# A stability result for the first Robin-Neumann eigenvalue:
a double perturbation approach

Simone Cito1, Gloria Paoli2, Gianpaolo Piscitelli3 1Dipartimento di Matematica
e Fisica “E. De Giorgi“, Università del Salento, Via per Arnesano, 73100
Lecce, Italy<EMAIL_ADDRESS>2 Friedrich-Alexander-Universität
Erlangen-Nürnberg, Cauerstraße 11, 51098, Erlangen, Germany
<EMAIL_ADDRESS>(corresponding author) 3Dipartimento di Matematica e
Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II,
Via Cinthia n. 26, Complesso Universitario Monte Sant’Angelo, 81026 Napoli,
Italy<EMAIL_ADDRESS>
###### Abstract.

Let $\Omega=\Omega_{0}\setminus\overline{\Theta}\subset\mathbb{R}^{n}$, $n\geq
2$, where $\Omega_{0}$ and $\Theta$ are two open, bounded and convex sets such
that $\overline{\Theta}\subset\Omega_{0}$ and let $\beta<0$ be a given
parameter. We consider the eigenvalue problem for the Laplace operator
associated to $\Omega$, with Robin boundary condition on $\partial\Omega_{0}$
and Neumann boundary condition on $\partial\Theta$. In [42] it is proved that
the spherical shell is the only maximizer for the first Robin-Neumann
eigenvalue in the class of domains $\Omega$ with fixed outer perimeter and
volume.

We establish a quantitative version of the afore-mentioned isoperimetric
inequality; the main novelty consists in the introduction of a new type of
hybrid asymmetry, that turns out to be the suitable one to treat the different
conditions on the outer and internal boundary. Up to our knowledge, in this
context, this is the first stability result in which _both_ the outer and the
inner boundary are perturbed.

MSC 2020: 35J25, 35P15, 47J30.
Keywords and phrases: Laplace operator; Robin-Neumann eigenvalue; Quantitative
spectral inequality.

##  1\. Introduction

Let $\Omega=\Omega_{0}\setminus\overline{\Theta}$, a subset of
$\mathbb{R}^{n}$, with $n\geq 2$, where $\Omega_{0}$ and $\Theta$ are two
open, bounded and convex sets such that $\overline{\Theta}\subset\Omega_{0}$
and let $\beta<0$ be a given parameter.

In this paper, we study the first Robin-Neumann eigenvalue of the the Laplace
operator, defined as:

(1) $\lambda_{1}(\beta,\Omega)=\min_{\begin{subarray}{c}\psi\in
H^{1}(\Omega)\\\ \psi\not\equiv 0\end{subarray}}R_{\Omega}(\psi),$

where

(2)
$R_{\Omega}(\psi)=\dfrac{\displaystyle\int_{\Omega}|\nabla\psi|^{2}\;dx+\beta\displaystyle\int_{\partial\Omega_{0}}\psi^{2}\;d\mathcal{H}^{n-1}}{\displaystyle\int_{\Omega}\psi^{2}\;dx}.$

If $w\in H^{1}(\Omega)$ is a minimizer of (1), then it satisfies

(3) $\begin{cases}-\Delta w=\lambda_{1}(\beta,\Omega)w&\mbox{in}\
\Omega\vspace{0.2cm}\\\ \dfrac{\partial w}{\partial\nu}+\beta w=0&\mbox{on}\
\partial\Omega_{0}\vspace{0.2cm}\\\ \dfrac{\partial
w}{\partial\nu}=0&\mbox{on}\ \partial\Theta,\end{cases}$

where we denote by $\nu$ the unit outer normal to the boundary and $\beta$ is
usually called the Robin boundary parameter associated to the problem.

We denote by $B_{R}$ is the open ball of $\mathbb{R}^{n}$ of radius $R$
centered at the origin and we set
$A_{R_{1},R_{2}}:=B_{R_{2}}\setminus\overline{B_{R_{1}}}$, with
$0<R_{1}<R_{2}$. Moreover, we denote, as usual, by $P(\cdot)$ and $|\cdot|$,
respectively, the perimeter and the volume of a set. In [42] it is proved that

(4) $\lambda_{1}(\beta,\Omega)\leq\lambda_{1}(\beta,A),$

where $A=A_{R_{1},R_{2}}$ is the spherical shell such that
$P(B_{R_{2}})=P(\Omega_{0})$ and $|\Omega|=|A|$ and equality holds if and only
if $\Omega=A$.

The aim of the present paper is to prove a quantitative estimate of the
isoperimetric inequality (4). More precisely, we want to improve (4),
obtaining an inequality of the following type:

$\lambda_{1}(\beta,A)-\lambda_{1}(\beta,\Omega)\geq g(\alpha(\Omega)),$

where $g(\cdot)$ is a modulus of continuity, i.e. a positive continuous
increasing function, vanishing only at $0$ and $\alpha(\cdot)$ is a suitable
asymmetry functional, i.e. a non negative functional such that
$\alpha(\Omega)=0$ if and only if $\Omega=A$.

