#  On a Steklov-Robin eigenvalue problem

Nunzia Gavitone, Rossano Sannipoli  Dipartimento di Matematica e Applicazioni
“R. Caccioppoli”, Università degli studi di Napoli Federico II
Via Cintia, Complesso Universitario Monte S. Angelo, 80126 Napoli, Italy.
<EMAIL_ADDRESS>Dipartimento di Matematica e Applicazioni “R.
Caccioppoli”, Università degli studi di Napoli Federico II
Via Cintia, Complesso Universitario Monte S. Angelo, 80126 Napoli, Italy.
<EMAIL_ADDRESS>
###### Abstract.

AbstractAbstract

###### Abstract.

In this paper we study a Steklov-Robin eigenvalue problem for the Laplacian in
annular domains. More precisely, we consider
$\Omega=\Omega_{0}\setminus\overline{B}_{r}$, where $B_{r}$ is the ball
centered at the origin with radius $r>0$ and
$\Omega_{0}\subset\mathbb{R}^{n}$, $n\geq 2$, is an open, bounded set with
Lipschitz boundary, such that $\overline{B}_{r}\subset\Omega_{0}$. We impose a
Steklov condition on the outer boundary and a Robin condition involving a
positive $L^{\infty}$ function $\beta(x)$ on the inner boundary. Then, we
study the first eigenvalue $\sigma_{\beta}(\Omega)$ and its main properties.
In particular, we investigate the behaviour of $\sigma_{\beta}(\Omega)$ when
we let vary the $L^{1}$-norm of $\beta$ and the radius of the inner ball.
Furthermore, we study the asymptotic behaviour of the corresponding
eigenfunctions when $\beta$ is a positive parameter that goes to infinity.

MSC 2020: 35B40, 35J25, 35P15.
Keywords: Laplacian eigenvalue, Steklov-Robin boundary conditions.

AbstractAbstract

##  1\. Introduction

The study of eigenvalue problems in annular domains is, actually, a topic
widely studied. In this kind of problems, usually, the shape of the hole is
spherical and different conditions are imposed on the two components of the
boundary. More precisely this problems are defined in annular domains having
the following form $\Omega=\Omega_{0}\setminus\overline{B_{r}}$. Here
$\Omega_{0}\subset\mathbb{R}^{n}$, $n\geq 2$, is an open, bounded, connected
set with Lipschitz boundary and $B_{r}$ is the ball of radius $r>0$ centered
at the origin such that $\overline{B_{r}}\subset\Omega_{0}$. Recently, the
following Steklov-Dirichlet Laplacian eigenvalue problem has attracted the
attention of many mathematicians ( see for instance [20, 21, 8, 16, 11, 18, 9]
and the references therein)

$\begin{cases}\Delta u=0&\mbox{in}\ \Omega\vspace{0.2cm}\\\ \dfrac{\partial
u}{\partial\nu}=\sigma u&\mbox{on}\ \partial\Omega_{0}\vspace{0.2cm}\\\
u=0&\mbox{on}\ \partial B_{r},\end{cases}$ (1.1)

where $\nu$ is the outer unit normal to $\partial\Omega$. The first eigenvalue
to the problem (1.1) has the following variational characterization

$\sigma_{D}(\Omega)=\inf_{\begin{subarray}{c}v\in H^{1}_{\partial
B_{r}}(\Omega)\\\ v\not\equiv 0\end{subarray}}\frac{\int_{\Omega}\left|\nabla
v\right|^{2}\,dx}{\int_{\partial\Omega_{0}}u^{2}\,d\mathcal{H}^{n-1}},$ (1.2)

where $H^{1}_{\partial B_{r}}(\Omega)$ is the set of those Sobolev functions
vanishing on the boundary of the hole. Since both Steklov and Dirichlet
boundary conditions are imposed, it is interesting to understand what
properties $\sigma_{D}(\Omega)$ inherits from them. In [16, 3, 11], the
authors prove that the infimum is actually a minimum and that it is achieved
by a function $u\in H^{1}_{\partial B_{r}}(\Omega)$ which is a weak solution
to problem (1.1) with constant sign and that $\sigma_{D}(\Omega)$ is simple.
This means that, even if a Steklov condition appears in (1.1),
$\sigma_{D}(\Omega)$ has the same properties of the first Dirichlet
eigenvalue. Nevertheless, in [16, 21, 11, 6, 18, 9] several shape optimization
problems related to $\sigma_{D}(\Omega)$ are studied. In this case it is
possible to observe that the Steklov condition influences the behavior of
$\sigma_{D}(\Omega)$. Indeed, one of the main property of $\sigma_{D}(\Omega)$
for instance is that it is bounded from above when both the measure of
$\Omega_{0}$ and the radius of the hole are fixed (see for instance [16, 8]),
unlike the first Dirichlet Laplacian eigenvalue. Our aim is to study the
eigenvalue problem obtained replacing the Dirichlet boundary condition in
(1.1) with a Robin one. More precisely we deal with the following Steklov-
Robin eigenvalue problem

$\begin{cases}\Delta u=0&\mbox{in}\ \Omega\vspace{0.2cm}\\\ \dfrac{\partial
u}{\partial\nu}=\sigma u&\mbox{on}\ \partial\Omega_{0}\vspace{0.2cm}\\\
\dfrac{\partial u}{\partial\nu}+\beta(x)u=0&\mbox{on}\ \partial
B_{r},\end{cases}$ (1.3)

where $\beta(x)\in L^{\infty}(\partial B_{r})$ is a positive function such
that $\|\beta\|_{L^{1}(\partial B_{r})}=m>0$.

######  Definition 1.1.

A real number $\sigma(\Omega)$ and a function $u\in H^{1}(\Omega)$ are,
respectively, called eigenvalue of (1.3) and associated eigenfunction to
$\sigma(\Omega)$, if and only if

$\int_{\Omega}\langle\nabla u,\nabla\varphi\rangle\;dx+\int_{\partial
B_{r}}\beta(x)u\varphi\,d\,\mathcal{H}^{n-1}=\sigma(\Omega)\int_{\partial\Omega_{0}}u\varphi\;d\mathcal{H}^{n-1}$

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

The aim of this paper is to study the first eigenvalue
$\sigma_{\beta}(\Omega)$ of (1.3) defined as (see Section 3 for the details)

$\sigma_{\beta}(\Omega)=\inf_{v\in
H^{1}(\Omega)\setminus\\{0\\}}\dfrac{\displaystyle\int_{\Omega}|\nabla
v|^{2}\;dx+\int_{\partial
B_{r}}\beta(x)v^{2}\,d\mathcal{H}^{n-1}}{\displaystyle\int_{\partial\Omega_{0}}v^{2}\;d\mathcal{H}^{n-1}}.$
(1.4)

We prove that similarly to problem (1.1), $\sigma_{\beta}(\Omega)$ is actually
a minimum, it is simple, and that the corresponding eigenfunctions have
constant sign. Hence, also in this case, despite the Steklov condition,
$\sigma_{\beta}(\Omega)$ is formally a Robin type eigenvalue. When $\Omega$ is
a spherical shell, that is $\Omega=B_{R}\setminus\overline{B_{r}}$, and
$\beta(x)=\beta$ is a positive constant, $\sigma_{\beta}(\Omega)$ and the
corresponding eigenfunctions can be explicitly computed; in particular, in the
radial case, it is straightforward to prove the asymptotic behaviors of
$\sigma_{\beta}(\Omega)$ and the corresponding eigenfunctions when $\beta\to
0$ and $\beta\to\infty$ (see Section 4 for the details). We observe that
$\sigma_{\beta}(\Omega)$ depends clearly also on $\beta$ and consequently on
its $L^{1}$-norm $m$. When $\Omega$ is a general open, bounded set with
Liptschitz boundary, we expect that the same asymptotic behavior of
$\sigma_{\beta}(\Omega)$ and the corresponding eigenfunctions still holds,
when we let vary the $L^{1}-$norm $m$ of $\beta(x)$. In order to show this, we
will prove some estimates on $\sigma_{\beta}(\Omega)$ in the spirit of the
ones contained in [12] for the first Robin Laplacian eigenvalue (see also [2]
for a more general case). More precisely, let us define the following
quantities

$\mu_{1}(\Omega):=\inf_{\begin{subarray}{c}v\in
H^{1}(\Omega)\setminus\\{0\\}\\\ \int_{\partial
B_{r}}v\,d\mathcal{H}^{n-1}=0\end{subarray}}\displaystyle\frac{\int_{\Omega}|\nabla
v|^{2}\,dx}{\int_{\partial\Omega_{0}}v^{2}\,d\mathcal{H}^{n-1}},$ (1.5)

and if $u$ is the eigenfunction corresponding to $\sigma_{\beta}(\Omega)$

$q_{\beta}(\Omega)=\inf_{\displaystyle\begin{subarray}{c}\Delta w=0,\,w\neq
0\\\ \frac{\partial
w}{\partial\nu}=0\,\textit{on}\,\,\partial\Omega_{0}\end{subarray}}\frac{\displaystyle\int_{\partial
B_{r}}\beta(x)w^{2}\,d\mathcal{H}^{n-1}}{\displaystyle\int_{\partial\Omega_{0}}w^{2}\,d\mathcal{H}^{n-1}},$
(1.6)

