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# On the optimal shape of a thin insulating layer

P. Acampora, E. Cristoforoni, C. Nitsch, C. Trombetti

###### Abstract

We are interested in the thermal insulation of a bounded open set $\Omega$
surrounded by a set whose thickness is locally described by $\varepsilon h$,
where $h$ is a non-negative function defined on the boundary $\partial\Omega$.
We study the problem in the limit for $\varepsilon$ going to zero using a
first-order asymptotic development by $\Gamma$-convergence.

Keywords: Robin boundary condition, thermal insulation, reinforcement,
$\Gamma$-convergence

MSC 2020: 49J45, 35J25, 35B06, 80A19

##  1 Introduction

Energy efficiency has emerged as one of the most pressing issues in recent
years, as it is critical to achieving sustainable development, lowering
greenhouse gas emissions, and mitigating climate change, all while boosting
economic growth and increasing quality of life. Thermal insulation is
important in the context of energy efficiency because it helps to reduce heat
transfer and energy losses in buildings and industrial processes, resulting in
significant energy and cost savings.

This paper addresses the topic of thermal insulation of a solid, kept at
constant temperature, by displacing around it an insulator. Roughly speaking,
the optimal insulation is the one minimizing, at thermal equilibrium, the heat
rate loss per unit time across the exterior boundary. More specifically, in
our model heat exchange with the environment occurs through convection, which
is by far the most common mechanism in real world applications.

In order to contain costs of insulation, the volume of the insulator is
prescribed. The resulting mathematical model gives rise to a free boundary
problem which has been previously studied in [7] and for more general heat
transfer mechanisms (including for instance also radiation) in [5]. In
particular, the former paved the way on how to prove the existence of an
optimal distribution of insulator and the regularity of its a priori unknown
boundary.

However, although a solution always exists for every set to be insulated and
every amount of insulator, very little is known about the optimal shape of the
insulator, and the only completely solved case happens to be the radial one.

This work addresses the problem of qualitatively and quantitatively describing
how to displace the insulator, and to do so, it restricts the analysis to
"small insulation thickness", an approximation reasonable in many contexts,
such as in the case of buildings and other big structures.

Throughout the paper, $\Omega$ represents the body to insulate, in which the
temperature is fixed. Since the problem is invariant under temperature scaling
and translations, without loss of generality the temperature of the body will
be $1$, while the environment temperature will be $0$. If convection is the
leading mechanism of thermal exchange with the environment, then the heat rate
loss per unit time and unit surface is proportional to the temperature jump
across the surface element separating the insulator (or the body) with the
environment. The constant of proportionality will be denoted by $\beta>0$. If
$\Sigma$ denotes the insulator (see figure), then at equilibrium the
temperature distribution in $\Sigma$ is an harmonic function which we denote
by $u$.

Figure 1: Body $\Omega$ with a thin insulating layer $\Sigma$$\Omega$$\Sigma$

On the portion of boundary that $\Sigma$ shares with $\Omega$ the function $u$
is equal to $1$. While on the portion of boundary which $\Sigma$ shares with
the environment, $u$ satisfies a Robin condition

$\frac{\partial u}{\partial\nu}+\beta u=0,$

where $\nu$ is outer normal of the boundary of $\Sigma$. In fact, according to
Fourier law, which holds inside $\Sigma$, the heat flux per unit time is
$-Du$, and therefore the flux across the surface element at the boundary is
$-\frac{\partial u}{\partial\nu}$. On the other hand, according to convection
law, the heat flux per unit time across the surface element is equal to $\beta
u$ ($\beta$ times the jump of $u$). Continuity of the heat flux enforces to
equalize these expressions and the Robin b.c. naturally arises.

In summary, $u$ is a continuous function in $\Omega\cup\Sigma$ which solves

$\begin{cases}\Delta u=0&\text{in }\Sigma,\\\\[5.0pt] u=1&\text{in
}\Omega,\\\\[5.0pt] \dfrac{\partial u}{\partial\nu}+\beta u=0&\text{on
}\partial\Sigma\setminus\partial\Omega.\end{cases}$

We can also characterize the function $u$ as the minimizer of the energy
functional

$\mathcal{E}(v)=\int_{\Sigma}\lvert\nabla
v\rvert^{2}\,dx+\beta\int_{\partial(\Omega\cup\Sigma)}v^{2}\,d\mathcal{H}^{n-1},$

among all functions $v\in H^{1}(\Omega\cup\Sigma)$ such that $v\equiv 1$ in
$\Omega$.

For a given $\Omega$, our ultimate goal would be to find the shape of $\Sigma$
which minimizes

$\int_{\partial(\Omega\cup\Sigma)}u\,d\mathcal{H}^{n-1}$

among all $\Sigma$ of prescribed measure.

So far, such a problem seems to be out of reach, so we restricted our analysis
to the case where the layer of the insulating material and the conductivity of
the insulator are both very small.

More precisely, let $\Omega\subset\mathbb{R}^{n}$ be a smooth bounded, open
set, and let $h\colon\partial\Omega\to\mathbb{R}$ be a non-negative function.
Denoting by $\nu$ the exterior unit normal to the boundary of $\Omega$, we
define

$\Sigma_{\varepsilon}=\Set{\sigma+t\nu(\sigma)}{\sigma\in\partial\Omega,\,0<t<\varepsilon
h(\sigma)}$

and we denote by
$\Omega_{\varepsilon}=\overline{\Omega}\cup\Sigma_{\varepsilon}$. Here, the
non negative parameter $\varepsilon$ is meant to be small and we want to
investigate the limit as $\varepsilon$ vanishes. But in order to have this
limit non trivial, we also assume that the conductivity of the insulator is
small, namely, the heat flux inside $\Sigma_{\varepsilon}$ is $-\varepsilon
Du$. In practice, we are assuming that we use a "small quantity" of a "very
good" insulator.

All at once we can consider the minimization of the following energy
functional

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

where $v\in H^{1}(\Omega_{\varepsilon})$, with $v=1$ in $\Omega$. Here, the
small parameter $\varepsilon$ in front of the first integral is encoding the
fact that the conductivity of the insulator is small. For given $h$, a minimum
$u_{\varepsilon,h}$ of

$\min\Set{\mathcal{F}_{\varepsilon}(v,h)}{v\in
H^{1}(\Omega_{\varepsilon}),\,v=1\,\text{in }\Omega}$ (1.1)

solves the boundary value problem:

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

where $\nu_{\varepsilon}$ is the exterior unit normal to the boundary of
$\Omega_{\varepsilon}$.

Similar problems have been studied before in the context of thermal insulation
in [3], [11], [1], and more recently in [4] and [9]. The limit has been
performed in several ways. In our case, we are going to use
$\Gamma$-convergence. But in order to extract as much information as possible
about the problem we are going to perform a first order expansion in
$\varepsilon$ [2], which, to our knowledge, has never been exploited in this
context.

The volume of insulator we displace is $\varepsilon m$, for some $m>0$, and we
define the volume constraint by defining the space

$\mathcal{H}_{m}=\mathcal{H}_{m}(\partial\Omega)=\Set{h\in
L^{1}(\partial\Omega)}{\begin{aligned}
&\int_{\partial\Omega}h\,d\mathcal{H}^{n-1}\leq m\\\\[3.0pt] &\,h\geq
0\end{aligned}}.$ (1.2)

Our problem reduces to finding the best configuration of insulating material
surrounding $\Omega$, that is

$\min\Set{\mathcal{F}_{\varepsilon}(v,h)}{\begin{aligned} &v\in
H^{1}(\Omega_{\varepsilon}),\\\ &v=1\,\text{in }\Omega,\\\
&h\in\mathcal{H}_{m}\end{aligned}}.$ (1.3)

Following argument similar to those used in [9], it can be proved that, for
any fixed Lipschitz function $h:\partial\Omega\to[0,+\infty)$, as
$\varepsilon\to 0^{+}$, the functional $\mathcal{F}_{\varepsilon}(\cdot,h)$
$\Gamma$-converges, in the strong $L^{2}(\mathbb{R}^{n})$ topology, to the
functional

$\mathcal{F}_{0}(h)=\beta\int_{\partial\Omega}\dfrac{1}{1+\beta
h}\,d\mathcal{H}^{n-1}.$

Then, in view of the convexity of the functional with respect to $h$,

$\min\Set{\mathcal{F}_{0}(h)}{h\in\mathcal{H}_{m}}$

is achieved by the constant $h=m/P(\Omega)$, where $P(\Omega)$ denotes the
perimeter of $\Omega$.

Displacing the insulator uniformly around the boundary is somehow the trivial
solution, the one suggested by common sense, and very common when insulating
buildings. However, it is mathematically not satisfactory at all. As we expect
that portions of boundary with higher (mean) curvature are less convenient to
insulate with respect to those with lower curvature. Such an idea is strongly
suggested by the radial cases (see for example [10, Proposition 5.1]).

But such a kind of evidence is lost when performing the $\Gamma$-limit and
therefore we decided to push our analysis a bit further. Let

$K_{0}=\Set{v\in L^{2}(\mathbb{R}^{n})}{\,v=1\,\text{in }\Omega},$

our main result is a first-order asymptotic development by
$\Gamma$-convergence (see Subsection 2.1) for the functional
$\mathcal{F}_{\varepsilon}$. We denote by $H$ the mean curvature of $\Omega$
(see Subsection 2.2) and we prove the following

######  Theorem 1.1.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open set with $C^{3}$
boundary, and fix a $C^{2}$ function $h\colon\partial\Omega\to(0,+\infty)$.
Then the functional

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

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

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

The paper is planned as follows. In Subsection 3.2 we prove Theorem 1.1.
Thereafter, in Section 4 we fix $\Omega$ and we deal with the minimum problem

$\inf\Set{\mathcal{F}_{0}(h)+\varepsilon\mathcal{F}^{(1)}(h)}{h\in\mathcal{H}_{m}},$

where $\mathcal{F}^{(1)}(h)=\mathcal{F}^{(1)}(\chi_{\Omega},h)$. As we already
mentioned, the problem above, is a first-order approximation of the problem
(1.3) with respect to $\varepsilon>0$. Indeed we have that (see Subsection
2.1)

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

where $u_{\varepsilon}$ is the minimizer to (1.1), and

$\lim_{\varepsilon\to 0^{+}}\dfrac{R(\Omega,h,\varepsilon)}{\varepsilon}=0.$

In particular, we will prove that, as the intuition suggests, if $\varepsilon$
is small enough then the optimal configuration for the insulating layer
concentrates close to the points of $\partial\Omega$ where the mean curvature
is relatively small. Finally, in Subsection 4.2 we discuss the behaviour of
the functional $\mathcal{F}_{0}+\varepsilon\mathcal{F}^{(1)}$ under various
geometrical constraints (volume, perimeter, quermassintegral) on the set
$\Omega$.