In order to get this kind of quantitative result, we need to introduce the
following class of sets, in which (4) turns out to be stable. Let us fix
$0<R_{1}<R_{2}<+\infty$ and let us denote by $\vartheta_{R_{1},R_{2}}$ a
positive constant such that
$\vartheta_{R_{1},R_{2}}<\min\\{R_{1},R_{2}-R_{1}\\}$. We set

(5)
$\begin{split}\mathcal{T}_{R_{1},R_{2}}=\\{\Omega\subseteq\mathbb{R}^{n}:&\
\Omega=\Omega_{0}\setminus\overline{\Theta},\ \Omega_{0},\Theta\
\text{convex},\ |\Omega|=|A_{R_{1},R_{2}}|,\ P(\Omega_{0})=P(B_{R_{2}}),\\\
&d_{\mathcal{H}}(\Theta,\Omega_{0})\geq\vartheta_{R_{1},R_{2}}\ \text{and}\
\rho(\Theta)\geq\vartheta_{R_{1},R_{2}}\\},\end{split}$

where $\rho(\cdot)$ is the inradius of a set $d_{\mathcal{H}}(\cdot,\cdot)$ is
the Hausdorff distance between two convex sets. More precisely, given
$E,F\subseteq\mathbb{R}^{n}$ convex sets, the Hausdorff distance is defined as
follows

$d_{\mathcal{H}}(E,F)=\inf\\{\varepsilon>0\ :\ E\subset F+B_{\varepsilon},\
F\subset E+B_{\varepsilon}\\}.$

Thus, as a direct consequence of (4), we have that the spherical shell
$A_{R_{1},R_{2}}$ is the only maximizer for the shape optimization problem

(6)
$\max\left\\{\lambda_{1}(\beta,\Omega):\Omega\in\mathcal{T}_{R_{1},R_{2}}\right\\}.$

In order to study the stability of the reverse Faber-Krahn type inequality (4)
in the class $\mathcal{T}_{R_{1},R_{2}}$, we need to introduce a suitable
asymmetry functional, that takes into account both the perturbations on the
outer and inner boundary with the different boundary conditions.

To treat the perturbation of the outer boundary, we use this standard notion
of asymmetry for a convex set:

(7)
$\mathcal{A}_{\mathcal{H}}(\Omega_{0})=\inf_{x_{0}\in\mathbb{R}^{n}}\left\\{\frac{d_{\mathcal{H}}(\Omega_{0},B_{R}(x_{0}))}{\omega_{n}R^{n}}\
:\ P(\Omega_{0})=P(B_{R}(x_{0}))\right\\},$

where $\omega_{n}$ is the measure of the unit ball in $\mathbb{R}^{n}$.

On the other hand, we also need to take into account the perturbation of the
inner hole $\Theta$. In order to do that, we recall the definition of the test
function $w_{\Omega_{0}}$ used in [42] to prove (4):

(8) $w_{\Omega_{0}}(x):=\begin{cases}G(d(x))\quad&\text{if}\
d(x)<R_{2}-R_{1}\\\ z_{m}&\text{if}\ d(x)\geq R_{2}-R_{1}\end{cases}\
x\in\Omega_{0},$

where $z$ be the first positive eigenfunction on the spherical shell
$A_{R_{1},R_{2}}$ with $\|z\|_{L^{2}(A_{R_{1},R_{2}})}=1$, $d(x)$ is the
distance function from the boundary of $\Omega_{0}$ and $G$ is defined as

$G^{-1}(t)=\int_{t}^{z_{M}}\dfrac{1}{\ell(\tau)}\;d\tau,$

with $\ell(t)=|\nabla z|_{z=t}$, determined for $z_{m}<t\leq z_{M}$ with
$z_{m}:=\min_{A}z$ and $z_{M}:=\max_{A}z$. We point out that $w_{\Omega_{0}}$
is a map depending only on the distance from the outer boundary (see Section
2.3 for further details) and, moreover, it is constant on the boundaries of
the inner parallels to $\Omega_{0}$. This type of functions are usually called
web-function and this name appeared for the first time in [34], since their
level lines recall the web of a spider.

Therefore, $w_{\Omega_{0}}$ can be used as a test function for every
admissible holed set with outer box $\Omega_{0}$. Now, we are in position to
give the following definition (see also Figure 1).

######  Definition 1.1 (Hybrid asymmetry).

We define the hybrid asymmetry of
$\Omega=\Omega_{0}\setminus\overline{\Theta}$ as:

(9)
$\alpha(\Omega):=\max\left\\{g(\mathcal{A}_{\mathcal{H}}(\Omega_{0})),\tilde{\mathcal{A}}(\Theta;\Omega_{0})\right\\}.$

Here, $\mathcal{A}_{\mathcal{H}}(\cdot)$ is the Hausdorff asymmetry defined in
(7) and $g(\cdot)$ is the positive increasing function defined as

$g(s)=\begin{cases}s^{2}&\text{if $n=2$}\\\ f^{-1}(s^{2})&\text{if $n=3$}\\\
s^{\frac{n+1}{2}}&\text{if $n\geq 4$},\end{cases}$

with $f(t)=\sqrt{t\log\left(\frac{1}{t}\right)}$ for $0<t<e^{-1}$.

