D^{2}v(y)\rvert^{2}\right).$ (5.31)

Using (5.30) and the uniform bounds on $k_{\varepsilon}$, we get

$\begin{split}\int_{{\\{y_{n}<\varepsilon\\}}\cap\tilde{Q}_{\delta}}\varepsilon(y_{n})\lvert
D^{2}v\rvert^{2}\,dy&=\int_{\Omega_{\varepsilon}\cap
Q_{\delta}}\frac{\varepsilon(d(x))}{k_{\varepsilon}}\lvert
D^{2}v\big{(}\Phi(x)\big{)}\rvert^{2}\,dx\\\\[7.0pt] &\leq
C\int_{\Omega_{\varepsilon}\cap Q_{\delta}}\varepsilon(d(x))\big{(}\lvert
D^{2}u\rvert^{2}+\lvert\nabla
u\rvert^{2}\big{)}\,d\mathcal{H}^{n-1}(x).\end{split}$ (5.32)

Analogously, using (5.31),

$\begin{split}\int_{\Omega_{\varepsilon}\cap
Q_{\delta}}\varepsilon(d(x))\lvert D^{2}u\rvert^{2}\,dx\leq
C\int_{{\\{y_{n}<\varepsilon\\}}\cap\tilde{Q}_{\delta}}\varepsilon(y_{n})\left(\lvert
D^{2}v\rvert^{2}+\lvert\nabla
v\rvert^{2}\right)\,d\mathcal{H}^{n-1}.\end{split}$ (5.33)

The result now follows by joining (5.29), (5.33), and (5.32). ∎

######  Lemma .

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open set with $C^{1,1}$
boundary, fix a positive function $h\in C^{1,1}(\Gamma_{d_{0}})$ such that
$\varepsilon\lVert h\rVert_{C^{0,1}}\leq\varepsilon_{0}$ and $\nabla
h\cdot\nu_{0}=0$. If $\sigma_{0}\in\partial\Omega$, and $v$ is as in
subsection 5.2, then

$v\in
H^{2}\left(\set{y_{n}<\varepsilon}\cap\tilde{Q}_{\delta_{0}/2}\setminus\set{y_{n}=0}\right),$

and there exists a positive constant $C=C(\Omega,\lVert
h\rVert_{C^{1,1}},\beta,f,\sigma_{0})$ such that

$\tilde{I}_{\delta_{0}/2}(v)\leq C.$ (5.34)

###### Proof.

Let $\xi\in C^{\infty}_{c}(\tilde{Q}_{\delta_{0}})$ be a non negative function
with $\xi\equiv 1$ in $\tilde{Q}_{\delta_{0}/2}$. For $\lvert\eta\rvert$ small
enough, we have

$\operatorname{supp}\xi+\eta\mathbf{e}_{i}\subset\joinrel\subset\tilde{Q}_{\delta_{0}}$

for every $k=1,\dots,n-1$. In the following, for any $L^{2}$ function $\psi$,
we denote by

$\Delta_{k}^{\eta}\psi(x)=\frac{\psi(x+\eta\mathbf{e}_{k})-\psi(x)}{\eta}$

the difference quotients (see for instance [12, §5.8.2]). We recall that for
any couple of functions $\psi_{1}$ and $\psi_{2}$ we have

$\Delta_{k}^{\eta}(\psi_{1}\psi_{2})(x)=\Delta_{k}^{\eta}\psi_{1}(x)\psi_{2}(x)+\psi_{1}(x+\eta\mathbf{e}_{k})\Delta_{k}^{\eta}\psi_{2}(x),$

and if $\psi_{1}$ and $\psi_{2}$ are measurable and

$(\operatorname{supp}\psi_{1}\cap\operatorname{supp}\psi_{2})\pm\eta\mathbf{e}_{k}\subset\joinrel\subset\tilde{Q}_{\delta},$

then for every $k=1,\dots,n-1$, it holds

$\int_{\tilde{Q}_{\delta_{0}}}\psi_{1}(y)\Delta_{k}^{\eta}\psi_{2}(y)\,dy=-\int_{\tilde{Q}_{\delta_{0}}}\Delta_{k}^{-\eta}\psi_{1}(y)\>\psi_{2}(y)\,dy.$
(5.35)

Moreover, we recall that if $\psi\in
H^{1}({\\{y_{n}<\varepsilon\\}\cap\tilde{Q}_{\delta_{0}}})$, then for every
$U\subset\joinrel\subset{\tilde{Q}_{\delta_{0}}}$ we have that