##  2 Notation and tools

###  2.1 $\Gamma$-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 [8] and [2] 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 2.1 we have that
$\overline{x}\in\mathcal{U}_{0}$ and then

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

In particular, we have

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

###  2.2 Calculus on hypersurfaces

We refer to [12] for the notions in this section. Let
$\Omega\subset\mathbb{R}^{n}$ be a bounded open set with $C^{1}$ boundary and
let $\nu$ be the outer unit normal to its boundary. For every
$\sigma\in\partial\Omega$ let
$\tau=\\{\tau_{1}(\sigma),\dots,\tau_{n-1}(\sigma)\\}$ be an orthonormal basis
orthogonal to $\nu(\sigma)$, namely a basis for the tangent plane at
$\partial\Omega$ in $\sigma$.

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

Let $U\subseteq\mathbb{R}^{n}$ be an open set containing $\partial\Omega$ and
let $\phi=(\phi_{1},\dots,\phi_{n}):U\to\mathbb{R}^{n}$ be a $C^{1}$ function.
We define the tangential gradient of $\phi$ as the matrix-valued function
$D_{\tau}\phi\colon\partial\Omega\to\mathbb{R}^{n\times(n-1)}$ such that

$(D_{\tau}\phi)_{i,j}=\nabla\phi_{i}\cdot\tau_{j},$

where $i=1,\dots,n$ and $j=1,\dots,n-1$.

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

Let $U\subseteq\mathbb{R}^{n}$ be an open set containing $\partial\Omega$ and
let $\phi:U\to\mathbb{R}^{n}$ be a $C^{1}$ function. We define the tangential
Jacobian of $\phi$ as

$J_{\tau}\phi=\sqrt{\det\big{(}(D_{\tau}\phi)^{T}(D_{\tau}\phi)\big{)}\,}\,.$

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

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

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

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

Let $U\subseteq\mathbb{R}^{n}$ be an open set containing $\partial\Omega$ and
let $\phi:U\to\mathbb{R}^{n}$ be a $C^{1}$ function. We define the tangential
divergence of $\phi$ as

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

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

Let $\Omega$ be a bounded open set with $C^{2}$ boundary and let $\nu$ be the
outer unit normal to its boundary. Let $U\subseteq\mathbb{R}^{n}$ be an open
set containing $\partial\Omega$ and let $X$ be a $C^{1}(U)$ extension of
$\nu$. For any $\sigma\in\partial\Omega$ we define the _mean curvature of
$\partial\Omega$_ as

$H(x)=\operatorname{div}_{\tau}\nu.$

######  Remark .

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

$J_{\tau}\phi(\sigma)=1+t\operatorname{div}_{\tau}X(\sigma)+t^{2}R(t,\sigma)$

where the remainder $R$ is a bounded function. In particular, if $\Omega$ has
$C^{2}$ boundary and $X$ is an extension of $\nu$, we have

$J_{\tau}\phi(\sigma)=1+tH(\sigma)+t^{2}R(t,\sigma).$

######  Definition  (Set of Finite perimeter).

Let $E\subseteq\mathbb{R}^{n}$ be a measurable set. We define the _perimeter_
of $E$ as

$P(E)=\sup\Set{\int_{E}\operatorname{div}\varphi\,d\mathcal{L}^{n}}{\begin{aligned}
\varphi\in&\>C_{c}(\Omega,\mathbb{R}^{n})\\\ &\lvert\varphi\rvert\leq
1\end{aligned}}.$

If $P(E)<+\infty$ we say that $E$ is a _set of finite perimeter_.

######  Definition  (Generalized mean curvature).

Let $\Omega\subset\mathbb{R}^{n}$ be a set of finite perimeter, and let
$p\in[1,+\infty]$. We say that $\Omega$ has _generalized mean curvature in
$L^{p}$_ if there exists $H_{\Omega}\in L^{p}(\partial\Omega)$ such that

$\int_{\partial\Omega}\operatorname{div}_{\tau}F\,d\mathcal{H}^{n-1}=\int_{\partial\Omega}H_{\Omega}F\cdot\nu\,d\mathcal{H}^{n-1},$

for any $F\in C^{\infty}_{C}(A;\mathbb{R}^{n})$ with $A$ open set containing
$\Omega$.

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

Let $f\colon\mathbb{R}^{n}\to\mathbb{R}$ be a Lipschitz function, let
$g\colon\mathbb{R}^{n}\to\mathbb{R}$ be an $L^{1}(\mathbb{R}^{n})$ function
and let $U\subset\mathbb{R}^{n}$ be an open set, then

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

##  3 The $\Gamma$-limit

###  3.1 Setting of the problem

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open set with $C^{1,1}$
boundary, and fix a positive Lispchitz function
$h\colon\partial\Omega\to\mathbb{R}$. We recall that

$\Sigma_{\varepsilon}=\Set{\sigma+t\nu(\sigma)}{\sigma\in\partial\Omega,\,0<t<\varepsilon
h(\sigma)},$

and

$\Omega_{\varepsilon}=\overline{\Omega}\cup\Sigma_{\varepsilon}.$

Our assumptions on $\partial\Omega$ ensure that there exists
$\varepsilon_{0}=\varepsilon_{0}(\Omega,h)$ such that, if
$0<\varepsilon\leq\varepsilon_{0}$, the map

$(\sigma,t)\longmapsto\sigma+t\nu(\sigma)$

is invertible, that is, for every $x\in\Sigma_{\varepsilon}$ there exist
unique $\sigma(x)\in\partial\Omega$ and $t(x)$, with $0\leq
t(x)\leq\varepsilon h(\sigma(x))$, such that

$x=\sigma(x)+t(x)\nu(\sigma(x)).$

Therefore, we can extend $h$ and $\nu$ on $\Sigma_{\varepsilon}$ as
$h(x)=h(\sigma(x))$, and $\nu(x)=\nu(\sigma(x))$ respectively. Moreover, for
every $x\in\mathbb{R}^{n}\setminus\Omega$, let

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

be the distance from $\partial\Omega$, then we have that $d(x)=t(x)$ for every
$x\in\Sigma_{\varepsilon}$.

######  Remark .

By coarea formula (Theorem 2.2), the area formula on surfaces (Theorem 2.1)
and Subsection 2.2, we have that if $f:\Omega_{\varepsilon}\to\mathbb{R}$ is a
positive Borel function, then

$\int_{\Sigma_{\varepsilon}}f(x)\,dx=\int_{\partial\Omega}\int_{0}^{\varepsilon
h(\sigma)}f(\sigma+t\nu)\left(1+\varepsilon
R_{1}(\sigma,t,\varepsilon)\right)\,dt\,d\mathcal{H}^{n-1}$ (3.1)

and

$\int_{\partial\Omega_{\varepsilon}}f(\sigma)\,d\mathcal{H}^{n-1}=\int_{\partial\Omega}f(\sigma+\varepsilon
h\nu)\left(1+\varepsilon
R_{2}(\sigma,\varepsilon)\right)\,d\mathcal{H}^{n-1},$ (3.2)

where the remainder terms $R_{1}$ and $R_{2}$ are bounded functions, then
there exists $Q_{0}>0$ such that $\lvert R_{1}\rvert,\lvert R_{2}\rvert\leq
Q_{0}$. Moreover, if $\Omega$ is a bounded open set with $C^{3}$ boundary,
then we have

$\int_{\Sigma_{\varepsilon}}f(x)\,dx=\int_{\partial\Omega}\int_{0}^{\varepsilon
h(\sigma)}f(\sigma+t\nu)\left(1+tH(\sigma)+\varepsilon^{2}R_{3}(\sigma,t,\varepsilon)\right)\,dt\,d\mathcal{H}^{n-1}$
(3.3)

and

$\int_{\partial\Omega_{\varepsilon}}f(\sigma)\,d\mathcal{H}^{n-1}=\int_{\partial\Omega}f(\sigma+\varepsilon
h\nu)\left(1+\varepsilon
h(\sigma)H(\sigma)+\varepsilon^{2}R_{4}(\sigma,\varepsilon)\right)\,d\mathcal{H}^{n-1},$
(3.4)

where the remainder terms $R_{3}$ and $R_{4}$ are bounded functions, then
there exists $Q>0$ such that $\lvert R_{3}\rvert,\lvert R_{4}\rvert\leq Q$.

Let

$K_{\varepsilon}=\Set{v\in H^{1}(\Omega_{\varepsilon})}{\,v=1\,\text{in
}\Omega},$ (3.5)

and

$K_{0}=\Set{v\in L^{2}(\mathbb{R}^{n})}{\,v=1\,\text{in }\Omega},$ (3.6)

and consider the functional

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

denoting by

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

following the approach of [9], we have the following

######  Proposition .

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

###### Proof.

We start by proving the $\Gamma$-liminf inequality: Let $v\in
L^{2}(\mathbb{R}^{n})$ and let $v_{\varepsilon}\in L^{2}(\mathbb{R}^{n})$ such
that $v_{\varepsilon}$ converges to $v$ in $L^{2}(\mathbb{R}^{n})$ as
$\varepsilon\to 0^{+}$. Up to passing to a sub-sequence, we can assume that

$\liminf_{\varepsilon\to
0^{+}}\mathcal{F}_{\varepsilon}(v_{\varepsilon},h)=\lim_{\varepsilon\to
0^{+}}\mathcal{F}_{\varepsilon}(v_{\varepsilon},h),$

moreover, we can assume that such a limit is finite and that
$v_{\varepsilon}\in K_{\varepsilon}$. Therefore we have that $v\in K_{0}$ and,
by (3.1), (3.2) we have that

$\int_{\Sigma_{\varepsilon}}\lvert\nabla
v_{\varepsilon}\rvert^{2}\,dx\geq\int_{\partial\Omega}\int_{0}^{\varepsilon
h(\sigma)}\lvert\nabla v_{\varepsilon}(\sigma+t\nu)\rvert^{2}(1-\varepsilon
Q_{0})\,dt\,d\mathcal{H}^{n-1}$ (3.7)

and

$\int_{\partial\Omega_{\varepsilon}}v_{\varepsilon}^{2}\,d\mathcal{H}^{n-1}\geq\int_{\partial\Omega}v_{\varepsilon}^{2}(\sigma+\varepsilon
h(\sigma)\nu)(1-\varepsilon Q_{0})\,d\mathcal{H}^{n-1}.$ (3.8)

On the other hand, we have that, for $\mathcal{H}^{n-1}$-almost every
$\sigma\in\partial\Omega$,

$\begin{split}\int_{0}^{\varepsilon h(\sigma)}\lvert\nabla
v_{\varepsilon}(\sigma+t\nu)\rvert^{2}\,dt\geq&\dfrac{1}{\varepsilon
h(\sigma)}\left(\int_{0}^{\varepsilon h(\sigma)}\lvert\nabla
v_{\varepsilon}(\sigma+t\nu)\rvert\,dt\right)^{2}\\\\[10.0pt]
\geq&\dfrac{(v_{\varepsilon}(\sigma+\varepsilon h\nu)-1)^{2}}{\varepsilon
h(\sigma)},\end{split}$

then, by Young’s inequality, we have that, for every $\lambda>0$ and for
$\mathcal{H}^{n-1}$-almost every $\sigma\in\partial\Omega$,