We observe that $\mu_{1}(\Omega)$ is the first nontrivial Steklov Laplacian
eigenvalue in $\Omega$. Moreover if $\beta(x)=\beta$ is constant, then

$q_{\beta}(\Omega)=\beta q(\Omega),$ (1.7)

where

$q(\Omega)=\inf_{\displaystyle\begin{subarray}{c}\Delta w=0,\,w\neq 0\\\
\frac{\partial
w}{\partial\nu}=0\,\textit{on}\,\,\partial\Omega_{0}\end{subarray}}\frac{\displaystyle\int_{\partial
B_{r}}w^{2}\,d\mathcal{H}^{n-1}}{\displaystyle\int_{\partial\Omega_{0}}w^{2}\,d\mathcal{H}^{n-1}}.$
(1.8)

Then our result is the following

######  Theorem 1.2.

Let $\Omega_{0}\subset\mathbb{R}^{n}$ be an open, bounded set with Lipschitz
boundary and let $\Omega=\Omega_{0}\setminus\overline{B_{r}}$, where $B_{r}$
is the ball centered at the origin and with radius $r$ such that
$\overline{B_{r}}\subset\Omega_{0}$. Then the following estimates hold

$\frac{1}{\sigma_{\beta}(\Omega)}\leq\frac{1}{\mu_{1}(\Omega)}+\frac{P(\Omega_{0})}{m},\\\
$ (1.9)

and

$\frac{1}{\sigma_{\beta}(\Omega)}\leq\frac{1}{\sigma_{D}(\Omega)}+\frac{1}{q_{\beta}(\Omega)},$
(1.10)

where $\mu_{1}(\Omega)$ and $q_{\beta}(\Omega)$ are defined in (1.5) and
(1.6), and $\sigma_{\beta}(\Omega)$ and $\sigma_{D}(\Omega)$ are the first
Steklov-Robin and Steklov Dirichlet eigenvalue of $\Omega$ defined in (1.4)
and (1.2) respectively.

As a consequence of the above estimates we can obtain the quoted asymptotic
behaviour of $\sigma_{\beta}(\Omega)$ with respect to $m$. We stress the fact
that, when $\beta(x)=\beta$ is a positive constant, Theorem 1.2 is a key point
to prove the convergence of the Steklov-Robin eigenfunctions to the Steklov-
Dirichlet eigenfunction as $\beta$ goes to infinity. Indeed, our second main
result is the following

######  Theorem 1.3.

Let $\Omega_{0}\subset\mathbb{R}^{n}$ be an open, bounded set with Lipschitz
boundary and let $\Omega=\Omega_{0}\setminus\overline{B_{r}}$, where $B_{r}$
is the ball centered at the origin and with radius $r$ such that
$\overline{B_{r}}\subset\Omega_{0}$. Let $\beta(x)=\beta$ be a positive
constant. Let $u_{\beta}$ and $v$ be positive eigenfunctions corresponding to
$\sigma_{\beta}(\Omega)$ and $\sigma_{D}(\Omega)$ respectively. Then

$\lim_{\beta\to+\infty}\|u_{\beta}-v\|_{H^{1}(\Omega)}=0.$ (1.11)

######  Remark 1.4.

The previous result states that the first Steklov-Robin eigenfunctions
converge to the Steklov-Dirichlet ones in $H^{1}(\Omega)$, when $\beta$ goes
to $\infty$. Actually, since they are smooth functions, by Theorem 1.2 and
Morrey’s inequality we can say that the convergence result holds true in
$C^{0}(\Omega).$

The structure of the paper is the following: in Section 2 we recall some
preliminary result and fix the notation we will use in the rest of the paper.
In Section 3 we state and prove the main properties of the first eigenvalue
$\sigma_{\beta}(\Omega)$ and of the corresponding eigenfunctions. In Section 4
we focus on the radial case and finally Section 5 and 6 are devoted for the
computation of some upper and lower bounds for $\sigma_{\beta}(\Omega)$ and
the proof of Theorems 1.2 \- 1.3.

##  2\. Preliminaries and notations

Throughout this paper, we denote by $B_{r}(x_{0})$ and $B_{r}$ the balls in
$\mathbb{R}^{n}$ of radius $r>0$ centered at $x_{0}\in\mathbb{R}^{n}$ and at
the origin, respectively. Moreover with $B$, $\mathbb{S}^{n-1}$ and
$\omega_{n}$ we will denote respectively the unit ball of $\mathbb{R}^{n}$,
its boundary and its volume. Let $r,R$ be such that $0<r<R$, the spherical
shell will be denoted as follows:

$A_{r,R}=\\{x\in\mathbb{R}^{n}\colon r<|x|<R\\}.$

Moreover, the $(n-1)$-dimensional Hausdorff measure in $\mathbb{R}^{n}$ will
be denoted by $\mathcal{H}^{n-1}$ and the Euclidean scalar product in
$\mathbb{R}^{n}$ is denoted by $\langle\cdot,\cdot\rangle$.
Let $D\subseteq\mathbb{R}^{n}$ be an open bounded set and let
$E\subseteq\mathbb{R}^{n}$ be a measurable set. For the sake of completeness,
we recall here the definition of the perimeter of $E$ in $D$ (see for instance
[14]), that is

$P(E;D)=\sup\left\\{\int_{E}{\rm div}\varphi\>dx:\;\varphi\in
C^{\infty}_{c}(D;\mathbb{R}^{n}),\;||\varphi||_{\infty}\leq 1\right\\}.$

The perimeter of $E$ in $\mathbb{R}^{n}$ will be denoted by $P(E)$ and, if
$P(E)<\infty$, we say that $E$ is a set of finite perimeter. Moreover, if $E$
has Lipschitz boundary, it holds

$P(E)=\mathcal{H}^{n-1}(\partial E).$

The Lebesgue measure of a measurable set $E\subset\mathbb{R}^{n}$ will be
denoted by $V(E)$. Moreover, we define the inradius of
$E\subset\mathbb{R}^{n}$ as

$\rho(E)=\sup_{x\in E}\;\inf_{y\in\partial E}|x-y|,$ (2.1)

while the diameter of $E$ is

$\operatorname{diam}(E)=\sup_{x,y\in E}|x-y|.$

###  2.1. Some properties of convex sets

In what follows we recall some properties of the convex bodies, i.e. compact
convex sets without empty interior. We refer to [17] for further properties
and the details. Let $K\subset\mathbb{R}^{n}$ be a bounded convex body. The
support function $h_{K}$ of $K$ is defined as follows

$h_{K}\colon\mathbb{S}^{n-1}\to\mathbb{R},\quad h_{K}(x)=\sup_{y\in K}(x,y).$

If the origin belongs to $K$ then $h_{K}$ is non-negative and
$h_{K}(x)\leq\text{diam}(K)$ for every $x\in\mathbb{S}^{n-1}$. Let
$K\subset\mathbb{R}^{n}$ be a bounded convex body such that the origin is an
interior point of $K$. The radial function of $K$ is defined as follows

$\rho_{K}(x)=\sup\\{\lambda\geq 0\colon\lambda x\in K\\},\quad
x\in\mathbb{S}^{n-1},$ (2.2)

and it is a Lipschitz function. Moreover the boundary of $K$ can be written as

$\partial K=\\{x\,\rho_{K}(x),x\in\mathbb{S}^{n-1}\\}.$ (2.3)

Let us define the minimum and the maximum distance of $\partial K$ from the
origin as follows

$R_{m}=\min_{\mathbb{S}^{n-1}}\rho_{K}(x),\qquad\quad
R_{M}=\max_{\mathbb{S}^{n-1}}\rho_{K}(x).$ (2.4)

Finally, if $f\colon\partial K\to\mathbb{R}$ is $\mathcal{H}^{n-1}-$integrable
the following formula for the change of variable given by the radial map holds

$\int_{\partial
K}f\,d\mathcal{H}^{n-1}=\int_{\mathbb{S}^{n-1}}f(r_{K}(x))(\rho_{K}(x))^{n-1}\sqrt{1+\left(\frac{|\nabla_{\tau}\rho_{K}(x)|}{\rho_{K}(x)}\right)^{2}}\,d\mathcal{H}^{n-1}.$

where $\nabla_{\tau}$ denotes the tangential gradient, which is the projection
of the gradient on the tangent plane.