The term $\tilde{\mathcal{A}}(\Theta;\Omega_{0})$ is the weak weighted
Fraenkel-type asymmetry of $\Theta$ relative to the set $\Omega_{0}$, defined
as

(10) $\tilde{\mathcal{A}}(\Theta;\Omega_{0}):=\int_{\Theta\setminus
K_{\Omega_{0}}}(w_{\Omega_{0}}^{2}(x)-z_{m}^{2})\>dx,$

where

  * •

$w_{\Omega_{0}}\in H^{1}(\Omega_{0})$ is the test function defined in (8);

  * •

$K_{\Omega_{0}}$ is the inner parallel set relative to $\Omega_{0}$, i.e. the
set

$K_{\Omega_{0}}=\\{x\in\Omega_{0}\;:\;d(x)>t_{\Omega_{0}}\\},$

with $t_{\Omega_{0}}\in(0,\rho(\Omega_{0}))$ such that
$|K_{\Omega_{0}}|=|\Theta|$;

  * •

$z$ is the first positive eigenfunction of (1) on the $A_{R_{1},R_{2}}$ with
$\|z\|_{L^{2}(A_{R_{1},R_{2}})}=1$;

  * •

$z_{m}$ is the minimum of $z$ on $A_{R_{1},R_{2}}$.

Figure 1. $\mathcal{A}_{\mathcal{H}}(\Omega_{0})$ is drawn in orange; the
green set is the integration domain in the definition of
$\tilde{\mathcal{A}}(\Theta;\Omega_{0})$.

We point out that the hybrid asymmetry just introduced is zero if and only if
$\Omega=A_{R_{1},R_{2}}$, and so, (9) actually quantifies a deviation of an
admissible set from the optimal set. The term
$g(\mathcal{A}_{\mathcal{H}}(\Omega_{0}))$ quantifies the deviation with
respect to the Hausdorff distance of the domain $\Omega_{0}$ from the optimal
exterior shape $B_{R_{2}}$; $\tilde{\mathcal{A}}(\Theta;\Omega_{0})$ is
necessary to estimate the total deviation of $\Omega$ from $A_{R_{1},R_{2}}$.
Indeed, if $\Theta$ has a low Hausdorff distance from $K_{\Omega_{0}}$, then
the larger term in the hybrid asymmetry is $g(\mathcal{A}(\Omega_{0}))$; on
the other hand, if $\Omega_{0}$ is the ball $B_{R_{2}}$, any admissible hole
$\Theta\neq B_{R_{1}}$ satisfies $\tilde{\mathcal{A}}(\Theta;B_{R_{2}})>0$ and
thus in that case the largest term is $\tilde{\mathcal{A}}(\Theta;B_{R_{2}})$
(see Remark 2.10).

Moreover, $\alpha(\Omega)$ is a proper asymmetry because it holds (see
Proposition 2.11) that

$\lim_{j\to+\infty}d_{\mathcal{H}}(\Omega_{j},A_{R_{1},R_{2}})=0\iff\lim_{j\to+\infty}\alpha(\Omega_{j})=0.$

The key idea that inspired our definition of asymmetry is the following:
provided that $\Omega_{0}$ is the outer box, the most natural hole to compare
$\Theta$ with is neither a ball with the same measure of $\Theta$ nor
$B_{R_{1}}$, but the inner parallel $K_{\Omega_{0}}$ having the same measure
as $\Theta$. This behaviour seems to depend on the fact that $K_{\Omega_{0}}$
makes the set $\Omega_{0}\setminus K_{\Omega_{0}}$ a "shell" with outer "box"
$\Omega_{0}$ and fixed distance of the inner boundary $\partial
K_{\Omega_{0}}$ from the outer boundary $\partial\Omega_{0}$. Moreover, the
test function built via web function is constant on both $\partial
K_{\Omega_{0}}$ and $\partial\Omega_{0}$. Nevertheless, the set
$K_{\Omega_{0}}$ coincides with $B_{R_{1}}$ if and only if
$\Omega_{0}=B_{R_{2}}$.

The main result of the paper is that inequality (4) is stable in the class
$\mathcal{T}_{R_{1},R_{2}}$.

######  Main Theorem.

Let $0<R_{1}<R_{2}<+\infty$. There exist two positive constants
$C(n,R_{1},R_{2},\beta)$ and $\delta_{0}(n,R_{1},R_{2},\beta)$ such that, for
every $\Omega\in\mathcal{T}_{R_{1},R_{2}}$, if
$\lambda_{1}(\beta,A)-\lambda_{1}(\beta,\Omega)\leq\delta_{0}$, then

(11) $\lambda_{1}(\beta,A)-\lambda_{1}(\beta,\Omega)\geq
C(n,R_{1},R_{2},\beta)\alpha(\Omega),$

where $A=A_{R_{1},R_{2}}$ is the spherical shell in
$\mathcal{T}_{R_{1},R_{2}}$ and $\alpha(\Omega)$ is the asymmetry defined in
(9).