$\int_{0}^{\varepsilon h(\sigma)}\lvert\nabla
v_{\varepsilon}(\sigma+t\nu)\rvert^{2}\,dt\geq\dfrac{(1-\lambda)v_{\varepsilon}(\sigma+\varepsilon
h\nu)^{2}}{\varepsilon h(\sigma)}+\dfrac{1}{\varepsilon
h(\sigma)}\left(1-\dfrac{1}{\lambda}\right).$ (3.9)

Putting together (3.7), (3.8) and (3.9) we finally have

$\mathcal{F}_{\varepsilon}(v_{\varepsilon},h)\geq\int_{\partial\Omega}\left(\left(\dfrac{1-\lambda}{h}+\beta\right)v_{\varepsilon}(\sigma+\varepsilon
h\nu)^{2}+\dfrac{1}{h}\left(1-\dfrac{1}{\lambda}\right)\right)\,d\mathcal{H}^{n-1}-\varepsilon
Q_{0}R_{\varepsilon}(\varepsilon,v_{\varepsilon}),$

where, if $\varepsilon$ is sufficiently small, using again (3.1) and (3.2), we
have

$R_{\varepsilon}(\varepsilon,v_{\varepsilon})\leq
2\mathcal{F}_{\varepsilon}(v_{\varepsilon},h).$

Finally, letting $\lambda=\lambda(\sigma)=1+\beta h(\sigma)$, and passing to
the limit as $\varepsilon\to 0^{+}$ we have that

$\liminf_{\varepsilon\to
0^{+}}\mathcal{F}_{\varepsilon}(v_{\varepsilon},h)\geq\beta\int_{\partial\Omega}\dfrac{1}{1+\beta
h}\,d\mathcal{H}^{n-1}=\mathcal{F}_{0}(v,h)$

and the $\Gamma$-liminf inequality is proved.

$\Gamma$-limsup inequality: Let $v\in L^{2}(\mathbb{R}^{n})$, if $v\notin
K_{0}$ the $\Gamma$-limsup inequality is trivial, therefore let $v\in K_{0}$.
Let

$v_{\varepsilon}(x)=\begin{dcases}1&\text{if }x\in\Omega,\\\\[5.0pt]
1-\dfrac{\beta d(x)}{\varepsilon(1+\beta h(x))}&\text{if
}x\in\Sigma_{\varepsilon},\\\\[5.0pt] v(x)&\text{if
}x\notin\Omega_{\varepsilon},\end{dcases}$

where we recall that, if $x=\sigma+t\nu(\sigma)$, then $h(x)=h(\sigma)$.
Trivially $v_{\varepsilon}$ converges to $v$ in $L^{2}(\mathbb{R}^{n})$ and
$v_{\varepsilon}\in K_{\varepsilon}$. For every $x\in\Sigma_{\varepsilon}$,

$\nabla v_{\varepsilon}(x)=-\dfrac{\beta\nabla d(x)}{\varepsilon(1+\beta
h(x))}+\dfrac{\beta^{2}d(x)\nabla h(x)}{\varepsilon(1+\beta h(x))^{2}}.$

Recalling that $0\leq d\leq\varepsilon h$, $\nabla d=\nu$ and $\nabla
h\cdot\nu=0$, we have

$\lvert\nabla
v_{\varepsilon}\rvert^{2}=\dfrac{\beta^{2}}{\varepsilon^{2}(1+\beta
h)^{2}}+\dfrac{\beta^{4}d^{2}\lvert\nabla h\rvert^{2}}{\varepsilon^{2}(1+\beta
h)^{4}}\leq\dfrac{\beta^{2}}{\varepsilon^{2}(1+\beta
h)^{2}}+\dfrac{\beta^{4}h^{2}\lvert\nabla h\rvert^{2}}{(1+\beta h)^{4}},$

where the second term is bounded since $h$ is Lipschitz. Hence, substituting
$\varepsilon h\tau=t$ in (3.1), we get

$\begin{split}\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
v_{\varepsilon}\rvert^{2}\,dx&\leq\dfrac{\beta^{2}}{\varepsilon}\int_{\Sigma_{\varepsilon}}\dfrac{1}{(1+\beta
h)^{2}}\,dx+\varepsilon C\lvert\Sigma_{\varepsilon}\rvert\\\\[10.0pt]
&\leq\int_{\partial\Omega}\int_{0}^{1}\dfrac{\beta^{2}h}{(1+\beta
h)^{2}}(1+\varepsilon Q_{0})\,d\tau\,d\mathcal{H}^{n-1}+\varepsilon
C\lvert\Sigma_{\varepsilon}\rvert\\\\[10.0pt]
&=\int_{\partial\Omega}\dfrac{\beta^{2}h}{(1+\beta
h)^{2}}\,d\mathcal{H}^{n-1}+o(\varepsilon).\end{split}$

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

$v_{\varepsilon}(\sigma+\varepsilon h(\sigma)\nu(\sigma))=\dfrac{1}{1+\beta
h(\sigma)},$

from which we get

$\beta\int_{\partial\Omega_{\varepsilon}}v_{\varepsilon}^{2}\,d\mathcal{H}^{n-1}\leq\int_{\partial\Omega}\dfrac{\beta}{(1+\beta
h)^{2}}(1+\varepsilon
Q_{0})\,d\mathcal{H}^{n-1}=\int_{\partial\Omega}\dfrac{\beta}{(1+\beta
h)^{2}}\,d\mathcal{H}^{n-1}+o(\varepsilon).$

Hence we have

$\mathcal{F}_{\varepsilon}(v_{\varepsilon},h)\leq\beta\int_{\partial\Omega}\dfrac{1}{1+\beta
h}\,d\mathcal{H}^{n-1}+o(\varepsilon),$

so that

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

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

In the following, for simplicity, we will denote by

$\mathcal{F}_{0}(h)=\beta\int_{\partial\Omega}\frac{1}{1+\beta
h}\,d\mathcal{H}^{n-1}.$

It can be deduced from the more general results in [9] that the minimum in the
class of functions $h$ with a given mass of such functional is achieved when
$h$ is constant. We include a direct proof of this statement in the following

######  Proposition .

Let $\Omega$ be a bounded open set with Lipschitz boundary, let $P(\Omega)=P$,
and let $m>0$. Then the problem

$\min\Set{\mathcal{F}_{0}(h)}{h\in\mathcal{H}_{m}}$ (3.10)

admits

$h_{0}=\dfrac{m}{P}$

as the unique solution.

###### Proof.

Let $h\in\mathcal{H}_{m}$. By Holder’s inequality, we have that

$\begin{split}P&=\int_{\partial\Omega}\,d\mathcal{H}^{n-1}\leq\left(\int_{\partial\Omega}\dfrac{1}{1+\beta
h}\,d\mathcal{H}^{n-1}\right)^{1/2}\left(\int_{\partial\Omega}(1+\beta
h)\,d\mathcal{H}^{n-1}\right)^{1/2}\\\\[10.0pt]
&\leq\left(\int_{\partial\Omega}\dfrac{1}{1+\beta
h}\,d\mathcal{H}^{n-1}\right)^{1/2}\left(P+\beta m\right)^{1/2},\end{split}$

so that

$\mathcal{F}_{0}(h)\geq\dfrac{\beta P^{2}}{P+\beta m}=\mathcal{F}_{0}(h_{0}).$

Finally, the uniqueness of the solution is given by the strict convexity of
the function

$x\longmapsto\dfrac{1}{1+\beta x}$

for $x\geq 0$. ∎

Let $H$ be the mean curvature of $\Omega$, we aim to show that

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

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

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

where $K_{0}$ is the set defined in (3.6).

###  3.2 Proof of Theorem 1.1

Let $\Omega$ be a bounded, open set with $C^{3}$ boundary, and fix a positive
$C^{2}$ function $h\colon\partial\Omega\to\mathbb{R}$. In this section, we
study the $\Gamma$-convergence of the family of functionals

$\delta\mathcal{F}_{\varepsilon}(v)=\dfrac{\mathcal{F}_{\varepsilon}(\cdot,h)-\mathcal{F}_{0}(h)}{\varepsilon},$
(3.11)

and we prove Theorem 1.1. In the following we consider the functions
$h,H\colon\partial\Omega\to\mathbb{R}$ extended on the set
$\Sigma_{\varepsilon}$ as $h(\sigma+t\nu)=h(\sigma)$ and
$H(\sigma+t\nu)=H(\sigma)$.

For every $\varepsilon>0$ let $u_{\varepsilon}\in K_{\varepsilon}$ be the
minimizer to $\mathcal{F}_{\varepsilon}$, where $K_{\varepsilon}$ is defined
in (3.5). By the assumptions on $\Omega$ and $h$, we have that
$u_{\varepsilon}$ is a $C^{2}(\Sigma_{\varepsilon})$ function and it is a
solution to

$\begin{cases}-\Delta u_{\varepsilon}=0&\text{in
}\Sigma_{\varepsilon},\\\\[5.0pt] u_{\varepsilon}=1&\text{on
}\partial\Omega,\\\\[5.0pt] \varepsilon\dfrac{\partial
u_{\varepsilon}}{\partial\nu_{\varepsilon}}+\beta u_{\varepsilon}=0&\text{on
}\partial\Omega_{\varepsilon}.\end{cases}$ (3.12)

Let $\alpha\in(0,\alpha_{0})$, where

$\alpha_{0}=1-\max_{\sigma\in\partial\Omega}\frac{\beta h(\sigma)}{1+\beta
h(\sigma)},$

and let $\varepsilon>0$, then on $\Sigma_{\varepsilon}$ we define

$w_{\varepsilon,\alpha}(x)=\begin{dcases}1-\left(\frac{d(x)}{\varepsilon
h(x)}\right)^{1-\alpha}\frac{\beta h(x)}{(1-\alpha)(1+\beta h(x))}&\text{if
}H(x)\geq 0,\\\\[7.0pt] 1-\left(\frac{d(x)}{\varepsilon
h(x)}\right)^{1+\alpha}\frac{\beta h(x)}{(1+\alpha)(1+\beta h(x))}&\text{if
}H(x)<0.\end{dcases}$

Then we have that $w_{\varepsilon,\alpha}>0$ and the following holds

######  Proposition .

For every $\alpha\in(0,\alpha_{0})$ there exists $\varepsilon_{\alpha}>0$ such
that if $0<\varepsilon<\varepsilon_{\alpha}$, then

$H(x)u_{\varepsilon}(x)\geq H(x)w_{\varepsilon,\alpha}(x)\qquad\text{for
}x\in\Sigma_{\varepsilon}.$