###  2.2. Trace and Friedrich inequalities

Here we recall some known inequalities that will be useful in the sequel. Let
$\Omega$ be an open bounded subset of $\mathbb{R}^{n}$ with Lipschitz
boundary, then by the classical Sobolev trace inequality (see [4]) we have
that

$\|u\|_{L^{2}(\partial\Omega)}\leq C\|u\|_{H^{1}(\Omega)},$ (2.5)

for some positive constant $C>0$. Moreover the embedding operator of
$H^{1}(\Omega)$ into $L^{2}(\partial\Omega)$ is compact.
Another important embedding theorem is a consequence of the so-called
Friedrich’s inequality (see for instance [5, 15] and for a more general case
[1]). Let $H^{1}(\Omega,\partial\Omega)$ the completion of the set of
functions in $C^{\infty}(\Omega)\cap C(\bar{\Omega})$ which have weak gradient
in $L^{2}(\Omega)$, equipped with the following norm (see [15] for the
details)

$\|u\|_{H^{1}(\Omega,\partial\Omega)}=\|\nabla
u\|_{L^{2}(\Omega)}+\|u\|_{L^{2}(\partial\Omega)}.$

Fridrich’s inequality states that

$\|u\|_{L^{2}(\Omega)}\leq C(\|\nabla
u\|_{L^{2}(\Omega)}+\|u\|_{L^{2}(\partial\Omega)})$ (2.6)

for some positive constant $C>0$. Also in this case the embedding operator of
$H^{1}(\Omega,\partial\Omega)$ into $L^{2}(\Omega)$ is compact (see Corollary
3, p. 392 in [15]).

##  3\. Existence and basic properties of $\sigma_{\beta}(\Omega)$

In this section we define and study the main properties of the first Steklov-
Robin Laplacian eigenvalue in $\Omega=\Omega_{0}\setminus\overline{B}_{r}$.
Let $\sigma_{\beta}(\Omega)$ be the following quantity

$\sigma_{\beta}(\Omega)=\inf_{\begin{subarray}{c}v\in H^{1}(\Omega)\\\
v\not\equiv 0\end{subarray}}J[v],$ (3.1)

where

$J[v]=\dfrac{\displaystyle\int_{\Omega}|\nabla v|^{2}\;dx+\int_{\partial
B_{r}}\beta(x)v^{2}\,d\mathcal{H}^{n-1}}{\displaystyle\int_{\partial\Omega_{0}}v^{2}\;d\mathcal{H}^{n-1}}$
(3.2)

and $\beta(x)\in L^{\infty}(\partial B_{r})$ is a positive function.
We observe that by (3.1) we immediately get

$\sigma_{\beta}(\Omega)\leq\sigma_{D}(\Omega),$ (3.3)

where $\sigma_{D}(\Omega)$ is the first Steklov-Dirichlet eigenvalue defined
in (1.2). In the next result we prove that $\sigma_{\beta}(\Omega)$ is the
first eigenvalue of problem (1.3) and we show some basic properties of
$\sigma_{\beta}(\Omega)$ and its corresponding eigenfunctions.

######  Theorem 3.1.

Let $n\geq 2$ and $\Omega=\Omega_{0}\setminus\overline{B}_{r}$, where
$\Omega_{0}$ is an open, bounded and connected set with Lipschitz boundary in
$\mathbb{R}^{n}$ and $B_{r}$ a ball centered at the origin of radius $r>0$
such that $\overline{B}_{r}\subset\Omega_{0}$. Then $\sigma_{\beta}(\Omega)$
is actually a minimum, that is

$\sigma_{\beta}(\Omega)=\min_{\begin{subarray}{c}v\in H^{1}(\Omega)\\\
v\not\equiv 0\end{subarray}}J[v],$ (3.4)

where $J[v]$ is defined in (3.2). Moreover $\sigma_{\beta}(\Omega)$ is the
first eigenvalue of (1.3), it is strictly positive and any minimizer has
constant sign.

###### Proof.

Let us notice that the Rayleigh quotient $J[w]$ defined in the previous
proposition is always non-negative and $0$-homogeneous. Let us consider a
minimizing normalized sequence $\\{u_{n}\\}_{n\in\mathbb{N}}$ such that
$\|u_{n}\|_{L^{2}(\partial\Omega_{0})}=1$, i.e.
$\lim_{n\to\infty}J[u_{n}]=\sigma_{\beta}(\Omega)$. By (3.3),
$J[u_{n}]\leq\sigma_{D}(\Omega)$, then by Friedrich’s inequality (2.6),
$\|u_{n}\|_{L^{2}(\Omega)}$ is uniformly bounded from above. By the
compactness of the embedding $H^{1}(\Omega,\partial\Omega)\subset
L^{2}(\Omega)$, there exists a subsequence, still denoted by $u_{n}$, and a
function $u\in H^{1}(\Omega)$ with $||u||_{L^{2}(\partial\Omega_{0})}=1$, such
that $u_{n}\to u$ strongly in $L^{2}(\Omega)$, hence also almost everywhere,
and $\nabla u_{n}\rightharpoonup\nabla u$ weakly in $L^{2}(\Omega)$. Moreover,
by the compactness of the trace embedding theorem (2.5),  $u_{n}$ converges
strongly to $u$ also in $L^{2}(\partial\Omega)$ and almost everywhere on
$\partial\Omega$ to $u$. Then, by weak lower semicontinuity we have

$\lim\limits_{n\to+\infty}J[u_{n}]\geq J[u].$

Hence the existence of a minimizer $u\in H^{1}(\Omega)$ follows.
It is obvious the fact that $\sigma_{\beta}(\Omega)\geq 0$. By contradiction
let us suppose that $\sigma_{\beta}(\Omega)=0$. This means that

$\int_{\Omega}|\nabla u|^{2}\,dx+\int_{\partial
B_{r}}\beta(x)u^{2}\,d\mathcal{H}^{n-1}=0.$

It follows that $\|\nabla u\|_{L^{2}(\Omega)}$ and
$\|\sqrt{\beta(x)}u\|_{L^{2}(\partial B_{r})}$ are both zero. From the first
we have that $u$ is constant a.e. in $\Omega$ and then it must be $\beta(x)=0$
a.e. on $\partial B_{r}$, which is an absurd. Therefore
$\sigma_{\beta}(\Omega)>0$.
By classical arguments of Calculus of Variation it is easy to prove that (1.3)
is the Euler-Lagrange equation corresponding to (3.4). Here we write down the
proof for completeness. Let $u\in H^{1}(\Omega)$ be minimum of the Rayleigh
quotient (3.2) and let $\lambda\in\mathbb{R}$ its value, i.e. $J[u]=\lambda$.
Let us now consider the first variation of $J[\cdot]$. If $v\in
H^{1}(\Omega),$ we define the following function

$f(\varepsilon)=J[u+\varepsilon v].$

It is clear that $f(0)=\lambda$ and in particular we have that

$\begin{split}\int_{\partial\Omega_{0}}u^{2}\,d\mathcal{H}^{n-1}\cdot
f^{\prime}(0)&=2\bigg{(}\int_{\Omega}\langle\nabla u,\nabla
v\rangle\,dx+\int_{\partial
B_{r}}\beta(x)uv\,d\mathcal{H}^{n-1}\bigg{)}\int_{\partial\Omega_{0}}u^{2}\,d\mathcal{H}^{n-1}\\\
&-\bigg{(}\int_{\Omega}|\nabla u|^{2}\,dx+\int_{\partial
B_{r}}\beta(x)u^{2}\,d\mathcal{H}^{n-1}\bigg{)}\int_{\partial\Omega_{0}}uv\,d\mathcal{H}^{n-1}=0\end{split}$

if and only if

$\dfrac{\int_{\Omega}\langle\nabla u,\nabla v\rangle\,dx+\int_{\partial
B_{r}}\beta(x)uv\,d\mathcal{H}^{n-1}}{\int_{\partial\Omega_{0}}uv\,d\mathcal{H}^{n-1}}=\dfrac{\int_{\Omega}|\nabla
u|^{2}\,dx+\int_{\partial
B_{r}}\beta(x)u^{2}\,d\mathcal{H}^{n-1}}{\int_{\partial\Omega_{0}}u^{2}\,d\mathcal{H}^{n-1}}=\lambda.$

Since the relation written above is valid for every $v\in H^{1}(\Omega)$, the
proposition is proved by definition of weak solution.
In particular it follows that $\sigma_{\beta}(\Omega)$ is the smallest
eigenvalue of problem (1.3). Indeed let us suppose that $v$ is another
eigenfunction of (1.3) with corresponding eigenvalue $\tilde{\sigma}$. Then an
integration by parts gives

$\sigma_{\beta}(\Omega)\leq\dfrac{\int_{\Omega}|\nabla
v|^{2}\,dx+\int_{\partial
B_{r}}\beta(x)v^{2}\,d\mathcal{H}^{n-1}}{\int_{\partial\Omega_{0}}v^{2}\,d\mathcal{H}^{n-1}}=\dfrac{\int_{\partial\Omega}\frac{\partial
u}{\partial\nu}u\,d\mathcal{H}^{n-1}+\int_{\partial
B_{r}}\beta(x)u^{2}\,d\mathcal{H}^{n-1}}{\int_{\partial\Omega_{0}}u^{2}\,d\mathcal{H}^{n-1}}=\tilde{\sigma}.$

It only remains to show that any minimizer has constant sign. If $u$ be an
eigenfunction corresponding to $\sigma_{\beta}(\Omega)$, then $J[u]=J[|u|]$.
This implies that $u=|u|$ on $\Omega$ and therefore $u\geq 0$ on $\Omega$. By
Harnack inequality (see [19, Thm 1.1]), $u$ is strictly positive on $\Omega$.
∎

Next propositions concern the simplicity of $\sigma_{\beta}(\Omega)$ and sign
properties of the corresponding eigenfunctions.