Our proof is based on two estimates. Since on $\partial\Omega_{0}$ a Robin
boundary condition holds, we can use a Fuglede type technique on an auxiliary
Steklov type problem to obtain the first estimate relative to the perturbation
of the outer boundary. For this type of approach we refer to [27, 29] for the
stability of the isoperimetric inequality and to [24, 30, 21] for the
quantitative estimates of some spectral functionals. Then, once the outer
boundary is fixed, we can estimate the deviation of $\Theta$ from
$K_{\Omega_{0}}$ using the weak asymmetry
$\tilde{\mathcal{A}}(\Theta;\Omega_{0})$, defined in (10) and inspired by [11,
2].

The structure of the paper is the following. In Section 2, we give some
historical insights, we fix the notation, we recall some known results on the
Robin-Neumann eigenvalue problem and recall some useful results on the nearly
spherical sets. In Section 3 we study an auxiliary problem of Steklov type; in
Section 4, we provide some quantitative estimates for
$\lambda_{1}(\beta,\Omega)$ and we prove of the main result of the paper.
Finally, in Section 5, a list of open problems is presented.

##  2\. Background and Preliminary results

We start this Section by describing the historical background on problems with
different boundary conditions on the outer and inner boundary. Then, we fix
the notation, we recall some results on the Robin-Neumann eigenvalue problem
and some properties of nearly spherical sets.

###  2.1.  A glimpse of history

The study of spectral problems for the Robin-Laplace operator with negative
parameter $\beta$ on holed domains arises from the fact that the optimal shape
for the first eigenvalue, among domains with fixed volume, is expected to be
the spherical shell for ${\left|\beta\right|}$ large enough. More precisely,
in 1977, Bareket [7] conjectured that, among all smooth bounded domains of
given volume, the first eigenvalue is maximized when $\Omega$ is a ball and,
in [24], the conjecture has been proved for domains close to a ball in the
$L^{\infty}$ topology. Subsequently, in [25, 38], it has been proved that the
conjecture is true for small values of $\left|\beta\right|$; on the other
hand, for $\left|\beta\right|$ great enough, the first Robin eigenvalue on the
spherical shell is greater than the eigenvalue on the disk, among domains of
fixed volume.

More recently, many authors have investigated the Laplace eigenvalue problem
with different boundary conditions on the outer and inner boundary: for
instance positive Robin-Neumann [42, 43], Neumann-positive Robin [23, 36],
Dirichlet-Neumann [4, 5], Dirichlet-Dirichlet [9, 18], Steklov-Dirichlet [26,
31, 32, 37, 41, 46], Steklov-Robin [33].

In particular, in this paper we study a stability result for this kind of
problems.

To the best of our knowledge, the quantitative improvement of spectral
inequalities firstly appeared in [35, 40]; quantitative improvements of the
Pólya-Szegö principle have been proven in [19]. It is worth noticing that,
when $\Theta=\emptyset$, the quantitative inequalities for the Laplace
eigenvalue have been studied when Dirichlet ([8, 29] and [13]), Neumann [14],
positive Robin [17], negative Robin [2, 21] or Steklov [12, 30] boundary
condition holds.

On the other hand, the quantitative versions of the spectral inequalities on
holed domain are still in the early stages of development. We mention [41],
where a result in this direction is proved for the first Steklov-Dirichlet
Laplacian eigenvalue among the class of holed domains where the inner hole
$\Theta$ is a given ball well cointained in $\Omega_{0}$. The inner ball can
be only translated and so the only perturbation of the boundary acts on
$\partial\Omega_{0}$.

Nevertheless, it seems that no results are available if a perturbation of
_both_ inner and outer boundary is kept into account.

###  2.2. Notations

Throughout this paper, we denote by $|\cdot|$, $P(\cdot)$ and
$\mathcal{H}^{k}(\cdot)$, the $n-$dimensional Lebesgue measure, the perimeter
and the $k-$dimensional Hausdorff measure in $\mathbb{R}^{n}$, respectively.
The unit open ball in $\mathbb{R}^{n}$ will be denoted by $B_{1}$ and
$\omega_{n}:=|B_{1}|$ and the unit sphere in $\mathbb{R}^{n}$ by
$\mathbb{S}^{n-1}$. More generally, we denote with $B_{r}(x_{0})$ the set
$x_{0}+rB_{1}$, that is the ball centered at $x_{0}$ with measure
$\omega_{n}r^{n}$, and, for any $0<R_{1}<R_{2}<+\infty$, we denote by
$A_{R_{1},R_{2}}$ the spherical shell
$B_{R_{2}}\setminus\overline{B_{R_{1}}}$.