###### Proof.

Fix $\alpha\in(0,\alpha_{0})$. For simplicity, we denote by

$v_{\gamma}:=1-\left(\frac{d(x)}{\varepsilon h(x)}\right)^{\gamma}\frac{\beta
h(x)}{\gamma(1+\beta h(x))},$

and we aim to show that there exists an $\varepsilon_{\alpha}>0$ such that for
any $0<\varepsilon<\varepsilon_{\alpha}$, we have that $v_{1-\alpha}$ is a
subsolution to (3.12), while $v_{1+\alpha}$ is a supersolution to the same
problem. Namely,

$\begin{cases}-\Delta v_{1-\alpha}\leq 0&\text{in
}\Sigma_{\varepsilon},\\\\[5.0pt] v_{1-\alpha}=1&\text{on
}\partial\Omega,\\\\[5.0pt] \varepsilon\dfrac{\partial
v_{1-\alpha}}{\partial\nu_{\varepsilon}}+\beta v_{1-\alpha}\leq 0&\text{on
}\partial\Omega_{\varepsilon},\end{cases}\qquad\qquad\begin{cases}-\Delta
v_{1+\alpha}\geq 0&\text{in }\Sigma_{\varepsilon},\\\\[5.0pt]
v_{1+\alpha}=1&\text{on }\partial\Omega,\\\\[5.0pt] \varepsilon\dfrac{\partial
v_{1+\alpha}}{\partial\nu_{\varepsilon}}+\beta v_{1+\alpha}\geq 0&\text{on
}\partial\Omega_{\varepsilon}.\end{cases}$ (3.13)

In the following, we will always assume that $\varepsilon<1$. Let us recall
that

$\Omega_{\varepsilon}=\Set{x\in\mathbb{R}^{n}}{\dfrac{d(x)}{h(x)}\leq\varepsilon},\qquad\partial\Omega_{\varepsilon}=\Set{x\in\mathbb{R}^{n}}{d(x)-\varepsilon
h(x)=0}.$

By standard computations we get

$\nabla\left(\frac{d}{h}\right)=\frac{\nabla d}{h}-\frac{d\nabla
h}{h^{2}},\qquad\left\lvert\nabla\left(\frac{d}{h}\right)\right\rvert=\frac{1}{h}\sqrt{1+\left(\frac{d}{h}\right)^{2}\lvert\nabla
h\rvert^{2}}.$

Then, recalling that $\nabla d=\nu$ and that $\nabla h\cdot\nu=0$, the normal
$\nu_{\varepsilon}$ to the set $\Omega_{\varepsilon}$ is given by

$\nu_{\varepsilon}=\frac{1}{\sqrt{1+\varepsilon^{2}\lvert\nabla
h\rvert^{2}}}(\nu-\varepsilon\nabla h).$

By direct computations, for any $\gamma\in(0,2)\setminus\\{1\\}$ we have

$\Delta v_{\gamma}=-\frac{\beta h}{\gamma\varepsilon^{\gamma}(1+\beta
h)}\Delta\\!\left[\left(\frac{d}{h}\right)^{\gamma}\right]-\frac{2}{\gamma\varepsilon^{\gamma}}\nabla\\!\left[\left(\frac{d}{h}\right)^{\gamma}\right]\cdot\nabla\\!\left[\frac{\beta
h}{1+\beta
h}\right]-\frac{1}{\gamma\varepsilon^{\gamma}}\left(\frac{d}{h}\right)^{\gamma}\Delta\\!\left[\frac{\beta
h}{1+\beta h}\right].$ (3.14)

We then compute

$\nabla\\!\left[\left(\frac{d}{h}\right)^{\gamma}\right]=\gamma\left(\frac{d}{h}\right)^{\gamma-1}\left(\frac{\nu}{h}-\frac{d\nabla
h}{h^{2}}\right),\qquad\nabla\\!\left[\frac{\beta h}{1+\beta
h}\right]=\frac{\beta\nabla h}{(1+\beta h)^{2}},$ (3.15)

from which we get

$\nabla\\!\left[\left(\frac{d}{h}\right)^{\gamma}\right]\cdot\nabla\\!\left[\frac{\beta
h}{1+\beta
h}\right]=-\gamma\left(\frac{d}{h}\right)^{\gamma}\frac{\beta\lvert\nabla
h\rvert^{2}}{h(1+\beta h)^{2}}.$ (3.16)

In addition, we have

$\begin{split}\Delta\\!\left[\left(\frac{d}{h}\right)^{\gamma}\right]&=\gamma(\gamma-1)\left(\frac{d}{h}\right)^{\gamma-2}\left\lvert\nabla\left(\frac{d}{h}\right)\right\rvert^{2}+\gamma\left(\frac{d}{h}\right)^{\gamma-1}\left(\frac{\Delta
d}{h}-\frac{d\Delta h}{h^{2}}+2\frac{d\lvert\nabla
h\rvert^{2}}{h^{3}}\right)\\\\[10.0pt]
&=\gamma\left(\frac{d}{h}\right)^{\gamma-2}\left(-\frac{1-\gamma}{h^{2}}-\left(\frac{d}{h}\right)^{2}\frac{(1-\gamma)\lvert\nabla
h\rvert^{2}}{h^{2}}+\frac{d}{h}\frac{\Delta
d}{h}-\left(\frac{d}{h}\right)^{2}\frac{\Delta
h}{h}+\left(\frac{d}{h}\right)^{2}\frac{2\lvert\nabla
h\rvert^{2}}{h^{2}}\right),\end{split}$ (3.17)

so that, by (3.14), (3.17), and (3.16), we get

$\left(\frac{d(x)}{h(x)}\right)^{2-\gamma}\varepsilon^{\gamma}\Delta
v_{\gamma}(x)=\frac{\beta(1-\gamma)}{h(x)(1+\beta
h(x))}+R_{1}(x,\varepsilon,\gamma),$ (3.18)

where $R_{1}(x,\varepsilon,\gamma)$ is a suitable remainder term. Since
$d\leq\varepsilon h$,

$0<\inf_{\Sigma_{\varepsilon}}h\leq\sup_{\Sigma_{\varepsilon}}h<+\infty,$

and $\lvert\nabla h\rvert$, $\Delta h$, $\Delta d$ are bounded, then there
exist $C_{\gamma},\varepsilon_{0}>0$ such that

$\lvert R_{1}(x,\varepsilon,\gamma)\rvert\leq C_{\gamma}\varepsilon$ (3.19)

for any $\varepsilon<\varepsilon_{0}$. Thus, using (3.19) in (3.18) we have
that there exists $\varepsilon_{\alpha}>0$ such that if
$0<\varepsilon<\varepsilon_{\alpha}$, then

$-\Delta v_{1-\alpha}<0,\qquad-\Delta v_{1+\alpha}>0.$ (3.20)

On the other hand, for every $x\in\partial\Omega_{\varepsilon}$, since
$d(x)=\varepsilon h(x)$ and (3.15) hold true, we get

$\nabla v_{\gamma}(x)=-\frac{\beta}{\varepsilon(1+\beta
h(x))}(\nu(x)-\varepsilon\nabla h(x))-\frac{\beta\nabla h(x)}{\gamma(1+\beta
h(x))^{2}},$

which yields

$\begin{split}\frac{\partial
v_{\gamma}}{\partial\nu_{\varepsilon}}(x)=&-\frac{\beta\sqrt{1+\varepsilon^{2}\lvert\nabla
h\rvert^{2}}}{\varepsilon(1+\beta
h)}-\frac{\beta}{\gamma}\nu_{\varepsilon}\cdot\frac{\nabla h}{(1+\beta
h)^{2}}\\\\[7.0pt] =&-\frac{\beta\sqrt{1+\varepsilon^{2}\lvert\nabla
h\rvert^{2}}}{\varepsilon(1+\beta h)}+\frac{\beta\varepsilon\lvert\nabla
h\rvert^{2}}{\gamma(1+\beta h)^{2}\sqrt{1+\varepsilon^{2}\lvert\nabla
h\rvert^{2}}},\end{split}$ (3.21)

while

$v_{\gamma}(x)=1-\frac{\beta h(x)}{\gamma(1+\beta h(x))}.$ (3.22)

Hence, we get by (3.21) and (3.22)

$\varepsilon\frac{\partial v_{\gamma}}{\partial\nu_{\varepsilon}}+\beta
v_{\gamma}=-(1-\gamma)\frac{\beta^{2}h}{\gamma(1+\beta
h)}+R_{2}(\sigma,\varepsilon,\gamma)\quad\text{on
}\partial\Omega_{\varepsilon},$

where, as before, up to choosing a smaller $\varepsilon_{0}$,

$\lvert R_{2}(\sigma,\varepsilon,\gamma)\rvert\leq C_{\gamma}\varepsilon.$

Again, for small enough $\varepsilon$, on $\partial\Omega_{\varepsilon}$ we
get

$\varepsilon\frac{\partial v_{1-\alpha}}{\partial\nu_{\varepsilon}}+\beta
v_{1-\alpha}<0,\quad\qquad\varepsilon\frac{\partial
v_{1+\alpha}}{\partial\nu_{\varepsilon}}+\beta v_{1+\alpha}>0.$ (3.23)

Finally, joining (3.20) and (3.23), by standard comparison results for
elliptic operators the proposition is proved. ∎

We can now prove Theorem 1.1.

###### Proof of Theorem 1.1.

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
$h$.

$\mathcal{F}_{\varepsilon}(u_{\varepsilon})=\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla
u_{\varepsilon}\rvert^{2}\,dx+\beta\int_{\partial\Omega_{\varepsilon}}u_{\varepsilon}^{2}\,d\mathcal{H}^{n-1},$
(3.24) $\mathcal{F}_{0}=\beta\int_{\partial\Omega}\dfrac{1}{1+\beta
h}\,d\mathcal{H}^{n-1}.$ (3.25)

By (3.3) and (3.4) we have

$\int_{\Sigma_{\varepsilon}}\lvert\nabla
u_{\varepsilon}\rvert^{2}\,dx\geq\int_{\partial\Omega}\int_{0}^{\varepsilon
h(\sigma)}\lvert\nabla
u_{\varepsilon}(\sigma+t\nu)\rvert^{2}\left(1+tH(\sigma)-\varepsilon^{2}Q\right)\,dt\,d\mathcal{H}^{n-1}$
(3.26)

and

$\dfrac{\beta}{\varepsilon}\int_{\partial\Omega_{\varepsilon}}u_{\varepsilon}^{2}\,\mathcal{H}^{n-1}\geq\dfrac{\beta}{\varepsilon}\int_{\partial\Omega}u_{\varepsilon}^{2}(\sigma+\varepsilon
h(\sigma)\nu(\sigma))\left(1+\varepsilon
h(\sigma)H(\sigma)-\varepsilon^{2}Q\right)\,d\mathcal{H}^{n-1}.$ (3.27)

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

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

Up to choosing a smaller $\varepsilon$, we can apply Young’s inequality,
having that for every $\lambda>0$