######  Proposition 3.2.

$\sigma_{\beta}(\Omega)$ is simple, which means that there exists a unique
corresponding eigenfunction up to multiplicative constants.

###### Proof.

Let us now suppose that $v$ is another eigenfunction corresponding to
$\sigma_{\beta}(\Omega)$. Since $v>0$ in $\Omega$, it follows

$\int_{\Omega}v\,dx\neq 0.$

Hence there exists a positive number $\xi>0$, such that

$\int_{\Omega}(u-\xi v)\,dx=0.$

Since $u-\xi v$ is another eigenfunction corresponding to the same eigenvalue,
it is necessary that $u=\xi v$ in $\Omega$, which proves the simplicity. ∎

######  Proposition 3.3.

Let $n\geq 2$ and $\Omega=\Omega_{0}\setminus\overline{B_{r}}$, where
$\Omega_{0}$ is an open, bounded and connected set with Lipschitz boundary in
$\mathbb{R}^{n}$ and $B_{r}$ a ball centered at the origin of radius $r>0$
such that $\overline{B_{r}}\subset\Omega_{0}$. Any nonnegative function $v\in
H^{1}(\Omega)$ that satisfies in the sense of definition (1.1)

$\begin{cases}\Delta u=0&\mbox{in}\ \Omega\vspace{0.2cm}\\\ \dfrac{\partial
u}{\partial\nu}=\sigma u&\mbox{on}\ \partial\Omega_{0}\vspace{0.2cm}\\\
\dfrac{\partial u}{\partial\nu}+\beta(x)u=0&\mbox{on}\ \partial
B_{R},\end{cases}$ (3.5)

is a first eigenfunction of (3.5), that is $\sigma=\sigma_{\beta}(\Omega)$,
and $v=u$ (up to multiplicative constants), where $u$ is the eigenfunction
corresponding to the first eigenvalue $\sigma_{\beta}(\Omega)$.

###### Proof.

Since $u$ is a positive eigenfunction corresponding to
$\sigma_{\beta}(\Omega)$, it satisfies

$\int_{\Omega}|\nabla u|^{2}\;dx+\int_{\partial
B_{r}}\beta(x)u^{2}\,d\,\mathcal{H}^{n-1}=\sigma_{\beta}(\Omega)\int_{\partial\Omega_{0}}u^{2}\;d\mathcal{H}^{n-1}.$
(3.6)

Let us use the following function $z=u^{2}/(v+\varepsilon)$, for $\epsilon>0$,
as a test function in the definition 1.1 of weak solution for $v$. We get

$\int_{\Omega}\bigg{[}\frac{2u\langle\nabla u,\nabla
v\rangle}{v+\varepsilon}-\frac{u^{2}|\nabla
v|^{2}}{(v+\varepsilon)^{2}}\bigg{]}\,dx+\int_{\partial
B_{r}}\frac{v}{v+\varepsilon}\beta(x)u^{2}\,d\,\mathcal{H}^{n-1}=\sigma\int_{\partial\Omega_{0}}\frac{v}{v+\varepsilon}u^{2}\;d\mathcal{H}^{n-1}.$
(3.7)

If we subtract (3.6) by (3.7), since $v/(v+\varepsilon)<1$, we get

$\begin{split}0\leq\int_{\Omega}\left|\nabla u-\frac{u\nabla
v}{v+\varepsilon}\right|^{2}\,dx=\int_{\Omega}\bigg{[}|\nabla
u|^{2}-&\frac{2u\langle\nabla u,\nabla
v\rangle}{v+\varepsilon}+\frac{u^{2}|\nabla
v|^{2}}{(v+\varepsilon)^{2}}\bigg{]}\,dx\\\
&\leq\int_{\partial\Omega_{0}}\bigg{[}\sigma_{\beta}(\Omega)-\sigma\frac{v}{v+\varepsilon}\bigg{]}u^{2}\;d\mathcal{H}^{n-1}.\end{split}$

Passing to the limit as $\varepsilon\to 0$, we get

$[\sigma_{\beta}(\Omega)-\sigma]\int_{\partial\Omega_{0}}u^{2}\,d\mathcal{H}^{n-1}\geq
0.$

Since $\sigma_{\beta}(\Omega)$ is the smallest eigenvalue, the only
possibility is that $\sigma=\sigma_{\beta}(\Omega)$ and by the simplicity of
the first eigenvalue, it must be $v=u$ up to multiplicative constants. ∎

##  4\. The first Steklov-Robin eigenvalue in the spherical shell

Let us consider now $A_{r,R}=B_{R}\setminus\overline{B_{r}}$, where $B_{R}$
and $B_{r}$ are balls centered at the origin with radii $R>r>0$. Let
$\beta(x)=\beta>0$ be positive and constant on $\partial B_{r}$ and let us
consider the Steklov-Robin eigenvalue problem for the Laplacian in the
spherical shell

$\begin{cases}\Delta u=0&\text{in}\;A_{r,R}\\\ \frac{\partial
u}{\partial\nu}+\beta u=0&\text{on}\;\partial B_{r}\\\ \frac{\partial
u}{\partial\nu}=\sigma u&\text{on}\;\partial B_{R},\end{cases}$ (4.1)

where $\nu$ is the outer unit normal to $\partial A_{r,R}$.
We are going to compute the solutions to problem (4.1).

######  Theorem 4.1.

The first Steklov-Robin eigenvalue of the problem (4.1) is

$\sigma_{\beta}(A_{r,R})\begin{cases}\frac{\displaystyle
1}{\displaystyle\frac{R}{\beta r}+R\log\frac{R}{r}}&n=2\\\ \frac{\displaystyle
n-2}{\displaystyle\frac{\displaystyle
n-2}{\beta}\bigg{(}\frac{R}{r}\bigg{)}^{n-2}+R\bigg{[}\bigg{(}\frac{R}{r}\bigg{)}^{n-2}-1\bigg{]}}&n\geq
3,\end{cases}$ (4.2)

and the corresponding eigenfunctions are the following

$u(x)=\begin{cases}\displaystyle\log\frac{\left|x\right|}{r}+\frac{1}{\beta
r}&n=2\\\
\displaystyle\frac{1}{r^{n-2}}-\frac{1}{\left|x\right|^{n-2}}+\frac{n-2}{\beta}\frac{1}{Rr^{n-2}}&n\geq
3.\end{cases}$ (4.3)

###### Proof.

Since the radial symmetry of the problem and the rotational invariance of the
Laplacian, we look forward to a solution which is of the type
$u(x)=v(\left|x\right|)=v(s)$, where $s=\left|x\right|$. Computing the
Laplacian of $v$ we get

$v^{\prime\prime}+\frac{n-1}{s}v^{\prime}=0,$

which is equivalent to

$(s^{n-1}v^{\prime})^{\prime}=0.$

So integrating twice we get

$v(s)=\begin{cases}c_{1}\log s+c_{2}&n=2\\\
\displaystyle\frac{c_{1}}{s^{n-2}}+c_{2}&n\geq 3.\end{cases}$ (4.4)

We are going to find the solution to (4.1) by using the boundary conditions on
$B_{r}$ and $B_{R}$. Let us begin by the bidimensional case. Using the
boundary condition we get the following system in the unknown variables
$c_{1}$ and $c_{2}$

$\begin{cases}-\frac{c_{1}}{r}+\beta(c_{1}\log r+c_{2})=0\\\
\frac{c_{1}}{R}-\sigma(c_{1}\log R+c_{2})=0.\end{cases}$ (4.5)

Since this is a homogeneous system, the only way not to have $c_{1}=c_{2}=0$
is that