$\int_{U}(\Delta_{k}^{\eta}\psi)^{2}\,dy\leq\int_{\tilde{Q}_{\delta_{0}}}\lvert\nabla\psi\rvert^{2}\,dy,$
(5.36)

for every $\lvert\eta\rvert<d(U,\partial\tilde{Q}_{\delta_{0}})$. We aim to
prove some uniform $L^{2}$ estimates for $\Delta_{k}^{\eta}(\nabla v)$, which
will imply weak differentiability and uniform $L^{2}$ estimates for
$\partial_{k}\nabla v$.

Given the equation (5.20), we use as a test function

$\varphi=-\Delta_{k}^{-\eta}(\xi^{2}\Delta_{k}^{\eta}v).$

For small $\lvert\eta\rvert$, the function $\varphi$ is admissible. Here we
recall the weak equation

$\underbrace{\int_{\\{y_{n}<\varepsilon\\}\cap\tilde{V}}\varepsilon(y_{n})A_{\varepsilon}\nabla
v\cdot\nabla\varphi\,dy}_{\mathcal{I}_{1}}+\underbrace{\beta\int_{\\{y_{n}=\varepsilon\\}\cap\tilde{V}}v\varphi
J_{\varepsilon}\,d\mathcal{H}^{n-1}}_{\mathcal{I}_{2}}=\underbrace{\int_{\\{y_{n}<0\\}\cap\tilde{V}}\tilde{f}_{\varepsilon}\varphi\,dy}_{\mathcal{I}_{3}}.$

We now estimate the integrals separately. In the following, we use the
Einstein notation on repeated indices, where $i,j=1,\dots,n$, and for
simplicity we drop the dependence on $\varepsilon$ in $A_{\varepsilon}$, whose
components are denoted by $a_{ij}$. We recall that $a_{ij}=a_{ij}(y)$ and
$J_{\varepsilon}=J_{\varepsilon}(y)$. When necessary, we will also write
$a_{ij}^{\eta}:=a_{ij}(y+\eta\mathbf{e}_{k})$ and
$J_{\varepsilon}^{\eta}:=J_{\varepsilon}(y+\eta\mathbf{e_{k}})$.

Let us start with the higher-order term $\mathcal{I}_{1}$: by direct
computation, we get

$\begin{split}A\nabla
v\cdot\nabla\varphi&=-a_{ij}\>\partial_{j}v\>\Delta_{k}^{-\eta}\,\partial_{i}(\xi^{2}\Delta_{k}^{\eta}v)\\\\[5.0pt]
&=-a_{ij}\,\partial_{j}v\,\left[\Delta_{k}^{-\eta}(\partial_{i}(\xi^{2})\>\Delta_{k}^{\eta}v)+\Delta_{k}^{-\eta}(\xi^{2}\Delta_{k}^{\eta}(\partial_{i}v))\right]\end{split}$

Multiplying by $\varepsilon(y_{n})$, integrating over
$\\{y_{n}<\varepsilon\\}\cap\tilde{Q}_{\delta_{0}}$, and using property
(5.35), we get (notice that $\varepsilon(y_{n})$ is constant along the
direction $\mathbf{e}_{k}$)

$\begin{split}\mathcal{I}_{1}&=\int_{\\{y_{n}<\varepsilon\\}\cap\tilde{Q}_{\delta_{0}}}\varepsilon(y_{n})\Delta_{k}^{\eta}(a_{ij}\,\partial_{j}v)\,\partial_{i}(\xi^{2})\Delta_{k}^{\eta}v\,dy\\\\[7.0pt]
&\hphantom{=\,}+\int_{\\{y_{n}<\varepsilon\\}\cap\tilde{Q}_{\delta_{0}}}\varepsilon(y_{n})\Delta_{k}^{\eta}(a_{ij}\partial_{j}v)\,\xi^{2}\,\Delta_{k}^{\eta}(\partial_{i}v)\,dy\\\\[7.0pt]
&=\int_{\\{y_{n}<\varepsilon\\}\cap\tilde{Q}_{\delta_{0}}}\varepsilon(y_{n})\xi^{2}\,a_{ij}^{\eta}\,\Delta_{k}^{\eta}(\partial_{j}v)\,\Delta_{k}^{\eta}(\partial_{i}v)\,dy+R_{1}\\\\[7.0pt]
&\geq
C_{1}\int_{\\{y_{n}<\varepsilon\\}\cap\tilde{Q}_{\delta_{0}}}\varepsilon(y_{n})\xi^{2}\,\sum_{j=1}^{n}\big{(}\Delta_{k}^{\eta}(\partial_{j}v)\big{)}^{2}\,dy-\lvert
R_{1}\rvert,\end{split}$ (5.37)

where $C_{1}$ is the ellipticity constant of $A$ estimated uniformly in
subsection 5.2, and