$\begin{split}\int_{0}^{\varepsilon h(\sigma)}\lvert\nabla
u_{\varepsilon}^{2}(\sigma+t\nu)\rvert\left(1+tH(\sigma)\right)\,dt\geq&\dfrac{(1-\lambda)(1+\varepsilon
hH)u_{\varepsilon}(\sigma+\varepsilon h\nu)^{2}}{\varepsilon h}\\\\[10.0pt]
&+\dfrac{1}{\varepsilon
h}\left(1-\dfrac{1}{\lambda}\right)\left(1+\int_{0}^{\varepsilon
h(\sigma)}\dfrac{Hu_{\varepsilon}(\sigma+t\nu)}{2\sqrt{1+tH}}\,dt\right)^{2}.\end{split}$
(3.28)

We then have, joining (3.24),(3.26), (3.28), (3.27), and (3.25),

$\begin{split}\delta\mathcal{F}_{\varepsilon}(u_{\varepsilon})=\dfrac{\mathcal{F}_{\varepsilon}(u_{\varepsilon})-\mathcal{F}_{0}}{\varepsilon}\geq&\int_{\partial\Omega}\dfrac{1}{\varepsilon
h(\sigma)}\left((1-\lambda)(1+\varepsilon hH)+\beta h(1+\varepsilon
hH)\right)u_{\varepsilon}^{2}(\sigma+\varepsilon
h\nu)\,d\mathcal{H}^{n-1}\\\\[10.0pt]
&+\int_{\partial\Omega}\dfrac{1}{\varepsilon
h}\left(\left(1-\dfrac{1}{\lambda}\right)\left(1+\int_{0}^{\varepsilon
h}\dfrac{Hu_{\varepsilon}(\sigma+t\nu)}{2\sqrt{1+tH}}\,dt\right)^{2}-\dfrac{\beta
h}{1+\beta h}\right)\,d\mathcal{H}^{n-1}\\\\[10.0pt] &-Q\varepsilon
R(\varepsilon,u_{\varepsilon})\end{split}$ (3.29)

where, if $\varepsilon$ is small enough,

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

Letting $\lambda=\lambda(\sigma)=1+\beta h(\sigma)$ in (3.29), and using the
inequality $(1+x)^{2}\geq 1+2x$,

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

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

$\delta\mathcal{F}_{\varepsilon}(u_{\varepsilon})\geq\beta\int_{\partial\Omega}\dfrac{hH}{(1+\beta
h)}\fint_{0}^{\varepsilon}u_{\varepsilon}(\sigma+th\nu)\,dt\,d\mathcal{H}^{n-1}+O(\varepsilon).$
(3.30)

Finally, let $\alpha\in(0,1)$ and let

$\gamma=\gamma(\sigma)=\begin{cases}1-\alpha&\text{if }H(\sigma)\geq
0\\\\[5.0pt] 1+\alpha&\text{if }H(\sigma)<0.\end{cases}$

Let us recall that

$w_{\varepsilon,\alpha}(\sigma+th(\sigma)\nu(\sigma))=1-t^{\gamma}\frac{\beta
h(\sigma)}{\varepsilon^{\gamma}\gamma(1+\beta h(\sigma))}.$

By Subsection 3.2 we have that for every $0<\varepsilon<\varepsilon_{\alpha}$

$\begin{split}\delta\mathcal{F}_{\varepsilon}(u_{\varepsilon})&\geq\beta\int_{\partial\Omega}\dfrac{hH}{(1+\beta
h)}\fint_{0}^{\varepsilon}w_{\varepsilon,\alpha}(\sigma+th\nu)\,dt\,d\mathcal{H}^{n-1}+O(\varepsilon)\\\\[10.0pt]
&=\int_{\partial\Omega}\left(1-\frac{\beta h}{(1+\beta
h)\gamma(\gamma+1)}\right)\frac{\beta hH}{1+\beta
h}\,d\mathcal{H}^{n-1}+O(\varepsilon),\end{split}$

so that

$\liminf_{\varepsilon\to
0^{+}}\delta\mathcal{F}_{\varepsilon}(u_{\varepsilon})\geq\int_{\partial\Omega}\left(1-\frac{\beta
h}{(1+\beta h)\gamma(\gamma+1)}\right)\frac{\beta hH}{1+\beta
h}\,d\mathcal{H}^{n-1}.$

Letting $\alpha$ go to $0$, we have that $\gamma$ tends to $1$, and

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

and the $\Gamma$-Liminf is proved.

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

Let

$\varphi_{\varepsilon}(x)=\begin{dcases}1&\text{if }x\in\Omega,\\\\[5.0pt]
1-\dfrac{\beta d(x)}{\varepsilon(1+\beta h(x))}-\frac{\beta
d(x)^{2}H(x)}{2\varepsilon(1+\beta h(x))^{2}}&\text{if
}x\in\Sigma_{\varepsilon},\\\\[5.0pt] 0&\text{if
}x\in\mathbb{R}^{n}\setminus\Omega,\end{dcases}$

where we recall that if $x=\sigma+t\nu(\sigma)$, then $h(x)=h(\sigma)$ and
$H(x)=H(\sigma)$. We have that $\varphi_{\varepsilon}\in H^{1}(\Omega)$ and
$\varphi_{\varepsilon}$ converges in $L^{2}(\mathbb{R}^{n})$, to the
characteristic function of $\Omega$. Computing the gradient of
$\varphi_{\varepsilon}$, for any $x\in\Sigma_{\varepsilon}$,

$\nabla\varphi_{\varepsilon}(x)=-\frac{\beta\nu(x)}{\varepsilon(1+\beta
h(x))}-\frac{\beta d(x)H(x)\nu(x)}{\varepsilon(1+\beta
h(x))^{2}}+R_{\varepsilon}(x),$

where $R_{\varepsilon}$ is a remainder term which is bounded, uniformly in
$\varepsilon$, since $d\leq\varepsilon h$, and $h,\nabla h,H,\nabla H$ are
bounded. Moreover, $\nu\cdot R_{\varepsilon}=0$, since both $h$ and $H$ are
defined in such a way that $\nabla h$ and $\nabla H$ are orthogonal to $\nu$.
Therefore, for sufficiently small $\varepsilon$,

$\begin{split}\lvert\nabla\varphi_{\varepsilon}\rvert^{2}&\leq\frac{\beta^{2}}{\varepsilon^{2}}\underbracket{\left(\dfrac{1}{(1+\beta
h)^{2}}+2\dfrac{Hd}{(1+\beta h)^{3}}\right)}_{\geq 0}+C,\end{split}$

where we used again the boundedness of $h$ and $H$, and the fact that
$d\leq\varepsilon h$. Hence, substituting $\varepsilon h\tau=t$ in (3.3), and
noticing that $d(\sigma+\varepsilon\tau h(\sigma)\nu(\sigma))=\varepsilon
h\tau$, we get

$\begin{split}\varepsilon\int_{\Sigma_{\varepsilon}}\lvert\nabla\varphi_{\varepsilon}\rvert^{2}\,dx\leq&\dfrac{\beta^{2}}{\varepsilon}\int_{\Sigma_{\varepsilon}}\left(\dfrac{1}{(1+\beta
h)^{2}}+2\dfrac{Hd}{(1+\beta h)^{3}}\right)\,dx+\varepsilon
C\left\lvert\Sigma_{\varepsilon}\right\rvert\\\\[10.0pt]
\leq&\int_{\partial\Omega}\int_{0}^{1}\dfrac{\beta^{2}h}{(1+\beta
h)^{2}}(1+\varepsilon\tau
hH+\varepsilon^{2}Q)\,d\tau\,d\mathcal{H}^{n-1}+\\\\[10.0pt]
&+\int_{\partial\Omega}\int_{0}^{1}\dfrac{2\varepsilon\beta^{2}h^{2}H\tau}{(1+\beta
h)^{3}}(1+\varepsilon\tau
hH+\varepsilon^{2}Q)\,d\tau\,d\mathcal{H}^{n-1}+\varepsilon
C\left\lvert\Sigma_{\varepsilon}\right\rvert\\\\[10.0pt]
=&\int_{\partial\Omega}\dfrac{\beta^{2}h}{(1+\beta
h)^{2}}\,d\mathcal{H}^{n-1}+\varepsilon\int_{\partial\Omega}\left(\dfrac{\beta^{2}h^{2}H}{2(1+\beta
h)^{2}}+\dfrac{\beta^{2}h^{2}H}{(1+\beta
h)^{3}}\right)\,d\mathcal{H}^{n-1}+O(\varepsilon^{2}).\end{split}$ (3.31)

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

$\varphi_{\varepsilon}(\sigma+\varepsilon
h(\sigma)\nu(\sigma))=\frac{1}{1+\beta h(\sigma)}-\frac{\varepsilon\beta
h(\sigma)^{2}H}{2(1+\beta h(\sigma))^{2}},$

from which we get

$\begin{split}\beta\int_{\partial{\Omega_{\varepsilon}}}\varphi_{\varepsilon}^{2}\,d\mathcal{H}^{n-1}\leq\int_{\partial\Omega}\dfrac{\beta}{(1+\beta
h)^{2}}(1+\varepsilon
hH)\,d\mathcal{H}^{n-1}-\int_{\partial\Omega}\dfrac{\varepsilon\beta^{2}h^{2}H}{(1+\beta
h)^{3}}\,d\mathcal{H}^{n-1}+O(\varepsilon^{2}).\end{split}$ (3.32)

Finally, joining (3.24), (3.31), (3.32), and (3.25) we have

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

so that

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

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

##  4 Properties of the first order development

Let $\Omega$ be a bounded, open set with $C^{1,1}$ boundary. Consider the
functional

$\mathcal{G}_{\varepsilon}(\Omega,h)=\beta\int_{\partial\Omega}\left(\dfrac{1}{1+\beta
h}+\varepsilon H\dfrac{h(2+\beta h)}{2(1+\beta
h)^{2}}\right)\,d\mathcal{H}^{n-1}.$

In the following, we drop the dependence on the set $\Omega$ and we write
$\mathcal{G}_{\varepsilon}(h)$ in place of
$\mathcal{G}_{\varepsilon}(\Omega,h)$. For every $m>0$, we will consider the
problem

$\inf\Set{\mathcal{G}_{\varepsilon}(h)}{h\in\mathcal{H}_{m}},$ (4.1)

where the set $\mathcal{H}_{m}$ is defined in (1.2).

######  Remark .

Assume that a non-zero continuous solution $\mu\in L^{1}(\partial\Omega)$ to
problem (4.1) exists, so that the set $U=\set{\mu>0}$ is open. Then for every
$\psi\in C_{c}^{\infty}(U)$ with zero mean, and for every $\eta\in\mathbb{R}$
sufficiently small, we can consider the variation $\mu+\eta\psi$ which leads
to the Euler-Lagrange equation

$\int_{\partial\Omega}\left(-\dfrac{\beta}{(1+\beta\mu)^{2}}+\dfrac{\varepsilon
H}{1+\beta\mu}-\varepsilon
H\dfrac{\beta\mu(2+\beta\mu)}{(1+\beta\mu)^{3}}\right)\psi\,d\mathcal{H}^{n-1}=0.$

The previous equation yields

$\dfrac{c}{\beta}(1+\beta\mu)^{3}-(1+\beta\mu)+\dfrac{\varepsilon
H}{\beta}=0,$

for some constant $c\in\mathbb{R}$.