$\det\begin{pmatrix}-1/r+\beta\log r&\beta\\\ 1/R-\sigma\log
R&-\sigma\end{pmatrix}=0.$

From this we get that

$\sigma_{\beta}(A_{r,R})=\frac{1}{\frac{R}{\beta r}+R\log\frac{R}{r}}.$

Since this choice of $\sigma$, $c_{1}$ and $c_{2}$ must be linearly
dependents. Hence if we chose $c_{1}=1$, by using the second equation in (4.5)
we have that

$c_{2}=\frac{1}{\sigma R}-\log R=\frac{1}{\beta r}+\log\frac{R}{r}-\log
R=\frac{1}{\beta r}-\log r.$

Hence inserting $c_{1}$ and $c_{2}$ in (4.4), we have

$u(x)=\log\frac{\left|x\right|}{r}+\frac{1}{\beta r}.$

In higher dimensions the system becomes

$\begin{cases}\frac{n-2}{r^{n-1}}c_{1}+\beta(\frac{c_{1}}{r^{n-2}}+c_{2})=0\\\
-\frac{n-2}{R^{n-1}}c_{1}-\sigma(\frac{c_{1}}{R^{n-2}}+c_{2})=0.\end{cases}$
(4.6)

Proceding in the same way as before, we find that

$\sigma=\sigma_{\beta}(A_{r,R})=\frac{n-2}{\displaystyle\frac{n-2}{\beta}\bigg{(}\frac{R}{r}\bigg{)}^{n-2}+R\bigg{[}\bigg{(}\frac{R}{r}\bigg{)}^{n-2}-1\bigg{]}}.$

Hence chosing $c_{1}=-1$

$c_{2}=\frac{1}{R^{n-2}}+\frac{n-2}{\sigma
R^{n-1}}=\frac{1}{r^{n-2}}+\frac{n-2}{\beta}\frac{1}{Rr^{n-2}},$

and

$u(x)=\frac{1}{r^{n-2}}-\frac{1}{\left|x\right|^{n-2}}+\frac{n-2}{\beta}\frac{1}{Rr^{n-2}}.$

Eventually, in any dimension, with these choices of the constants
$c_{1},c_{2}$, the corresponding eigenfunctions do not change sign and so they
must be eigenfunctions corresponding to the first Steklov-Robin eigenvalue
$\sigma_{\beta}(A_{r,R})$. ∎

By the explicit form of $\sigma_{\beta}(A_{r,R})$ and the corresponding
eigenfunctions in (4.2)-(4.3), we deduce the following properties when we let
vary $\beta$ or the radii of the spherical shell.

  * •

$\lim_{\beta\to 0}\sigma_{\beta}(A_{r,R})=0$, and in particular

$\lim_{\beta\to
0}\frac{\sigma_{\beta}(A_{r,R})}{\beta}=\frac{P(B_{r})}{P(B_{R})}.$ (4.7)

  * •

Recalling the explicit value of the first Steklov-Dirichlet eigenvalue of
spherical shells (see [21, 8, 16]), we have

$\lim_{\beta\to\infty}\sigma_{\beta}(A_{r,R})=\sigma_{D}(A_{r,R}).$ (4.8)

  * •

Finally we have

$\lim_{r\to 0}\sigma_{\beta}(A_{r,R})=\lim_{R\to 0}\sigma_{\beta}(A_{r,R})=0,$
(4.9)

We will see in the next section that all of these behaviours will persist in
the case of a generic $\Omega$.

##  5\. Asymptotic estimates of $\sigma_{\beta}(\Omega)$ with respect to
$\beta$ and $r$

In this section we will study the behaviour of $\sigma_{\beta}(\Omega)$ when
$m$ (or $\beta$ constant) and $r$ vary.

###  5.1. Behaviour with respect to the inner radius

We will prove that (4.9) continues to hold for a general annular domain
$\Omega$ by proving some suitable estimates in terms of the radius of the
hole. Indeed let us consider the spherical shell $A_{r,R_{m}}$, where $R_{m}$
is defined in (2.4), which is contained in $\Omega$. If we choose as a test
function in the variational characterization of $\sigma_{\beta}(\Omega)$

$\varphi=\begin{cases}\displaystyle v(|x|)&\text{in}\,A_{r,R_{m}}\\\
\displaystyle
v(R_{m})&\text{in}\,\Omega\setminus\overline{A_{r,R_{m}}},\end{cases}$

where $v$ is the first eigenfunction in $A_{r,R_{m}}$ with constant Robin
parameter $\overline{\beta}=1/m$, then

$\sigma_{\beta}(\Omega)\leq\sigma_{1}(A_{r,R_{m}}).$

As a consequence when $r\to 0$,

$\sigma_{\beta}(\Omega)\to 0.$

A natural question, now, is asking if we have a lower bound in terms of the
first Steklov-Robin eigenvalue of an opportune spherical shell. In the
following result we prove that for starshaped set is possible to have an
optimal lower bound $\sigma_{\beta}(\Omega)$.

######  Theorem 5.1.

Let $r>0$ and $\Omega_{0}\subset\mathbb{R}^{n}$ be an open, bounded starshaped
set such that $\overline{B_{r}}\subset\Omega_{0}$ and let
$\Omega=\Omega_{0}\setminus\overline{B_{r}}$. Then, if
$\tilde{\beta}=\inf_{x\in\partial B_{r}}\beta(x)$, it holds

$\sigma_{\beta}(\Omega)\geq\displaystyle\frac{\displaystyle\sigma_{\tilde{\beta}}\left(A_{r,R_{m}}\right)}{R_{M}^{n-1}\max_{\mathbb{S}^{n-1}}\left(\sqrt{1+\frac{|\nabla_{\tau}\rho_{0}|^{2}}{\rho_{0}^{2}}}\right)},$
(5.1)

where $R_{m}$ and $R_{M}$ are defined in (2.4), $\rho_{0}$ is the radial
function of $\Omega_{0}$ defined in (2.2) and $A_{r,R_{m}}$ is the spherical
shell with radii $r$ and $R_{m}$.
Moreover, the equality case holds if and only if $\Omega_{0}$ is also a ball
$B_{R}$ centered at the origin of radius $R>0$.

###### Proof.

We will follow an idea used in [13] for the planar case and in [7, 20] for any
dimension. Let $u\in H^{1}(\Omega)$ By using spherical coordinates and the
notation introduced in Section $2$:

$\partial\Omega_{0}=\\{x\,\rho_{0}(x),x\in\mathbb{S}^{n-1}\\}$

and

$\partial B_{r}=\\{xr,\,x\in\mathbb{S}^{n-1}\\}.$

The integrals over the boundaries $\partial\Omega_{0}$ and $\partial B_{R}$ of
$u^{2}$ become

$\displaystyle\int_{\partial\Omega_{0}}u^{2}\;d\mathcal{H}^{n-1}=\displaystyle\int_{\mathbb{S}^{n-1}}u^{2}\,\,\sqrt{1+\left(\frac{|\nabla_{\tau}\rho_{0}|}{\rho_{0}}\right)^{2}}\,(\rho_{0})^{n-1}\,d\mathcal{H}^{n-1},$
$\int_{\partial
B_{r}}\beta(x)u^{2}\,d\mathcal{H}^{n-1}\geq\tilde{\beta}r^{n-1}\int_{\mathbb{S}^{n-1}}u^{2}\,\mathcal{H}^{n-1}.$
(5.2)

In particular we have

$\displaystyle\int_{\partial\Omega_{0}}u^{2}\;d\mathcal{H}^{n-1}\leq(R_{M})^{n-1}\max_{\mathbb{S}^{n-1}}\left(\sqrt{1+\frac{|\nabla_{\tau}\rho_{0}|^{2}}{\rho_{0}^{2}}}\right)\int_{\mathbb{S}^{n-1}}u^{2}\,\,d\mathcal{H}^{n-1}.$
(5.3)

We can parametrize

$\Omega=\\{s\in\mathbb{R}^{n}\colon
s=x\,\tilde{r},\,x\in\mathbb{S}^{n-1},\,\tilde{r}\leq\tilde{r}\leq\rho_{0}(x)\\},$

by using spherical coordinates, where we denote by $R(y)=\rho_{0}(x(y))$, and
$x\colon y\in U\subset\mathbb{R}^{n-1}\to x(y)\in\mathbb{S}^{n-1}$ is a
standard parametrization of the boundary of the unit ball in $\mathbb{R}^{n}$.
Then we get