Throughout this paper, $\Omega=\Omega_{0}\setminus\overline{\Theta}$ is a
subset of $\mathbb{R}^{n}$, $n\geq 2$, such that $\Omega_{0}$ is an open,
bounded and convex set and $\Theta$ is a convex set such that
$\overline{\Theta}\subset\Omega_{0}$.

The distance of a point from the boundary $\partial\Omega_{0}$ is the function

$d(x)=\inf_{y\in\partial\Omega_{0}}|x-y|,\quad x\in\overline{\Omega_{0}},$

moreover, the inradius of $\Omega_{0}$ is the radius of the largest ball
contained in $\Omega_{0}$, i.e.

$\rho({\Omega_{0}})=\max\\{d(x),\;x\in\overline{\Omega_{0}}\\}.$

We denote by $J_{\nu}(r)$, $Y_{\nu}(r)$, $I_{\nu}(r)$ and $K_{\nu}(r)$ the
Bessel function of first and second kind, the modified Bessel functions of the
first and second kind of order $\nu$, respectively. We recall the following
recurrence and derivation formulas, see [1, 47]:

(12) $\displaystyle I_{\nu-1}(r)=\dfrac{2\nu}{r}I_{\nu}(r)+I_{\nu+1}(r),$ (13)
$\displaystyle K_{\nu-1}(r)=-\dfrac{2\nu}{r}K_{\nu}(r)+K_{\nu+1}(r),$ (14)
$\displaystyle\left(z^{-\nu}J_{\nu}(r)\right)^{\prime}=-z^{-\nu}J_{\nu+1}(r),$
(15)
$\displaystyle\left(z^{-\nu}Y_{\nu}(r)\right)^{\prime}=-z^{-\nu}Y_{\nu+1}(r),$
(16)
$\displaystyle\left(z^{-\nu}I_{\nu}(r)\right)^{\prime}=z^{-\nu}I_{\nu+1}(r),$
(17)
$\displaystyle\left(z^{-\nu}K_{\nu}(r)\right)^{\prime}=-z^{-\nu}K_{\nu+1}(r),$
(18) $\displaystyle
I_{\nu}^{\prime}(r)=\dfrac{1}{2}\left(I_{\nu-1}(r)+I_{\nu+1}(r)\right),$ (19)
$\displaystyle
K_{\nu}^{\prime}(r)=-\frac{1}{2}\left(K_{\nu-1}(r)+K_{\nu+1}(r)\right).$

###  2.3. The Robin-Neumann eigenvalue problem

We collect now the definitions and the basic properties of the first Robin-
Neumann eigenvalue; for the proofs refer to [42].

We start by recalling the weak formulation of problem (3).

######  Definition 2.1.

A real number $\lambda$ is an eigenvalue of (3) if and only if there exists a
function $w\in H^{1}(\Omega)$, not identically zero, such that

$\int_{\Omega}\nabla\psi\nabla\varphi\;dx+\beta\int_{\partial\Omega_{0}}\psi\varphi\;d\mathcal{H}^{n-1}=\lambda\int_{\Omega}\psi\varphi\;dx$

for every $\varphi\in H^{1}(\Omega)$. The function $\psi$ is called the
eigenfunction associated to $\lambda$.

In the following definition, we list the main properties of (1).

######  Proposition 2.2.

Let $\beta<0$.

  * •

There exists a minimizer $w\in H^{1}(\Omega)$ of (1), which is a weak solution
to (3).

  * •

$\lambda_{1}(\beta,\Omega)$ is simple, i.e. all the associated eigenfunctions
are scalar multiple of each other and can be taken to be positive. Moreover,
$\lambda_{1}(\beta,\Omega)$ is negative.

  * •

The map $\beta\mapsto\lambda_{1}(\beta,\Omega)$ is Lipschitz continuous, non-
decreasing, concave and surjective onto $(-\infty,0)$.

  * •

Let $R_{1},R_{2}$ be two nonnegative real numbers such that $R_{2}>R_{1}$, and
let $z$ be the minimizer of problem (1) on the spherical shell
$A_{R_{1},R_{2}}$. Then, $z$ is strictly positive and radially symmetric, in
the sense that $z(x)=:\Psi(|x|)$, with $\Psi^{\prime}(r)>0$.

We recall now explicitly Theorem $3.1$ in [42], where it is proved that the
spherical shell is the optimal shape for $\lambda_{1}(\beta,\Omega)$ when both
the volume and the outer perimeter are fixed. This result holds for all
$\beta\in\mathbb{R}\setminus\\{0\\}$, but for our purposes we state it only in
the case of negative parameter.