$\begin{split}R_{1}&=2\int_{\\{y_{n}<\varepsilon\\}\cap\tilde{Q}_{\delta_{0}}}\varepsilon(y_{n})\left(\Delta_{k}^{\eta}(a_{ij})\partial_{j}v+a_{ij}^{\eta}\,\Delta_{k}^{\eta}(\partial_{j}v)\right)\,\xi\,\partial_{i}\xi\,\Delta_{k}^{\eta}v\,dy\\\\[7.0pt]
&\hphantom{=\,}+\int_{\\{y_{n}<\varepsilon\\}\cap\tilde{Q}_{\delta_{0}}}\varepsilon(y_{n})\Delta_{k}^{\eta}(a_{ij})\partial_{j}v\,\xi^{2}\,\Delta_{k}^{\eta}(\partial_{j}v)\,dy.\end{split}$

Using Young’s inequality

$ab\leq\dfrac{\lambda}{2}a^{2}+\dfrac{1}{2\lambda}b^{2}\qquad\quad\forall
a,b\geq 0,$

with a suitable choice of $\lambda>0$, and using the uniform bounds on
$\lVert\xi\rVert_{C^{1}}$ and $\lVert A_{\varepsilon}\rVert_{C^{1}}$ (see
subsection 5.2 and subsection 5.2), we may obtain a positive constant
$C=C(A,\xi)$ such that the following estimate holds

$\begin{split}\lvert
R_{1}\rvert&\leq\frac{C_{1}}{2}\int_{\\{y_{n}<\varepsilon\\}\cap\tilde{Q}_{\delta_{0}}}\varepsilon(y_{n})\xi^{2}\,\sum_{j=1}^{n}\big{(}\Delta_{k}^{\eta}(\partial_{j}v)\big{)}^{2}\,dy\\\\[7.0pt]
&\hphantom{=}+C\int_{\\{y_{n}<\varepsilon\\}\cap\,\operatorname{supp}\xi}\varepsilon(y_{n})\left((\Delta_{k}^{\eta}v)^{2}+\sum_{j=1}^{n}(\partial_{j}v)^{2}\right)\,dy.\end{split}$

This estimate, joint with (5.37), implies

$\mathcal{I}_{1}\geq\frac{C_{1}}{2}\int_{\\{y_{n}<\varepsilon\\}\cap\tilde{Q}_{\delta_{0}}}\varepsilon(y_{n})\xi^{2}\,\sum_{j=1}^{n}\big{(}\Delta_{k}^{\eta}(\partial_{j}v)\big{)}^{2}\,dy-C\int_{\\{y_{n}<\varepsilon\\}\cap\tilde{Q}_{\delta_{0}}}\varepsilon(y_{n})\lvert\nabla
v\rvert^{2}\,dy.$ (5.38)

Similarly, let us work with the boundary terms $\mathcal{I}_{2}$, where we
have

$v\varphi
J_{\varepsilon}=-vJ_{\varepsilon}\Delta_{k}^{-\eta}(\xi^{2}\Delta_{k}^{\eta}v).$

Since $k\neq n$, we can use an analogous of property (5.35) also on
$\set{y_{n}=\varepsilon}\cap\tilde{Q}_{\delta_{0}}$, so that

$\begin{split}\mathcal{I}_{2}&=\beta\int_{\set{y_{n}=\varepsilon}\cap\tilde{Q}_{\delta_{0}}}\xi^{2}\Delta_{k}^{\eta}(vJ_{\varepsilon})\Delta_{k}^{\eta}v\,d\mathcal{H}^{n-1}\\\\[7.0pt]
&=\beta\int_{\set{y_{n}=\varepsilon}\cap\tilde{Q}_{\delta_{0}}}\xi^{2}(\Delta_{k}^{\eta}v)^{2}J_{\varepsilon}^{\eta}\,d\mathcal{H}^{n-1}+R_{2}\\\\[7.0pt]
&\geq\beta
C_{2}\int_{\set{y_{n}=\varepsilon}\cap\tilde{Q}_{\delta_{0}}}\xi^{2}(\Delta_{k}^{\eta}v)^{2}\,d\mathcal{H}^{n-1}-\lvert
R_{2}\rvert,\end{split}$ (5.39)

where $C_{2}=\inf_{Q_{\delta}}J_{\varepsilon}$ uniformly estimated in
subsection 5.2, and