$\displaystyle\int_{\Omega}|\nabla
u|^{2}\;ds=\displaystyle\int_{U}\int_{r}^{R(y)}\left\\{\left(\frac{\partial
u}{\partial\tilde{r}}\right)^{2}+\frac{1}{\tilde{r}^{2}}|\nabla_{\tau}u|^{2}\right\\}\tilde{r}^{n-1}\sqrt{\tilde{g}}\,d\tilde{r}\,dy,\\\
$ (5.4)

where $\sqrt{\tilde{g}}$ is the determinant of the matrix $\tilde{g}_{ij}$,
that is the standard metric on $\mathbb{S}^{n-1}$ and $\nabla_{\tau}u$ is the
component of $\nabla u$ tangential to $\mathbb{S}^{n-1}$. Therefore

$\displaystyle\displaystyle\int_{\Omega}|\nabla
u|^{2}\;ds\geq\int_{U}\int_{r}^{R_{m}}\left\\{\left(\frac{\partial
u}{\partial\tilde{r}}\right)^{2}+\frac{1}{\tilde{r}^{2}}|\nabla_{\tau}u|^{2}\right\\}\tilde{r}^{n-1}\sqrt{\tilde{g}}\,d\tilde{r}\,dy,$
(5.5)

Combining (5.3), (5.2) and (5.5) and recalling (3.4), we get

$\sigma_{\beta}(\Omega)\geq\displaystyle\frac{\displaystyle\int_{U}\int_{r}^{R_{m}}\left\\{\left(\frac{\partial
u}{\partial\tilde{r}}\right)^{2}+\frac{1}{\tilde{r}^{2}}|\nabla_{\tau}u|^{2}\right\\}\tilde{r}^{n-1}\sqrt{\tilde{g}}\,d\tilde{r}\,dy+\tilde{\beta}r^{n-1}\int_{\mathbb{S}^{n-1}}u^{2}\,d\mathcal{H}^{n-1}}{\displaystyle(R_{M})^{n-1}\max_{\mathbb{S}^{n-1}}\left(\sqrt{1+\frac{|\nabla_{\tau}\rho_{0}|^{2}}{\rho_{0}^{2}}}\right)\displaystyle\int_{\mathbb{S}^{n-1}}u^{2}\,\,d\mathcal{H}^{n-1}}\geq\\\
\frac{\sigma_{\tilde{\beta}}(A_{r,R_{m}})}{\displaystyle(R_{M})^{n-1}\max_{\mathbb{S}^{n-1}}\left(\sqrt{1+\frac{|\nabla_{\tau}\rho_{0}|^{2}}{\rho_{0}^{2}}}\right)}.$
(5.6)

Finally, we stress that the equality case implies that all the inequalities
become equalities. So, we have that $\nabla_{\tau}\rho_{0}=0$ and
$\rho_{0}(x)=\overline{R}$, with $\overline{R}>R$ constant. ∎

Theorem 5.1 tells us that as long $\Omega_{0}$ is an open, bounded starshaped
set, and $r>0$, then $\sigma_{\beta}(\Omega)$ remains away from zero. Is this
property still true for any open, bounded set in $\mathbb{R}^{n}$ with
Lipschitz boundary? In general the answer is no, as showed in the following
bidimensional counterexample, that is contained in [10], that can be easily
generalized in any dimension.

######  Counterexample 5.2.

Let $\beta(x)=\beta$ be a positive constant and let us consider a sequence of
open, bounded and connected sets
$\\{\Omega_{\varepsilon}\\}\subset\mathbb{R}^{2}$ as follows

$\Omega_{\varepsilon}=B(x_{0})\cup
R_{\varepsilon}\cup(B(x_{1})\setminus\overline{B_{r}(x_{1})}).$

Here

$R_{\varepsilon}=\bigg{(}-\frac{\varepsilon}{2},\frac{\varepsilon}{2}\bigg{)}\times\bigg{(}-\frac{\varepsilon^{3}}{2},\frac{\varepsilon^{3}}{2}\bigg{)}$

is a rectangle centered at the origin with sides of length $\varepsilon$ and
$\varepsilon^{3}$ respectively, $B(x_{1})$, $B(x_{2})$ are two dimensional
balls of radius $1$ centered at the points $x_{1}$ and $x_{2}$, chosen such
that the rectangle $R_{\varepsilon}$ is well glued and eventually
$B_{r}(x_{2})$ is a concentric ball in $B(x_{2})$ of radius $0<r<1$.
Let us consider the following function

$u(x,y)=\begin{cases}\sin\bigg{(}\displaystyle\frac{2\pi
x}{\varepsilon}\bigg{)}&\text{in}\,R_{\varepsilon}\\\
0&\text{elsewhere},\end{cases}$

which is a continuous test function for the first Steklov-Robin eigenvalue.
Let us evaluate $u$ in the numerator and denominator the Rayleigh quotient.
The denominator becomes

$\begin{split}\int_{\partial\Omega_{\varepsilon}}u^{2}\,d\mathcal{H}^{n-1}&=\int_{\partial
R_{\varepsilon}}u^{2}\,d\mathcal{H}^{n-1}=2\int_{-\frac{\varepsilon}{2}}^{\frac{\varepsilon}{2}}\sin^{2}\bigg{(}\displaystyle\frac{2\pi
x}{\varepsilon}\bigg{)}\,dx\\\
&=4\int_{0}^{\frac{\varepsilon}{2}}\sin^{2}\bigg{(}\displaystyle\frac{4\pi
x}{\varepsilon}\bigg{)}\,dx=4\int_{0}^{\frac{\varepsilon}{2}}\frac{1-\cos\bigg{(}\displaystyle\frac{4\pi
x}{\varepsilon}\bigg{)}}{2}\,dx=\varepsilon\end{split}$

Since

$|\nabla u|^{2}=\bigg{(}\frac{\partial u}{\partial
x}\bigg{)}^{2}=\bigg{(}\frac{2\pi}{\varepsilon}\bigg{)}^{2}\cos^{2}\bigg{(}\displaystyle\frac{2\pi
x}{\varepsilon}\bigg{)},$

we have that the numerator is

$\begin{split}\int_{\Omega_{\varepsilon}}|\nabla
u|^{2}\,dx\,dy&=\int_{R_{\varepsilon}}|\nabla
u|^{2}\,dx\,dy=\bigg{(}\frac{2\pi}{\varepsilon}\bigg{)}^{2}\int_{-\frac{\varepsilon^{3}}{2}}^{\frac{\varepsilon^{3}}{2}}\int_{-\frac{\varepsilon}{2}}^{\frac{\varepsilon}{2}}\cos^{2}\bigg{(}\displaystyle\frac{4\pi
x}{\varepsilon}\bigg{)}\,dx\,dy\\\
&=2\bigg{(}\frac{2\pi}{\varepsilon}\bigg{)}^{2}\varepsilon^{3}\int_{0}^{\frac{\varepsilon}{2}}\frac{1+\cos\bigg{(}\displaystyle\frac{4\pi
x}{\varepsilon}\bigg{)}}{2}\,dx=2\pi^{2}\varepsilon^{2}.\end{split}$

In this way, since $u$ is zero on $\partial B_{r}(x_{2})$, we get

$\sigma_{\beta}(\Omega_{\varepsilon})\leq\frac{\displaystyle\int_{\Omega_{\varepsilon}}|\nabla
u|^{2}\,dx}{\displaystyle\int_{\partial\Omega_{\varepsilon}}u^{2}\,d\mathcal{H}^{n-1}}=2\pi^{2}\varepsilon,$

and

$\sigma_{\beta}(\Omega_{\varepsilon})\to 0$

as $\varepsilon\to 0$. We stress the fact that the same proof can be exhibited
even in the Steklov-Dirichlet case.

This counterexample gives us two information. The first is the one we already
mentioned: if $\Omega_{0}$ is not starshaped, the first eigenvalue could be
arbitrarily close to zero. The second one is that when $\Omega$ is not
connected, then $\sigma_{\beta}(\Omega)$ could be zero, even though $r>0$.

###  5.2. Behaviour with respect to $m$ and $\beta$

In this section we will give the proof of the Theorem 1.2.