######  Theorem 2.3 (Theorem $3.1$ in [42]).

Let $\beta<0$ and $A=A_{R_{1},R_{2}}$ be the spherical shell such that
$|A|=|\Omega|$ and $P(B_{R_{2}})=P(\Omega_{0})$. Then,

$\displaystyle\lambda_{1}(\beta,\Omega)\leq\lambda_{1}(\beta,A).$

The test functions used in the proof of Theorem 2.3 fall in the class of the
so-called web-functions (for more details refer to [34, 39, 44]), which depend
only on the distance from the boundary. In particular, the authors in [42] use
the following test function

$w_{\Omega_{0}}(x):=\begin{cases}G(d(x))\quad&\text{if}\ d(x)<R_{2}-R_{1}\\\
z_{m}&\text{if}\ d(x)\geq R_{2}-R_{1}\end{cases}\quad x\in\Omega_{0},$

where $z$ is the eigenfunction of (1) on $A_{R_{1},R_{2}}$ and $G$ is defined
as

$G^{-1}(t)=\int_{t}^{z_{M}}\dfrac{1}{\ell(\tau)}\;d\tau,$

and $\ell(t)=|\nabla z|_{z=t}$, determined for $z_{m}<t\leq z_{M}$ with
$z_{m}:=\min_{A}z$ and $z_{M}:=\max_{A}z$. Moreover, $w_{\Omega_{0}}$
satisfies the following properties: $w_{\Omega_{0}}\in H^{1}(\Omega_{0})$ and

$\displaystyle|\nabla w_{\Omega_{0}}|_{u=t}=|\nabla z|_{v=t},$
$\displaystyle(w_{\Omega_{0}})_{m}:=\min_{\Omega_{0}}z_{\Omega_{0}}\geq
z_{m},$
$\displaystyle(w_{\Omega_{0}})_{M}:=\max_{\Omega_{0}}z_{\Omega_{0}}=z_{M}=G(0).$

Moreover, we recall the expression of the eigenfunction $z$ associated to
$\lambda=\lambda_{1}(\beta,A_{R_{1},R_{2}})$:

$z(x)=|x|^{1-\frac{n}{2}}\left(Y_{\frac{n}{2}}(\sqrt{\lambda}R_{1})J_{\frac{n}{2}-1}(\sqrt{\lambda}|x|)-J_{\frac{n}{2}}(\sqrt{\lambda}R_{1})Y_{\frac{n}{2}-1}(\sqrt{\lambda}|x|)\right).$

So, by the radiality of $z$, we have $z(x)=\Psi(|x|)$ and

$\Psi(r)=r^{1-\frac{n}{2}}\delta(r),\quad\Psi^{\prime}(r)=r^{1-\frac{n}{2}}\gamma(r),$

once having set

$\begin{split}&\delta(r)=Y_{\frac{n}{2}}(\sqrt{\lambda}R_{1})J_{\frac{n}{2}-1}(\sqrt{\lambda}r)-J_{\frac{n}{2}}(\sqrt{\lambda}R_{1})Y_{\frac{n}{2}-1}(\sqrt{\lambda}r),\\\
&\gamma(r)=-Y_{\frac{n}{2}}(\sqrt{\lambda}R_{1})J_{\frac{n}{2}}(\sqrt{\lambda}r)-J_{\frac{n}{2}}(\sqrt{\lambda}R_{1})Y_{\frac{n}{2}}(\sqrt{\lambda}r),\end{split}$

and using (14)-(15).

Therefore, substituting $z$ in (1), we obtain

(20)
$\lambda_{1}(\beta,A_{R_{1},R_{2}})=\frac{\displaystyle\int_{R_{1}}^{R_{2}}r\gamma^{2}(r)dr+\beta
R_{2}\delta^{2}(R_{2})}{\displaystyle\int_{R_{1}}^{R_{2}}r\delta^{2}(r)dr}.$

Finally, we stress that it is possible to write the explicit expression of
(20) by using the following relations for Bessel functions (see [47, eq.
(10)-(11) pag. 134]):

$\begin{split}&\int
rF_{\nu}(\sqrt{\lambda}r)G_{\nu}(\sqrt{\lambda}r)dr=\frac{1}{4}r^{2}\left(2F_{\nu}(\sqrt{\lambda}r)G_{\nu}(\sqrt{\lambda}r)-F_{\nu-1}(\sqrt{\lambda}r)G_{\nu+1}(\sqrt{\lambda}r)-F_{\nu+1}(\sqrt{\lambda}r)G_{\nu-1}(\sqrt{\lambda}r)\right),\\\
&\int
rF^{2}_{\nu}(\sqrt{\lambda}r)dr=\frac{1}{2}r^{2}\left(F^{2}_{\nu}(\sqrt{\lambda}r)-F_{\nu-1}(\sqrt{\lambda}r)F_{\nu+1}(\sqrt{\lambda}r)\right),\end{split}$

where $F_{\nu}$ and $G_{\nu}$ are Bessel functions of any kind of order $\nu$.

###  2.4. Nearly spherical sets

In this Section, we give some definitions and useful properties related to
nearly spherical sets (see [27, 28]).