$R_{2}=\beta\int_{\set{y_{n}=\varepsilon}\cap\tilde{Q}_{\delta_{0}}}\xi^{2}\Delta_{k}^{\eta}(J_{\varepsilon})v\Delta_{k}^{\eta}v\,d\mathcal{H}^{n-1}$

As done for $\mathcal{I}_{1}$, using Young’s inequality and the uniform bounds
on $\lVert J_{\varepsilon}\rVert_{C^{1}}$ (see subsection 5.2 and subsection
5.2), we have that for some positive constant $C=C(J_{\varepsilon},\xi)$,

$\lvert R_{2}\rvert\leq\frac{\beta
C_{2}}{2}\int_{\set{y_{n}=\varepsilon}\cap\tilde{Q}_{\delta_{0}}}\xi^{2}(\Delta_{k}^{\eta}v)^{2}\,d\mathcal{H}^{n-1}+C\beta\int_{\set{y_{n}=\varepsilon}\cap\tilde{Q}_{\delta_{0}}}v^{2}\,d\mathcal{H}^{n-1}$

This estimate, joint with (5.39) ensures

$\mathcal{I}_{2}\geq\frac{\beta
C_{2}}{2}\int_{\set{y_{n}=\varepsilon}\cap\tilde{Q}_{\delta_{0}}}\xi^{2}(\Delta_{k}^{\eta}v)^{2}\,d\mathcal{H}^{n-1}-C\beta\int_{\set{y_{n}=\varepsilon}\cap\tilde{Q}_{\delta_{0}}}v^{2}\,d\mathcal{H}^{n-1}.$
(5.40)

Finally, we estimate the source term $\mathcal{I}_{3}$, recalling that

$\tilde{f}_{\varepsilon}\varphi=-\tilde{f}_{\varepsilon}\Delta_{k}^{-\eta}(\xi^{2}\Delta_{k}^{\eta}v).$

Using Young’s inequality and property (5.36) for
$\psi=\xi^{2}\Delta_{k}^{\eta}v$ (notice that
$\nabla\Delta_{k}^{\eta}v=\Delta_{k}^{\eta}(\nabla v)$), then for a suitable
positive constant $C=C(\tilde{f},\xi,\delta_{0})$,

$\begin{split}\mathcal{I}_{3}&\leq\frac{C_{1}}{4}\int_{\set{y_{n}<0}\cap\tilde{Q}_{\delta_{0}}}\xi^{2}\sum_{j=1}^{n}(\Delta_{k}^{\eta}\partial_{j}v)^{2}\,dy\\\\[7.0pt]
&\hphantom{=}+C\int_{\set{y_{n}<0}\cap\tilde{Q}_{\delta_{0}}}\tilde{f}_{\varepsilon}^{2}\,dy+C\int_{\set{y_{n}<0}\cap\tilde{Q}_{\delta_{0}}}\lvert\nabla
v\rvert^{2}\,dy\\\\[7.0pt]
&\leq\frac{C_{1}}{4}\int_{\set{y_{n}<\varepsilon}\cap\tilde{Q}_{\delta_{0}}}\varepsilon(y_{n})\xi^{2}\sum_{j=1}^{n}(\Delta_{k}^{\eta}\partial_{j}v)^{2}\,dy\\\\[7.0pt]
&\hphantom{=}+C\int_{\set{y_{n}<0}\cap\tilde{Q}_{\delta_{0}}}\tilde{f}_{\varepsilon}^{2}\,dy+C\int_{\set{y_{n}<\varepsilon}\cap\tilde{Q}_{\delta_{0}}}\varepsilon(y_{n})\lvert\nabla
v\rvert^{2}\,dy.\end{split}$ (5.41)

We can now turn back to the equation

$\mathcal{I}_{1}+\mathcal{I}_{2}=\mathcal{I}_{3}$

Joining (5.38), (5.40), (5.41), and using the fact that $\xi=1$ in
$\tilde{Q}_{\delta_{0}/2}$, then, for every $k=1,\dots,n-1$,