Let $\varepsilon>0$, in the following we will assume that

$\sup_{\partial\Omega}\dfrac{\varepsilon H}{\beta}\leq\dfrac{2}{3}.$ (4.2)

Let

$H_{0}=\inf_{\partial\Omega}H$

and

$k_{0}=1-\dfrac{\varepsilon H_{0}}{\beta}.$

For every $k\in(0,k_{0})$, let

$\Gamma_{k}=\Set{\sigma\in\partial\Omega}{\dfrac{\varepsilon
H(\sigma)}{\beta}<1-k}$

and consider

$P_{k}(y,\sigma)=ky^{3}-y+\dfrac{\varepsilon H(\sigma)}{\beta}.$

######  Proposition .

Let (4.2) hold true. Then, for every $k\in(0,k_{0})$, and
$\sigma\in\Gamma_{k}$, in the interval $(1,+\infty)$ there exists a unique
$y_{k}(\sigma)$ such that

$P_{k}(y_{k}(\sigma),\sigma)=0,$ (4.3)

and there exists $z_{k}>1$ such that

$\max\left\\{\dfrac{1}{\sqrt{3k}},1\right\\}\leq y_{k}(\sigma)\leq z_{k}.$
(4.4)

In particular, we have

$\lim_{k\to k_{0}^{-}}z_{k}=1.$ (4.5)

Moreover, for every $k_{1}<k_{2}$ and $\sigma\in\Gamma_{k_{2}}$ we have that

$y_{k_{2}}(\sigma)<y_{k_{1}}(\sigma).$ (4.6)

###### Proof.

For any fixed $\sigma\in\Gamma_{k}$ we have that $P_{k}(1,\sigma)<0$, and in
addition, for $k\geq 1/3$, the polynomial $P_{k}(y,\sigma)$ is strictly
increasing in $y\geq 1$, while for $k<1/3$ we have that

$\displaystyle\frac{\partial}{\partial y}P_{k}(y,\sigma)<0\qquad\text{if
}y\in\left[1,\frac{1}{\sqrt{3k}}\right),$
$\displaystyle\frac{\partial}{\partial y}P_{k}(y,\sigma)>0\qquad\text{if
}y\in\left(\frac{1}{\sqrt{3k}},+\infty\right).$

Therefore, in the interval $(1,+\infty)$ there exists a unique zero
$y_{k}(\sigma)$ of the polynomial $P_{k}(\cdot,\sigma)$, and

$y_{k}(\sigma)\geq\frac{1}{\sqrt{3k}}.$

Notice in addition that for every $y>1$, we have that $y<y_{k}(\sigma)$ if and
only if $P_{k}(y,\sigma)<0$. Hence, if we choose $z_{k}$ to be the unique real
number in $(1,+\infty)$ such that

$kz_{k}^{3}-z_{k}+\frac{\varepsilon H_{0}}{\beta}=0,$ (4.7)

then

$P_{k}(z_{k},\sigma)=\frac{\varepsilon H(\sigma)}{\beta}-\frac{\varepsilon
H_{0}}{\beta}\geq 0,$

and we have that (4.4) holds.

We now prove (4.5). We first observe that $z_{k}$ is decreasing in $k$: let
$k_{1}<k_{2}$, so that

$k_{1}z_{k_{2}}^{3}-z_{k_{2}}+\dfrac{\varepsilon
H_{0}}{\beta}<k_{2}z_{k_{2}}^{3}-z_{k_{2}}+\dfrac{\varepsilon
H_{0}}{\beta}=0=k_{1}z_{k_{1}}^{3}-z_{k_{1}}+\dfrac{\varepsilon
H_{0}}{\beta},$

which ensures

$z_{k_{2}}<z_{k_{1}},$

since the polynomial $k_{1}y^{3}-y+\varepsilon H_{0}/\beta$ is strictly
increasing in $[1/\sqrt{3k_{1}},+\infty)$. We now have that there exists

$z=\lim_{k\to k_{0}^{-}}z_{k},$

and, passing to the limit in (4.7) and recalling that by definition $\beta
k_{0}=\varepsilon H_{0}$, we get that $z$ solves the equation

$k_{0}(z^{3}-1)-z+1=0.$

From (4.2), we have that $k_{0}>1/3$, so that $z=1$ is the unique solution in
$[1,+\infty)$ to the previous equation, proving (4.5).

Finally, in order to prove (4.6), let $k_{1}<k_{2}$ and
$\sigma\in\Gamma_{k_{1}}\cap\Gamma_{k_{2}}=\Gamma_{k_{2}}$, then (4.3) ensures
that

$P_{k_{1}}(y_{k_{2}},\sigma)<P_{k_{2}}(y_{k_{2}},\sigma)=0,$

from which

$y_{k_{2}}(\sigma)<y_{k_{1}}(\sigma).$

∎

Let $k\in(0,k_{0})$, and let $y_{k}$ be as in Section 4, we define

$\mu_{k}(\sigma)=\begin{cases}\dfrac{1}{\beta}(y_{k}(\sigma)-1)&\text{if
}\sigma\in\Gamma_{k}.\\\\[10.0pt] 0&\text{if
}\sigma\in\partial\Omega\setminus\Gamma_{k},\end{cases}$

Notice that, by (4.6), $\mu_{k}$ is decreasing in $k$.

We have the following

######  Proposition .

Let (4.2) hold true. Then, for every $m>0$ ,there exists a unique
$k=k_{m}\in(0,k_{0})$ such that

$\int_{\partial\Omega}\mu_{k}\,d\mathcal{H}^{n-1}=m.$

###### Proof.

We first prove that the function

$M(k)=\int_{\partial\Omega}\mu_{k}\,d\mathcal{H}^{n-1}$

is continuous. Fix $k\in(0,k_{0})$ and let $\delta>0$, then

$\beta(M(k)-M(k+\delta))=\int_{\Gamma_{k+\delta}}(y_{k}-y_{k+\delta})\,d\mathcal{H}^{n-1}+\int_{\Set{1-k-\delta\leq\frac{\varepsilon
H}{\beta}<1-{k}}}(y_{{k}}-1)\,d\mathcal{H}^{n-1}.$ (4.8)

By definition, we have that for every $\sigma\in\partial\Omega$

$\lim_{\delta\to
0^{+}}\chi_{\Gamma_{k+\delta}}(\sigma)=\chi_{\Gamma_{k}}(\sigma).$

Let $\sigma\in\Gamma_{k}$, then the function $y_{k+\delta}(\sigma)$ is defined
for small enough $\delta<\delta_{\sigma}$, and by the implicit function
theorem and the regularity of $P_{k}(y,\sigma)$, we get

$\lim_{\delta\to 0^{+}}y_{k+\delta}(\sigma)=y_{k}(\sigma).$

Therefore, by (4.4), the monotonicity of $z_{k}$, and the dominated
convergence theorem we have

$\lim_{\delta\to
0^{+}}\int_{\Gamma_{k+\delta}}(y_{k}-y_{k+\delta})\,d\mathcal{H}^{n-1}=0.$
(4.9)

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

$\lim_{\delta\to 0^{+}}\chi_{\Set{1-k-\delta\leq\frac{\varepsilon
H}{\beta}<1-{k}}}(\sigma)=0,$

which entails

$\lim_{\delta\to 0^{+}}\int_{\Set{1-k-\delta\leq\frac{\varepsilon
H}{\beta}<1-{k}}}(y_{{k}}-1)\,d\mathcal{H}^{n-1}=0.$ (4.10)

Joining (4.8), (4.9), and (4.10), we get

$\lim_{\delta\to 0^{+}}M(k)-M(k+\delta)=0.$

We now fix $k\in(0,k_{0})$, $\delta>0$, and we compute

$\beta(M(k-\delta)-M(k))=\int_{\Gamma_{k}}(y_{k-\delta}-y_{k})\,d\mathcal{H}^{n-1}+\int_{\Set{1-k\leq\frac{\varepsilon
H}{\beta}<1-{k}+\delta}}(y_{{k-\delta}}-1)\,d\mathcal{H}^{n-1}.$ (4.11)

As in the previous case, by the implicit function theorem, for every
$\sigma\in\Gamma_{k}$,

$\lim_{\delta\to 0^{+}}y_{k-\delta}(\sigma)=y_{k}(\sigma).$

By the dominated convergence theorem,

$\lim_{\delta\to
0^{+}}\int_{\Gamma_{k}}(y_{k-\delta}-y_{k})\,d\mathcal{H}^{n-1}=0.$ (4.12)

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

$\lim_{\delta\to 0^{+}}\chi_{\Set{1-k\leq\frac{\varepsilon
H}{\beta}<1-{k}+\delta}}(\sigma)=\chi_{\Set{\frac{\varepsilon
H}{\beta}=1-k}}(\sigma),$

and, for every $\sigma$ such that ${\varepsilon H(\sigma)}=\beta(1-k)$, we may
use the monotonicity of $y_{k-\delta}$ and then we pass to the limit in (4.3),
having that

$\lim_{\delta\to 0^{+}}y_{k-\delta}(\sigma)=1,$

which entails

$\lim_{\delta\to 0^{+}}\int_{\Set{1-k\leq\frac{\varepsilon
H}{\beta}<1-{k}+\delta}}(y_{{k-\delta}}-1)\,d\mathcal{H}^{n-1}=0.$ (4.13)

Joining (4.11), (4.12), and (4.13), we get

$\lim_{\delta\to 0^{+}}M(k-\delta)-M(k)=0,$

thus concluding the proof of the continuity of $M$.

Finally, by monotonicity, and by (4.4), and (4.2), we have that

$\lim_{k\to 0^{+}}M(k)=+\infty,\quad\lim_{k\to k_{0}^{-}}M(k)=0.$

and the proposition is proved. ∎

######  Theorem 4.1.

Let (4.2) hold true. Then, for every $m>0$, the function $\mu_{k_{m}}$ is the
unique minimizer to problem (4.1).