###### Proof of Theorem 1.2.

Firstly we prove inequality (1.9). We observe that the claim is well posed
since, by proceeding analogously as in the existence theorem,
$\mu_{1}(\Omega)$ is positive. For any $w\in H^{1}(\Omega)$, for simplicity,
we will use the following notation

$D(w):=\displaystyle\int_{\Omega}|\nabla w|^{2}\,dx.$ (5.7)

Let $u$ be a positive eigenfunction corresponding to $\sigma_{\beta}(\Omega)$.
By the Minkowski inequality and the definition of $\mu_{1}(\Omega)$ we have

$\sqrt{\int_{\partial\Omega_{0}}u^{2}\,d\mathcal{H}^{n-1}}\leq\sqrt{\int_{\partial\Omega_{0}}(u-c)^{2}\,d\mathcal{H}^{n-1}}+\sqrt{c^{2}P(\Omega_{0})}\leq\sqrt{\frac{D(u)}{\mu_{1}(\Omega)}}+\sqrt{c^{2}P(\Omega_{0})},$

where $c$ is

$c=\dfrac{1}{m}\int_{\partial B_{r}}\beta(x)u\,d\mathcal{H}^{n-1},$ (5.8)

Squaring and using the arithmetic-geometric mean inequality, we have

$\begin{split}\int_{\partial\Omega_{0}}u^{2}d\mathcal{H}^{n-1}&\leq\frac{D(u)}{\mu_{1}(\Omega)}+c^{2}P(\Omega_{0})+2\sqrt{\frac{D(u)c^{2}P(\Omega_{0})}{\mu_{1}(\Omega)}}\\\
&=D(u)\left(\frac{1}{\mu_{1}(\Omega)}+\frac{P(\Omega_{0})}{m}\right)+c^{2}m\left(\frac{1}{\mu_{1}(\Omega)}+\frac{P(\Omega_{0})}{m}\right)\\\
&=\left(\frac{1}{\mu_{1}(\Omega)}+\frac{P(\Omega_{0})}{m}\right)(D(u)+c^{2}m).\end{split}$
(5.9)

Applying Hölder inequality in (5.8) we have

$c^{2}m=\frac{1}{m}\bigg{(}\int_{\partial
B_{r}}\beta(x)u\,d\mathcal{H}^{n-1}\bigg{)}^{2}\leq\int_{\partial
B_{r}}\beta(x)u^{2}\,d\mathcal{H}^{n-1},$

so that

$\begin{split}\int_{\partial\Omega_{0}}u^{2}d\mathcal{H}^{n-1}&\leq\left(\frac{1}{\mu_{1}(\Omega)}+\frac{P(\Omega_{0})}{m}\right)\left(D(u)+\int_{\partial
B_{r}}\beta(x)u^{2}\,d\mathcal{H}^{n-1}\right)\\\
&=\left(\frac{1}{\mu_{1}(\Omega)}+\frac{P(\Omega_{0})}{m}\right)\left(\sigma_{\beta}(\Omega)\int_{\partial\Omega_{0}}u^{2}d\mathcal{H}^{n-1}\right),\end{split}$
(5.10)

which gives (1.5).
Now we prove inequality (1.10). Let us stress the fact that the quantity
$q(\Omega)$ defined in (1.8) is strictly positive. Indeed if $q(\Omega)=0$,
then it would follow that $w=0$ a.e. on $\partial B_{r}$ and since $\partial
w\backslash\partial\nu=0$ on $\partial\Omega_{0}$, then it would imply
$w\equiv 0$ in $\Omega$, which is a contradiction. Let $u$ be the
eigenfunction corresponding to $\sigma_{\beta}(\Omega)$, solution to problem
(1.3). Let us observe that $u=v+h$, where $v$ and $h$ solve the following
problems

$\begin{cases}\Delta v=0&\text{in}\,\Omega\\\ v=0&\text{on}\,\partial B_{r}\\\
\frac{\partial v}{\partial\nu}=\frac{\partial
u}{\partial\nu}&\text{on}\,\partial\Omega_{0},\end{cases}\qquad\qquad\begin{cases}\Delta
h=0&\text{in}\,\Omega\\\ h=u&\text{on}\,\partial B_{r}\\\ \frac{\partial
h}{\partial\nu}=0&\text{on}\,\partial\Omega_{0}.\end{cases}$

It is easy to check that

$\int_{\Omega}\left|\nabla u\right|^{2}\,dx=\int_{\Omega}\left|\nabla
v\right|^{2}\,dx+\int_{\Omega}\left|\nabla h\right|^{2}\,dx.$ (5.11)

Then, proceeding as the proof of inequality (1.5), by applying Minkowski
inequality and using (5.11), (3.7) and (1.6), we get

$\begin{split}\sqrt{\displaystyle\int_{\partial\Omega_{0}}u^{2}\,\mathcal{H}^{n-1}}&\leq\sqrt{\displaystyle\int_{\partial\Omega_{0}}v^{2}\,\mathcal{H}^{n-1}}+\sqrt{\displaystyle\int_{\partial\Omega_{0}}h^{2}\,\mathcal{H}^{n-1}}\\\
&\leq\sqrt{\frac{1}{\sigma_{D}(\Omega)}\displaystyle
D(u)}+\sqrt{\frac{1}{q_{\beta}(\Omega)}\displaystyle\int_{\partial
B_{r}}\beta(x)u^{2}\,\mathcal{H}^{n-1}}.\end{split}$

Squaring both sides we have

$\begin{split}\int_{\partial\Omega_{0}}u^{2}\,d\mathcal{H}^{n-1}\leq\frac{D(u)}{\sigma_{D}(\Omega)}&+\frac{1}{q_{\beta}(\Omega)}\int_{\partial
B_{r}}\beta(x)u^{2}\,\mathcal{H}^{n-1}\\\
&+2\sqrt{\frac{D(u)}{\sigma_{D}(\Omega)}\frac{1}{q_{\beta}(\Omega)}\int_{\partial
B_{r}}\beta(x)u^{2}\,\mathcal{H}^{n-1}}\end{split}$

Applying the arithmetic-geometric mean inequality and Hölder inequality we get

$\begin{split}\int_{\partial\Omega_{0}}u^{2}\,d\mathcal{H}^{n-1}&\leq\frac{D(u)}{\sigma_{D}(\Omega)}+\frac{D(u)}{q_{\beta}(\Omega)}\\\
&+\frac{1}{q_{\beta}(\Omega)}\int_{\partial
B_{r}}\beta(x)u^{2}\,\mathcal{H}^{n-1}+\frac{1}{\sigma_{D}(\Omega)}\int_{\partial
B_{r}}\beta(x)u^{2}\,\mathcal{H}^{n-1}\\\
&=\bigg{(}\frac{1}{\sigma_{D}(\Omega)}+\frac{1}{q_{\beta}(\Omega)}\bigg{)}\bigg{(}D(u)+\int_{\partial
B_{r}}\beta(x)u^{2}\,\mathcal{H}^{n-1}\bigg{)}\end{split}$

This gives (1.10). ∎

######  Remark 5.3.

We stress that a rough but meaningful estimate can be obtained choosing as a
test function in (3.4) the constant function. In this case we have the
following upper bound

$\sigma_{\beta}(\Omega)\leq\frac{m}{P(\Omega_{0})},$ (5.12)

which immediately gives that when $m\to 0$, then $\sigma_{\beta}(\Omega)\to
0$. However, inequality (1.5) allows us to show that it holds

$\lim_{m\rightarrow
0}\frac{\sigma_{\beta}(\Omega)}{m}=\frac{1}{P(\Omega_{0})}.$ (5.13)

Indeed we have

$\frac{1}{P(\Omega_{0})}\geq\frac{\sigma_{\beta}(\Omega)}{m}\geq\dfrac{\mu_{1}(\Omega)}{m+P(\Omega_{0})\mu_{1}(\Omega)},$
(5.14)

where the first inequality follows by (5.12) and the second by using (1.9).
Taking in (5.14) the limit for $m$ which goes to zero one gets (5.13).
In particular, if $\beta(x)=\beta$ is the constant function we recover the
asymptotic formula valid in the radial case

$\lim_{\beta\to
0}\frac{\sigma_{\beta}(\Omega)}{\beta}=\frac{P(B_{r})}{P(\Omega_{0})}.$

######  Remark 5.4.

We observe that if $\beta(x)=\beta$ is constant, then inequality (1.10) gives

$\frac{1}{\sigma_{\beta}(\Omega)}-\frac{1}{\sigma_{D}(\Omega)}\leq\frac{1}{\beta
q(\Omega)}$

which immediately implies that

$\lim_{\beta\to\infty}\sigma_{\beta}(\Omega)=\sigma_{D}(\Omega).$

##  6\. Asymptotic behaviour of the eigenfunctions

In the radial case it is easy to check that the eigenfunctions corresponding
to $\sigma_{\beta}(\Omega)$ converge pointwise to the corresponding
eigenfunction of the Steklov-Dirichlet eigenvalue. In this section we will
prove that the same happens even for a generic $\Omega$ when $\beta$ is a
positive constant.