$\begin{split}\mathcal{I}_{4}&:=\frac{C_{1}}{4}\int_{\set{y_{n}<\varepsilon}\cap\tilde{Q}_{\delta_{0}/2}}\varepsilon(y_{n})\sum_{j=1}^{n}(\Delta_{k}^{\eta}\partial_{j}v)^{2}\,dy\\\\[7.0pt]
&\hphantom{=}+\frac{\beta
C_{2}}{2}\int_{\set{y_{n}=\varepsilon}\cap\tilde{Q}_{\delta_{0}/2}}(\Delta_{k}^{\eta}v)^{2}\,d\mathcal{H}^{n-1}\\\\[7.0pt]
&\leq
C\Bigg{(}\int_{\set{y_{n}<\varepsilon}\cap\tilde{Q}_{\delta_{0}}}\varepsilon(y_{n})\lvert\nabla
v\rvert^{2}\,dy\\\\[7.0pt]
&\hphantom{=C\Bigg{(}}+\beta\int_{\set{y_{n}=\varepsilon}\cap\tilde{Q}_{\delta_{0}}}v^{2}\,d\mathcal{H}^{n-1}+\int_{\set{y_{n}<0}\cap\tilde{Q}_{\delta_{0}}}\tilde{f}_{\varepsilon}^{2}\,dy\Bigg{)}.\end{split}$
(5.42)

Since we have assumed the uniform $H^{1}$ estimates (5.9), and we have uniform
estimates for $\tilde{f}_{\varepsilon}$, computed in subsection 5.2, we get
that for some positive constant $C=C(\delta_{0},\Omega,\xi)$

$\mathcal{I}_{4}\leq C,$

which implies that for every $k=1,\dots,n-1$, we have $\partial_{k}v\in
H^{1}(\\{y_{n}<\varepsilon\\}\cap\tilde{Q}_{\delta_{0}/2})$ and

$\begin{split}\int_{\set{y_{n}<\varepsilon}\cap\tilde{Q}_{\delta_{0}/2}}\varepsilon(y_{n})\sum_{j=1}^{n}(\partial_{k}\partial_{j}v)^{2}\,dy&\leq
C\Bigg{(}\int_{\set{y_{n}<\varepsilon}\cap\tilde{Q}_{\delta_{0}}}\varepsilon(y_{n})\lvert\nabla
v\rvert^{2}\,dy\\\\[7.0pt]
&\hphantom{=C\Bigg{(}}+\beta\int_{\set{y_{n}=\varepsilon}\cap\tilde{Q}_{\delta_{0}}}v^{2}\,d\mathcal{H}^{n-1}+\int_{\set{y_{n}<0}\cap\tilde{Q}_{\delta_{0}}}\tilde{f}_{\varepsilon}^{2}\,dy\Bigg{)}.\end{split}$

It only remains to get a uniform estimate for $\partial^{2}_{nn}v$. We notice
that since $v$ solves (5.20), then almost everywhere we have

$-\sum_{\begin{subarray}{c}1\leq i,j\leq n\\\
(i,j)\neq(n,n)\end{subarray}}\varepsilon(y_{n})\partial_{i}(a_{ij}\partial_{j}v)-\varepsilon(y_{n})\partial_{n}a_{nn}\partial_{n}v-\varepsilon(y_{n})a_{nn}\partial^{2}_{nn}v=\tilde{f}\chi_{\\{y_{n}<0\\}}.$

In particular, the fact that $a_{nn}=k_{\varepsilon}$ is uniformly bounded
from below gives estimates for $\partial^{2}_{nn}v$ in terms of
$A_{\varepsilon}$, $\tilde{f}$, and the other derivatives of $v$. Therefore,
so that, for every $0<\delta<\delta_{0}$, we can find a positive constant
$C=C(\Omega,h,f)$ such that

$\begin{split}\int_{\set{y_{n}<\varepsilon}\cap\tilde{Q}_{\delta}}\varepsilon(y_{n})\lvert\partial_{nn}^{2}v\rvert^{2}\leq
C\Bigg{(}&\int_{\set{y_{n}<\varepsilon}\cap\tilde{Q}_{\delta}}\varepsilon(y_{n})\sum_{k=1}^{n-1}{\lvert\nabla\partial_{k}v\rvert}^{2}\,dy\\\\[7.0pt]
&+\int_{\set{y_{n}<\varepsilon}\cap\tilde{Q}_{\delta}}\varepsilon(y_{n})\lvert\nabla
v\rvert^{2}\,d\mathcal{H}^{n-1}+1\Bigg{)}.\end{split}$ (5.43)

Finally, (5.42), joint with (5.43) and the bound (5.21), gives (5.34).