######  Definition 2.4.

Let $\Omega_{0}\subset\mathbb{R}^{n}$ be an open bounded set with
$0\in\Omega_{0}$. The set $\Omega_{0}$ is said a _$R-$ nearly spherical set
parametrized by $u$_ if there exists a constant $R>0$ and $u\in
W^{1,\infty}(\mathbb{S}^{n-1})$, with $||u||_{W^{1,\infty}}\leq R/2$, such
that

$\partial\Omega_{0}=\left\\{y\in\mathbb{R}^{n}\colon y=\xi(R+u(\xi)),\
\xi\in\mathbb{S}^{n-1}\right\\}.$

Let us observe that the volume of a $R-$nearly spherical set is given by

$|\Omega_{0}|=\dfrac{1}{n}\int_{\mathbb{S}^{n-1}}\left(R+u(\xi)\right)^{n}\;d\mathcal{H}^{n-1},$

and the perimeter by

$P(\Omega_{0})=\int_{\mathbb{S}^{n-1}}\left(R+u(\xi)\right)^{n-1}\sqrt{1+\dfrac{|\nabla_{\tau}u(\xi)|^{2}}{(R+u(\xi))^{2}}}d\mathcal{H}^{n-1},$

where $\nabla_{\tau}u$ denotes the tangential gradient of $u$ and the
tangential Jacobian of the map
$\phi:\xi\in\mathbb{S}^{n-1}\to y=R\xi(1+u(\xi))$ is

$J_{\phi}(\xi)=(R+u(\xi))^{n-2}\sqrt{(R+u(\xi))^{2}+|\nabla_{\tau}u(\xi)|^{2}}.$

Moreover, the unit outer normal to $\partial\Omega_{0}$ at $y=\xi(R+u(\xi))$
is given by

$\nu(\xi)=\dfrac{\xi(R+u(\xi))-\nabla_{\tau}u(\xi)}{\sqrt{(R+u(\xi))^{2}+|\nabla_{\tau}u(\xi)|^{2}}}.$

In the following, we deal with sets that are, in a suitable sense, close to a
spherical shell and here we set the needed notation.

######  Definition 2.5.

Let $\Omega=\Omega_{0}\setminus\Theta$, where $\Omega_{0}$ and $\Theta$ are
two convex sets such that $\Theta\subset\subset\Omega_{0}$ and $0\in\Theta$.
We say that $\Omega$ is a _$(R_{1},R_{2})-$ nearly annular set_ if there
exists $0<R_{1}<R_{2}<+\infty$ and $u,v\in W^{1,\infty}(\mathbb{S}^{n-1})$,
with $||u||_{W^{1,\infty}}<R_{2}/2$, $||v||_{W^{1,\infty}}\leq R_{1}/2$, such
that

$\partial\Omega_{0}=\left\\{y\in\mathbb{R}^{n}\colon
y=\xi(R_{2}+u(\xi)),\,\xi\in\mathbb{S}^{n-1}\right\\}$

and

$\partial\Theta=\left\\{y\in\mathbb{R}^{n}\colon
y=\xi(R_{1}+v(\xi)),\,\xi\in\mathbb{S}^{n-1}\right\\}.$

This definition and the outer perimeter and volume constraints yield to the
following Lemma, based on the Taylor expansions.

######  Lemma 2.6.

There exists $C=C(n)>0$ such that, for any parametrization couple $u,v\in
W^{1,\infty}(\mathbb{S}^{n-1})$ of a $(R_{1},R_{2})-$nearly annular set and
for every $0<\varepsilon<R_{1}/2$ such that $||u||_{W^{1,\infty}},$
$||v||_{W^{1,\infty}}\leq\varepsilon$, the following estimates hold:

(21)
$\left|\left(R_{2}+u\right)^{n-1}-\left(R_{2}^{n-1}+(n-1)R_{2}^{n-2}u+(n-1)(n-2)R_{2}^{n-3}\frac{u^{2}}{2}\right)\right|\leq
C\varepsilon R_{2}^{n-3}u^{2};$ (22)
$\left|\left(R_{1}+v\right)^{n-1}-\left(R_{1}^{n-1}+(n-1)R_{1}^{n-2}v+(n-1)(n-2)R_{1}^{n-3}\frac{v^{2}}{2}\right)\right|\leq
C\varepsilon R_{1}^{n-3}v^{2};$ (23)
$1+\frac{|\nabla_{\tau}u|^{2}}{2R_{2}^{2}}-\sqrt{1+\dfrac{|\nabla_{\tau}u|^{2}}{(R_{2}+u)^{2}}}\leq
C\varepsilon\left(u^{2}+|\nabla_{\tau}u|^{2}\right).$