###### Proof.

Let $h\colon\partial\Omega\to R$ with $h\geq 0$ and

$\int_{\partial\Omega}h\,d\mathcal{H}^{n-1}\leq m.$

for every $t\in[0,1]$ consider

$h_{t}=\mu_{k}+t(h-\mu_{k})$

and

$g(t)=\mathcal{G}_{\varepsilon}(h_{t}).$

We claim that $g(t)$ is increasing in $t$. By explicit computation, we have
that

$g^{\prime}(t)=\beta\int_{\partial\Omega}\dfrac{\varepsilon H-\beta(1+\beta
h_{t})}{(1+\beta h_{t})^{3}}(h-\mu_{k})\,d\mathcal{H}^{n-1}.$

From (4.2) we have that, for every $\sigma\in\partial\Omega$ the function

$x\mapsto\dfrac{\varepsilon H-\beta(1+\beta x)}{(1+\beta x)^{3}}$

is increasing on $[0,+\infty)$, so that

$\begin{split}g^{\prime}(t)&\geq\beta\int_{\partial\Omega}\dfrac{\varepsilon
H-\beta(1+\beta\mu_{k})}{(1+\beta\mu_{k})^{3}}(h-\mu_{k})\,d\mathcal{H}^{n-1}\\\\[10.0pt]
&=\beta\left(\int_{\partial\Omega\setminus\Gamma_{k}}(\varepsilon
H-\beta)h\,d\mathcal{H}^{n-1}+\int_{\Gamma_{k}}\dfrac{\varepsilon
H-\beta(1+\beta\mu_{k})}{(1+\beta\mu_{k})^{3}}(h-\mu_{k})\,d\mathcal{H}^{n-1}\right).\end{split}$

By (4.3) we have that on $\Gamma_{k}$

$\dfrac{\varepsilon H-\beta(1+\beta\mu_{k})}{(1+\beta\mu_{k})^{3}}=-\beta k$

while on $\partial\Omega\setminus\Gamma_{k}$

$\varepsilon H-\beta\geq-\beta k$

so that

$g^{\prime}(t)\geq\beta^{2}k\int_{\partial\Omega}(\mu_{k}-h)\,d\mathcal{H}^{n-1}\geq
0,$

and the claim is proven. In particular, we have that

$\mathcal{G}_{\varepsilon}(\mu_{k})=g(0)\leq
g(1)=\mathcal{G}_{\varepsilon}(h)$

that is, $\mu_{k}$ is a minimizer for problem (4.1). Finally, by (4.2), we
have that, for every $\sigma\in\partial\Omega$, the function

$x\in[0,+\infty)\mapsto\dfrac{1}{1+\beta x}+\varepsilon
H(\sigma)\dfrac{x(2+\beta x)}{2(1+\beta x)^{2}}$

is strictly convex, thus problem (4.1) admits a unique minimizer. ∎

######  Remark .

Notice that the optimal configuration $\mu_{k_{m}}$ concentrates where the
mean curvature is smaller: for simplicity, let us write $k=k_{m}$, and let us
take $\sigma_{1},\sigma_{2}\in\Gamma_{k}$ such that

$H(\sigma_{1})<H(\sigma_{2}).$

Noticing that

$P_{k}(y_{k}(\sigma_{1}),\sigma_{1})=0=P_{k}(y_{k}(\sigma_{2}),\sigma_{2}),$

we get

$ky_{k}(\sigma_{1})^{3}-y_{k}(\sigma_{1})>ky_{k}(\sigma_{2})^{3}-y_{k}(\sigma_{2}).$

By (4.4), we can use the monotonicity of the function $y\mapsto ky^{3}-y$ on
$[1/\sqrt{3k},+\infty)$, getting

$y_{k}(\sigma_{1})>y_{k}(\sigma_{2}),$

so that $\mu_{k}(\sigma_{1})>\mu_{k}(\sigma_{2})$.

######  Remark .

Notice that, if for every $\sigma\in\partial\Omega$

$\dfrac{\varepsilon H(\sigma)}{\beta}\geq 2,$ (4.14)

then the optimal configuration is given by $\mu\equiv 0$. Indeed if (4.14)
holds, for every $\sigma\in\partial\Omega$ the function

$x\in[0,+\infty)\mapsto\dfrac{1}{1+\beta x}+\dfrac{\varepsilon
H(\sigma)}{\beta}\dfrac{\beta x(2+\beta x)}{2(1+\beta x)^{2}}$

reaches its minimum for $x=0$.

###  4.1 Minimization with perimeter constraint

Fix $\varepsilon>0$. For every $P>0$ and $m>0$ we define,

$\mathcal{K}_{P}=\Set{\Omega\subset\mathbb{R}^{n}}{\begin{aligned}
&\Omega\text{ open and bounded with $C^{1,1}$ boundary}\\\ &H_{\Omega}\geq
0\\\ &P(\Omega)=P\end{aligned}},$

where $H_{\Omega}$ is the mean curvature of $\Omega$. Let
$\Omega\in\mathcal{K}_{P}$, and let $h\in\mathcal{H}_{m}(\partial\Omega)$,
where $\mathcal{H}_{m}$ is the set defined in (1.2), then we study the
functional

$\mathcal{G}_{\varepsilon}(\Omega,h)=\beta\int_{\partial\Omega}\left(\dfrac{1}{1+\beta
h}+\varepsilon H_{\Omega}\dfrac{h(2+\beta h)}{2(1+\beta
h)^{2}}\right)\,d\mathcal{H}^{n-1}.$

We will now consider the problem

$\inf\Set{\mathcal{G}_{\varepsilon}(\Omega,h)}{(\Omega,h)\in\mathcal{K}_{p}\times\mathcal{H}_{m}(\partial\Omega)}.$
(4.15)

######  Definition  (Cookie Shape).

For any $r,R>0$ we define the _cookie shape_

$C_{r,R}=\Set{(x^{\prime},x_{n})}{-f_{r,R}(x^{\prime})\leq x_{n}\leq
f_{r,R}(x^{\prime})},$

where

$f_{r,R}(x^{\prime})=\begin{cases}r&\lvert x^{\prime}\rvert\leq R,\\\
\sqrt{r^{2}-\left(\lvert x^{\prime}\rvert-R\right)^{2}}&R<\lvert
x^{\prime}\rvert\leq R+r.\end{cases}$

######  Remark .

For every $r,R>0$, We have that $C_{r,R}$ is a convex set with $C^{1,1}$
boundary and

$H=H_{C_{r,R}}\geq\frac{1}{r}\chi_{\set{H>0}}.$

Moreover,

$P(C_{r,R})=2\omega_{n-1}\left(R^{n-1}+(n-1)r\int_{0}^{1}\frac{\left(r\rho+R\right)^{n-2}}{\sqrt{1-\rho^{2}}}\,d\rho\right).$
(4.16)

We observe that the function $P(C_{r,R})$ is increasing in $r$ and $R$.

######  Theorem 4.2.

For every $P,m>0$ we have

$\inf\Set{\mathcal{G}_{\varepsilon}(\Omega,h)}{(\Omega,h)\in\mathcal{K}_{p}\times\mathcal{H}_{m}(\partial\Omega)}=\dfrac{\beta
P^{2}}{P+\beta m}$

and the infimum is asymptotically achieved by a sequence of thin cookie
shapes.

###### Proof.

For every $(\Omega,h)\in\mathcal{K}_{p}\times\mathcal{H}_{m}(\partial\Omega)$,

$\mathcal{G}_{\varepsilon}(\Omega,h)\geq\beta\int_{\partial\Omega}\dfrac{1}{1+\beta
h}\,d\mathcal{H}^{n-1}\geq\dfrac{\beta P^{2}}{P+\beta m}.$

Let $r_{k}>0$ be a decreasing sequence with

$\lim_{k}r_{k}=0;$

let $R_{k}$ be such that, for every $k$,

$P(C_{r_{k},R_{k}})=P.$

Then $R_{k}$ is increasing in $k$ and

$\lim_{k}R_{k}=\left(\dfrac{P}{2\omega_{n-1}}\right)^{\frac{1}{n-1}}.$

Consider

$h_{k}(\sigma)=\begin{cases}\dfrac{m}{2\omega_{n-1}R_{k}^{n-1}}&\text{if
}H(\sigma)=0,\\\\[5.0pt] 0&\text{if }H(\sigma)>0.\end{cases}$

We have that

$\mathcal{G}_{\varepsilon}(C_{r_{k},R_{k}},h_{k})=\dfrac{2\beta\omega_{n-1}R_{k}^{n-1}}{1+\frac{m}{2\omega_{n-1}R_{k}^{n-1}}}+\beta\left(P(C_{r_{k},R_{k}})-2\omega_{n-1}R_{k}^{n-1}\right).$

Passing to the limit for $k$ to infinity, we have

$\lim_{k}\mathcal{G}_{\varepsilon}(C_{r_{k},R_{k}},h_{k})=\dfrac{\beta
P^{2}}{P+\beta m}=\min_{h\in\mathcal{H}_{m}}\mathcal{F}_{0}(h).$

∎

###  4.2 Maximization with geometric constraints

Fix $\varepsilon,m>0$. For every bounded, open set
$\Omega\subseteq\mathbb{R}^{n}$ with $C^{1,1}$ boundary, we let

$\mathcal{G}_{\varepsilon}(\Omega)=\inf\Set{\mathcal{G}_{\varepsilon}(\Omega,h)}{\,h\in\mathcal{H}_{m}},$

where $\mathcal{H}_{m}$ is the set defined in (1.2). This section will study
the maximization of $\mathcal{G}_{\varepsilon}(\Omega)$ with fixed
quermassintegral. We refer to [13, 6] for the following

######  Definition  (Quermassintegrals).

Let $\Omega\subseteq\mathbb{R}^{n}$ be a nonempty, bounded, convex set. We
define the _quermassintegrals_ as the unique coefficients $W_{j}(\Omega)$ such
that

$\lvert\Omega+tB\rvert=\sum_{j=0}^{n}\binom{n}{j}W_{j}(\Omega)t^{j},$

where $B$ is the unit ball in $\mathbb{R}^{n}$, and

$\Omega+tB=\Set{x+ty}{x\in\Omega,\>y\in B}.$

In particular $W_{0}(\Omega)$ is the measure of $\Omega$ and
$W_{n}(\Omega)=\omega_{n}$, the measure of the unit ball.

######  Theorem 4.3 (Alexandrov-Fenchel Inequality).

Let $0\leq i<j\leq n-1$, and let $\Omega\subseteq\mathbb{R}^{n}$ be a
nonempty, bounded, convex set, then

$\left(\frac{W_{i}(\Omega)}{\omega_{n}}\right)^{\frac{1}{n-i}}\leq\left(\frac{W_{j}(\Omega)}{\omega_{n}}\right)^{\frac{1}{n-j}}.$

Moreover, the inequality holds as equality if and only if $\Omega$ is a ball.