∎

######  Remark .

Let $\sigma_{0}\in\partial\Omega$, by the previous lemma we have that
$u_{\varepsilon}\in H^{2}(Q_{\delta_{0}/2}\setminus\partial\Omega)$. Moreover,
putting together estimates (5.24) and (5.25) and the bounds on the first
derivative ((5.9) and (5.21)), we have

$I_{\delta}(u_{\varepsilon})\leq C(1+\tilde{I}_{\delta}(v_{\varepsilon}))\leq
C^{2}(1+I_{\delta}(u_{\varepsilon})).$ (5.44)

We can finally prove Theorem 3.2.

###### Proof of Theorem 3.2.

We recall that we have defined $\tilde{I}_{\delta,\sigma}$ and
$I_{\delta,\sigma}$ in (5.22) and (5.23) respectively. For every
$\sigma_{0}\in\partial\Omega$, using (5.34) from subsection 5.3, an and (5.44)
from subsection 5.3, we have that there exists a constant
$C=C(\Omega,h,\sigma_{0},\delta_{0})$ such that

$I_{\delta_{0}/2,\sigma_{0}}(u_{\varepsilon})\leq C.$ (5.45)

From the boundedness of $\Omega$ there exist
$\sigma_{1},\dots,\sigma_{m}\in\partial\Omega$ and associated
$\delta_{i}=\delta(\sigma_{i},\Omega)$ for which estimates of the type (5.45)
hold and such that, choosing
$\varepsilon_{0}=\min\\{\varepsilon_{0}(\sigma_{1}),\dots,\varepsilon_{0}(\sigma_{m})\\}>0$,
we have

$\Sigma_{\varepsilon}\subset\bigcup_{i=1}^{m}\Phi_{\sigma_{i}}^{-1}(\tilde{Q}_{\delta_{i}/2})=V_{0}.$

Let $U$ be an open set such that $\overline{U}\subset\Omega$ and
$\Omega_{\varepsilon}\subset U\cup V_{0}$ for every $\varepsilon$ such that
$\varepsilon\lVert h\rVert_{C^{0,1}}<\varepsilon_{0}$. Also in $U$ we may get
an estimate analogous to (5.45). Indeed, by standard elliptic regularity and
by estimate (5.10), we have that

$\int_{U}\lvert D^{2}u_{\varepsilon}\rvert^{2}\,dx\leq
C\left(\int_{\Omega}f^{2}\,dx+\int_{\Omega}u_{\varepsilon}^{2}\right)\leq C.$
(5.46)

Let

$I(u_{\varepsilon})=\int_{\Omega}\lvert
D^{2}u_{\varepsilon}\rvert^{2}\,dx+\varepsilon\int_{\Sigma_{\varepsilon}}\lvert
D^{2}u_{\varepsilon}\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}\lvert\nabla^{\partial\Omega_{\varepsilon}}u_{\varepsilon}\rvert^{2}\,d\mathcal{H}^{n-1},$

summing from $i=1$ to $m$ the estimates of the type (5.45) and (5.46), we have
that there exists $C=C(\Omega,h,f)$ such that

$I(u_{\varepsilon})\leq C,$

and the assertion is proven. ∎

## Acknowledgements

The authors are members of Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta
Matematica (INdAM). The authors were partially supported by the project GNAMPA
2023: "Symmetry and asymmetry in PDEs", CUP_E53C22001930001, and by the 2024
project GNAMPA 2024: "Modelli PDE-ODE nonlineari e proprieta’ di PDE su domini
standard e non-standard", CUP_E53C23001670001.

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Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli
studi di Napoli Federico II, Via Cintia, Complesso Universitario Monte S.
Angelo, 80126 Napoli, Italy.

E-mail address, P. Acampora<EMAIL_ADDRESS>
Mathematical and Physical Sciences for Advanced Materials and Technologies,
Scuola Superiore Meridionale, Largo San Marcellino 10, 80126, Napoli, Italy.

E-mail address, E. Cristoforoni<EMAIL_ADDRESS>