###### Proof of Theorem 1.2.

Without loss of generality, we suppose that $u_{\beta}$ and $v$ are normalized
in $L^{2}(\partial\Omega_{0})$ with unitary norm. By their weak formulations
we get

$\int_{\Omega}\langle\nabla u,\nabla\varphi\rangle\;dx+\beta\int_{\partial
B_{r}}u\varphi\,d\,\mathcal{H}^{n-1}=\sigma_{\beta}(\Omega)\int_{\partial\Omega_{0}}u\varphi\;d\mathcal{H}^{n-1},\qquad\forall\varphi\in
H^{1}(\Omega),$ (6.1)

and

$\int_{\Omega}\langle\nabla
v,\nabla\varphi\rangle\;dx=\sigma_{D}(\Omega)\int_{\partial\Omega_{0}}v\varphi\;d\mathcal{H}^{n-1},\qquad\forall\varphi\in
H^{1}_{\partial B_{r}}(\Omega).$ (6.2)

In particular $v$ is an admissible function for (6.1), hence

$\int_{\Omega}\langle\nabla u,\nabla
v\rangle\;dx=\sigma_{\beta}(\Omega)\int_{\partial\Omega_{0}}uv\;d\mathcal{H}^{n-1}.$
(6.3)

Now, by Friederich’s inequality (2.6), there exists a positive constant $C>0$
such that

$\|u_{\beta}-v\|_{L^{2}(\Omega)}\leq C\bigg{(}\|\nabla u_{\beta}-\nabla
v\|_{L^{2}(\Omega)}+\|u_{\beta}-v\|_{L^{2}(\partial\Omega)}\bigg{)}.$ (6.4)

Let us now estimate the terms inside the round brackets. Using the variational
characterization of $u_{\beta}$ and $v$, the fact that
$\|u_{\beta}\|_{L^{2}(\partial\Omega_{0})}=\|v\|_{L^{2}(\partial\Omega_{0})}=1$
and equation (6.3) we get

$\begin{split}\|\nabla u_{\beta}&-\nabla
v\|_{L^{2}(\Omega)}+\|u_{\beta}-v\|_{L^{2}(\partial\Omega)}=\\\
&=\int_{\Omega}|\nabla u_{\beta}|^{2}\,dx-2\int_{\Omega}\langle\nabla
u_{\beta},\nabla v\rangle\,dx+\int_{\Omega}|\nabla v|^{2}\,dx+\\\
&+\int_{\partial\Omega_{0}}|u_{\beta}-v|^{2}\,d\mathcal{H}^{n-1}+\int_{\partial
B_{r}}u_{\beta}^{2}\,d\mathcal{H}^{n-1}\\\ &=(1-\beta)\int_{\partial
B_{r}}u^{2}_{\beta}\,d\mathcal{H}^{n-1}+\int_{\partial\Omega_{0}}|u_{\beta}-v|^{2}\,d\mathcal{H}^{n-1}\\\
&+\sigma_{\beta}(\Omega)-2\sigma_{\beta}(\Omega)\int_{\partial\Omega_{0}}uv\,d\mathcal{H}^{n-1}+\sigma_{D}(\Omega).\end{split}$

Since we want $\beta\to+\infty$, we can chose $\beta>1$. Therefore

$\begin{split}\|\nabla u_{\beta}&-\nabla
v\|_{L^{2}(\Omega)}+\|u_{\beta}-v\|_{L^{2}(\partial\Omega)}<\\\
&<\int_{\partial\Omega_{0}}|u_{\beta}-v|^{2}\,d\mathcal{H}^{n-1}+\sigma_{\beta}(\Omega)-2\sigma_{\beta}(\Omega)\int_{\partial\Omega_{0}}uv\,d\mathcal{H}^{n-1}+\sigma_{D}(\Omega).\end{split}$

Since in the previous section we proved that
$\sigma_{\beta}(\Omega)\to\sigma_{D}(\Omega)$ as $\beta\to+\infty$, if we
prove that $\|u_{\beta}-v\|_{L^{2}(\partial\Omega_{0})}\to 0$ we have (1.11).
Let $\\{\beta_{k}\\}_{k\in\mathbb{N}}$ a sequence of Robin parameters such
that $\beta_{k}>1$ for every $k\geq 1$ and such that $\beta_{k}\to+\infty$.
Let $u_{k}\equiv u_{\beta_{k}}$ be the eigenfunctions corresponding to the
eigenvalues $\sigma_{\beta_{k}}(\Omega)$, such that
$\|u_{k}\|_{L^{2}(\partial\Omega_{0})}=1$. We already know that
$\sigma_{\beta_{k}}(\Omega)\to\sigma_{D}(\Omega)$ as $k\to+\infty$.
Applying once more Friederich’s inequality (2.6), we get

$\begin{split}\|u_{k}\|_{H^{1}(\Omega)}&=\|\nabla
u_{k}\|_{L^{2}(\Omega)}+\|u\|_{L^{2}(\Omega)}\\\ &\leq\|\nabla
u_{k}\|_{L^{2}(\Omega)}+C(\|\nabla
u_{k}\|_{L^{2}(\Omega)}+\|u\|_{L^{2}(\partial\Omega)})\\\ &\leq\|\nabla
u_{k}\|_{L^{2}(\Omega)}+C(\|\nabla
u_{k}\|_{L^{2}(\Omega)}+\beta_{k}\|u\|_{L^{2}(\partial B_{r})}+1)\\\
&\leq\sigma_{D}(\Omega)+C(\sigma_{D}(\Omega)+1)=(C+1)\sigma_{D}(\Omega)+C.\end{split}$

Hence we have that the sequence $\\{u_{k}\\}_{k\in\mathbb{N}}$ is uniformly
bounded in $H^{1}(\Omega)$ with respect to $\beta_{k}$. So there exists a
function $\bar{u}\in H^{1}(\Omega)$ such that $\nabla u_{k}\to\nabla\bar{u}$
weakly in $L^{2}(\Omega)$ and by the compactness of the trace operator we have
that $u_{k}\to\bar{u}$ in $L^{2}(\partial\Omega)$ and hence $u_{k}\to\bar{u}$
a.e. on $\partial\Omega$. In particular, since for every $k\in\mathbb{N}$

$\beta_{k}\|u_{k}\|_{L^{2}(\partial B_{r})}\leq\sigma_{D}(\Omega)<+\infty,$

it is necessary that $\|u_{k}\|_{L^{2}(\partial B_{r})}\to 0$ as
$k\to+\infty$. Using Fatou’s Lemma we have that for $k\to+\infty$

$0\leq\int_{\partial
B_{r}}\bar{u}^{2}\,d\mathcal{H}^{n-1}\leq\liminf_{k\to+\infty}\int_{\partial
B_{r}}u_{k}^{2}\,d\mathcal{H}^{n-1}\to 0.$

Therefore $\|\bar{u}\|_{L^{2}(\partial\Omega)}=0$, and hence $\bar{u}\in
H^{1}_{\partial B_{r}}(\Omega)$. We only need to show that $\bar{u}=v$. If we
use $\bar{u}$ as a test function in (6.1) we have that

$\int_{\Omega}\langle\nabla
u_{k},\nabla\bar{u}\rangle\;dx=\sigma_{\beta}(\Omega)\int_{\partial\Omega_{0}}u_{k}\bar{u}\;d\mathcal{H}^{n-1}.$

Now by the weak convergence of the gradients, the weak convergence of $u_{k}$
on $L^{2}(\partial\Omega_{0})$ and since $\sigma_{\beta_{k}}\to\sigma_{D}$, we
get

$\int_{\Omega}|\nabla\bar{u}|^{2}\,dx=\sigma_{D}(\Omega)\int_{\partial\Omega_{0}}\bar{u}^{2}\,d\mathcal{H}^{n-1}.$

By the simplicity of $\sigma_{D}$ and the fact that
$\|v\|_{L^{2}(\partial\Omega_{0})}=\|\bar{u}\|_{L^{2}(\partial\Omega_{0})}=1$,
it follows that $\bar{u}=v$. This implies that
$\|u_{\beta_{k}}-v\|_{L^{2}(\partial\Omega_{0})}\to 0$ and so

$\|u_{\beta}-v\|_{L^{2}(\Omega)}\to 0.$

In particular the following

$\begin{split}0&\leq\|u_{\beta}-v\|_{L^{2}(\Omega)}\leq C(\|\nabla
u_{\beta}-\nabla
v\|_{L^{2}(\Omega)}+\|u_{\beta}-v\|_{L^{2}(\partial\Omega)})\\\ &\leq
C\bigg{(}\sigma_{\beta}(\Omega)-2\sigma_{\beta}(\Omega)\int_{\partial\Omega_{0}}uv\,d\mathcal{H}^{n-1}+\sigma_{D}(\Omega)\bigg{)}\to
0\end{split}$

implies that $\|\nabla u_{\beta}-\nabla v\|_{L^{2}(\Omega)}\to 0$. This
conclude the proof. ∎

## Acknowledgements

This work has been partially supported by the MiUR-PRIN 2017 grant
“Qualitative and quantitative aspects of nonlinear PDEs”, by GNAMPA of INdAM
and by FRA 2020 “Optimization problems in Geometric-functional inequalities
and nonlinear PDEs”(OPtImIzE).

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