######  Remark .

Let $\Omega$ be a bounded, open, convex set with $C^{2}$ boundary and nonzero
Gaussian curvature, then the quermassintegral are related to the principal
curvatures of the boundary of $\Omega$. Indeed we have that for every
$j=1,\dots,n$

$W_{j}=\dfrac{1}{n}\int_{\partial\Omega}H_{j-1}(\sigma)\,d\mathcal{H}^{n-1}$

Here $H_{j}$ denotes the $j$-th normalized elementary symmetric function of
the principal curvatures of $\partial\Omega$, that is $H_{0}=1$ and, for every
$j=1,\dots n-1$,

$H_{j}(\sigma)=\binom{n-1}{j}^{-1}\sum_{1\leq i_{1}\leq\dots\leq i_{j}\leq
n-1}k_{i_{1}}(\sigma)\cdots k_{i_{j}}(\sigma),$

where $k_{1}(\sigma),\dots,k_{n-1}(\sigma)$ are the principal curvatures at a
point $\sigma\in\partial\Omega$. In particular, we have that

$W_{1}(\Omega)=\dfrac{1}{n}P(\Omega)$

and

$W_{2}(\Omega)=\dfrac{1}{n(n-1)}\int_{\partial\Omega}H_{\Omega}\,d\mathcal{H}^{n-1}.$

In the planar case, we have the following

######  Proposition .

Let $n=2$, and let $\Omega$ be a bounded, open, simply connected set $\Omega$
with $C^{2}$ boundary such that either

$P(\Omega)\geq 3\pi\dfrac{\varepsilon}{\beta}$

or

$P(\Omega)\leq\pi\dfrac{\varepsilon}{\beta}.$

Then

$\mathcal{G}_{\varepsilon}(\Omega)\leq\mathcal{G}_{\varepsilon}(\Omega^{*}),$
(4.17)

where $\Omega^{*}$ is the ball having the same perimeter as $\Omega$.

###### Proof.

If

$P(\Omega)\leq\pi\dfrac{\varepsilon}{\beta},$

then

$\dfrac{\varepsilon H_{\Omega^{*}}}{\beta}\geq 2,$

and, by Section 4, we get

$\mathcal{G}_{\varepsilon}(\Omega^{*})=\mathcal{G}_{\varepsilon}(\Omega^{*},0)=\beta
P(\Omega^{*}).$

On the other hand,

$\mathcal{G}_{\varepsilon}(\Omega)\leq\mathcal{G}_{\varepsilon}(\Omega,0)=\beta
P(\Omega)=\beta P(\Omega^{*}),$

which gives (4.17). If

$P(\Omega)\geq 3\pi\dfrac{\varepsilon}{\beta},$

then

$\dfrac{\varepsilon H_{\Omega^{*}}}{\beta}\leq\dfrac{2}{3},$

and, by Theorem 4.1, we get

$\mathcal{G}_{\varepsilon}(\Omega^{*})=\mathcal{G}_{\varepsilon}(\Omega^{*},m/P(\Omega^{*})).$

On the other hand,

$\begin{split}\mathcal{G}_{\varepsilon}(\Omega)&\leq\mathcal{G}_{\varepsilon}(\Omega,m/P(\Omega^{*}))\\\
&=\beta\left(\dfrac{P(\Omega^{*})P(\Omega)}{P(\Omega^{*})+\beta
m}+\dfrac{\varepsilon m(2P(\Omega^{*})+\beta m)}{2(P(\Omega^{*})+\beta
m)^{2}}\int_{\partial\Omega}H_{\Omega}\,d\mathcal{H}^{n-1}\right).\end{split}$

By Gauss-Bonnet theorem we have that

$\int_{\partial\Omega}H_{\Omega}\,d\mathcal{H}^{n-1}=2\pi=\int_{\partial\Omega^{*}}H_{\Omega^{*}}\,d\mathcal{H}^{n-1},$

so that

$\begin{split}\mathcal{G}_{\varepsilon}(\Omega)&\leq\beta\left(\dfrac{P(\Omega^{*})^{2}}{P(\Omega^{*})+\beta
m}+\dfrac{\varepsilon m(2P(\Omega^{*})+\beta m)}{2(P(\Omega^{*})+\beta
m)^{2}}\int_{\partial\Omega^{*}}H_{\Omega^{*}}\,d\mathcal{H}^{n-1}\right)\\\\[10.0pt]
&=\mathcal{G}_{\varepsilon}(\Omega^{*},m/P(\Omega^{*})),\end{split}$

which is (4.17). ∎

In the general case of possibly higher dimensions, we have

######  Proposition .

Let $n\geq 3$, $2\leq k\leq n-1$, and let $\Omega$ be a bounded, open, convex
set with $C^{2}$ boundary and nonzero Gaussian curvature such that either

$W_{k}(\Omega)\geq\omega_{n}\left(\dfrac{3(n-1)\varepsilon}{2\beta}\right)^{n-k}$
(4.18)

or

$W_{k}(\Omega)\leq\omega_{n}\left(\dfrac{(n-1)\varepsilon}{2\beta}\right)^{n-k}.$
(4.19)

Then

$\mathcal{G}_{\varepsilon}(\Omega)\leq\mathcal{G}_{\varepsilon}(\Omega^{*})$

where $\Omega^{*}$ is the ball such that $W_{k}(\Omega)=W_{k}(\Omega^{*})$.

###### Proof.

Since $W_{k}(\Omega)=W_{k}(\Omega^{*})$, from the Alexandrov-Fenchel
inequalities, we have that for every $0\leq i\leq k$

$\left(\frac{W_{i}(\Omega^{*})}{\omega_{n}}\right)^{\frac{1}{n-i}}=\left(\frac{W_{k}(\Omega^{*})}{\omega_{n}}\right)^{\frac{1}{n-k}}=\left(\frac{W_{k}(\Omega)}{\omega_{n}}\right)^{\frac{1}{n-k}}\geq\left(\frac{W_{i}(\Omega)}{\omega_{n}}\right)^{\frac{1}{n-i}},$

that is

$W_{i}(\Omega)\leq W_{i}(\Omega^{*}).$

In particular, we have that

$P(\Omega)\leq P(\Omega^{*})$

and

$\int_{\partial\Omega}H_{\Omega}\,d\mathcal{H}^{n-1}\leq\int_{\partial\Omega^{*}}H_{\Omega^{*}}\,d\mathcal{H}^{n-1}.$

Moreover, since

$\left(\frac{W_{k}(\Omega^{*})}{\omega_{n}}\right)^{\frac{1}{n-k}}=\left(\frac{W_{1}(\Omega^{*})}{\omega_{n}}\right)^{\frac{1}{n-1}}=\left(\dfrac{P(\Omega^{*})}{n\omega_{n}}\right)^{\frac{1}{n-1}}=\frac{n-1}{H_{\Omega^{*}}},$

the conditions (4.18) and (4.19) read as

$\dfrac{\varepsilon H_{\Omega^{*}}}{\beta}\leq\dfrac{2}{3},$

and

$\dfrac{\varepsilon H_{\Omega^{*}}}{\beta}\geq 2,$

respectively. Therefore, the result can be obtained following the proof of
Subsection 4.2. ∎

###  4.3 Final remarks

Let $\Omega,\Omega_{0}\subseteq\mathbb{R}^{n}$ be bounded open sets with
smooth boundary such that $\overline{\Omega}\subset\overline{\Omega}_{0}$ and
let

$\mathcal{I}_{\beta}(\Omega,\Omega_{0})=\inf\Set{\int_{\Omega_{0}}\lvert\nabla
u\rvert^{2}\,dx+\beta\int_{\partial\Omega_{0}}u^{2}\,d\mathcal{H}^{n-1}}{\begin{aligned}
&\,u\in H^{1}(\Omega_{0}),\\\ &\,u\geq 1\,\text{ in
}\overline{\Omega}\end{aligned}}.$

The results in Subsection 4.2 and Subsection 4.2 are coherent with the ones
proved in [10] for the functional

$\mathcal{I}_{\beta,\delta}(\Omega)=\mathcal{I}_{\beta}(\Omega,\Omega+\delta
B),$

where $\Omega_{0}=\Omega+\delta B$ is the Minkowski sum of $\Omega$ and the
unit ball $B=B_{1}(0)\subset\mathbb{R}^{n}$. Namely, in [10], the following
theorems are proved

######  Theorem 4.4.

Let $\Omega$ be a connected, bounded, open set in $\mathbb{R}^{2}$ with
piecewise $C^{1}$ boundary. Then

$\mathcal{I}_{\beta,\delta}(\Omega)\leq\mathcal{I}_{\beta,\delta}(\Omega^{*}),$

where $\Omega^{*}$ is the ball having the same perimeter as $\Omega$.

######  Theorem 4.5.

Let $n\geq 3$ and let $\Omega$ be a bounded, open, convex set in
$\mathbb{R}^{n}$. Then

$\mathcal{I}_{\beta,\delta}(\Omega)\leq\mathcal{I}_{\beta,\delta}(\Omega^{*}),$

where $\Omega^{*}$ is the ball such that
$W_{n-1}(\Omega^{*})=W_{n-1}(\Omega)$.

Finally, in [5], it is proved the following

######  Theorem 4.6.

The solution to the problem

$\min\Set{I_{\beta}(\Omega,\Omega_{0})}{\begin{aligned}
&\lvert\Omega\rvert=\omega_{n},\\\ &\lvert\Omega_{0}\rvert\leq
M\end{aligned}}$

consists of two concentric balls.

The previous theorem naturally leads to the following question

######  Open problem 1.

Prove or disprove that the problem

$\min\Set{\mathcal{G}_{\varepsilon}(\Omega,h)}{\begin{aligned}
&\lvert\Omega\rvert=\omega_{n},\\\ &h\in\mathcal{H}_{m}\end{aligned}}$

admits the couple $(B_{1},h^{*})$ as a solution, where $h^{*}$ is a constant
function.

## References

  * [1] Emilio Acerbi and Giuseppe Buttazzo  “Reinforcement problems in the calculus of variations”  In _Annales de l’Institut Henri Poincaré C, Analyse non linéaire_ 3.4, 1986, pp. 273–284
  * [2] Gabriele Anzellotti and Sisto Baldo  “Asymptotic development by $\Gamma$-convergence”  In _Applied mathematics and optimization_ 27 Springer, 1993, pp. 105–123
  * [3] Haïm Brézis, Luis A Caffarelli and Avner Friedman  “Reinforcement problems for elliptic equations and variational inequalities”  In _Annali di matematica pura ed applicata_ 123.1 Springer, 1980, pp. 219–246
  * [4] Dorin Bucur, Giuseppe Buttazzo and Carlo Nitsch  “Symmetry breaking for a problem in optimal insulation”  In _Journal de Mathématiques Pures et Appliquées_ 107.4 Elsevier, 2017, pp. 451–463
  * [5] Dorin Bucur, Mickaël Nahon, Carlo Nitsch and Cristina Trombetti  “Shape optimization of a thermal insulation problem”  In _Calculus of Variations and Partial Differential Equations_ 61.5 Springer, 2022, pp. 186
  * [6] Yu.. Burago and V.. Zalgaller  “Geometric inequalities” 285, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]  Springer-Verlag, Berlin, 1988, pp. xiv+331
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  * [8] Gianni Dal Maso  “An introduction to $\Gamma$-convergence” 8, Progress in Nonlinear Differential Equations and their Applications  Birkhäuser Boston, Inc., Boston, MA, 1993, pp. xiv+340
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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>
E-mail address, C. Nitsch<EMAIL_ADDRESS>
E-mail address, C. Trombetti<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>