Moreover,

(24)
$\left|R_{2}^{n-2}\int_{\mathbb{S}^{n-1}}u\;d\mathcal{H}^{n-1}+\dfrac{R^{n-3}_{2}}{2(n-1)}\int_{\mathbb{S}^{n-1}}|\nabla_{\tau}u|^{2}\;d\mathcal{H}^{n-1}+\dfrac{(n-2)}{2}R_{2}^{n-3}\int_{\mathbb{S}^{n-1}}u^{2}\;d\mathcal{H}^{n-1}\right|\leq
C\varepsilon,$

and

(25)
$\begin{split}\left|R_{1}^{n-1}\int_{\mathbb{S}^{n-1}}v\;d\mathcal{H}^{n-1}+\right.&\dfrac{(n-1)}{2}R_{1}^{n-1}\int_{\mathbb{S}^{n-1}}v^{2}\;d\mathcal{H}^{n-1}\\\
&\left.-R_{2}^{n-1}\int_{\mathbb{S}^{n-1}}u\;d\mathcal{H}^{n-1}-\dfrac{(n-1)}{2}R_{2}^{n-1}\int_{\mathbb{S}^{n-1}}u^{2}\;d\mathcal{H}^{n-1}\right|\leq
C\varepsilon.\end{split}$

###### Proof.

The proofs of (21)-(25) can be immediately adapted from [27]. Inequalities
(21) and (22) easily follow, respectively, from

(26)
$(R_{2}+u)^{n-1}=\sum_{k=0}^{n-1}\binom{n-1}{k}u^{k}R_{2}^{n-1-k}\quad\text{and}\quad(R_{1}+v)^{n-1}=\sum_{k=0}^{n-1}\binom{n-1}{k}v^{k}R_{1}^{n-1-k}.$

Inequality (23) is a consequence of inequality $\sqrt{1+x}\geq
1+\frac{x}{2}-\frac{x^{2}}{4}$. Furthermore, using the outer perimeter
constraint

$\frac{1}{n}\int_{\mathbb{S}^{n-1}}(R_{2}+u)^{n-1}\sqrt{1+\dfrac{|\nabla
u|^{2}}{(R_{2}+u)^{2}}}d\mathcal{H}^{n-1}=\omega_{n}R_{2}^{n-1},$

the first equality in (26) and (23), we obtain (24). Finally, inequality (25)
follows by using the volume constraint

$\frac{1}{n}\int_{\mathbb{S}^{n-1}}(R_{2}+u)^{n-1}-(R_{1}+v)^{n-1}d\mathcal{H}^{n-1}=\omega_{n}(R_{2}^{n-1}-R_{1}^{n-1})$

and the second equality in (26). ∎

The Poincaré inequality in this setting holds with constant $(n-1)$, but,
since we are working with functions parametrizing nearly spherical sets, this
constant can been improved to $2n$. For the proof we refer to [27] and [24,
Lem. 2.4].

######  Lemma 2.7.

There exists a constant $C>0$ such that, for any parametrization $u\in
W^{1,\infty}(\mathbb{S}^{n-1})$ of a $R-$nearly spherical set and for every
$0<\varepsilon<R/2$ with $||u||_{W^{1,\infty}}\leq\varepsilon$, it holds

(27) $||\nabla_{\tau}u||_{L^{2}}^{2}\geq 2n||u||_{L^{2}}^{2}-\varepsilon C.$

Finally, these last two Lemmata provide key estimates for the norms of
functions parametrizing the boundary of a nearly spherical set, see [27, 28].

######  Lemma 2.8.

Let $u\in W^{1,\infty}(\mathbb{S}^{n-1})$ such that
$\displaystyle\int_{\mathbb{S}^{n-1}}u\;d\mathcal{H}^{n-1}=0$, then

$\displaystyle||u||_{L^{\infty}}^{n-1}\leq\begin{cases}\pi\|\nabla_{\tau}u\|_{L^{2}}&n=2\\\
4||\nabla_{\tau}u||^{2}_{L^{2}}\log\frac{8e||\nabla_{\tau}u||_{L^{\infty}}^{n-1}}{||\nabla_{\tau}u||_{L^{2}}^{2}}&n=3\\\\[5.69046pt]
C(n)||\nabla_{\tau}u||_{L^{2}}^{2}||\nabla_{\tau}u||_{L^{\infty}}^{n-3}\,\,&n\geq
4.\end{cases}$

######  Lemma 2.9.

Let $u\in W^{1,\infty}(\mathbb{S}^{n-1})$ be a parametrization of a $R-$nearly
spherical set such that $||u||_{W^{1,\infty}}<\varepsilon$ for some
$\varepsilon<R/2$, then

$||\nabla_{\tau}u||_{\infty}\leq 3||u||_{\infty}^{\frac{1}{2}}.$

###  2.5. Some properties of the hybrid asymmetry

In this Section we focus on some properties of the hybrid asymmetry
$\alpha(\Omega)$, see Definition 1.1. First of all, we show that
$\alpha(\Omega)=0$ if and only if $\Omega=A_{R_{1},R_{2}}$.

######  Remark 2.10.

We point out that $\tilde{\mathcal{A}}(\Theta;\Omega_{0})\geq 0$, but it could
be zero even if $\Omega_{0}\neq B_{R_{2}}$ and $\Theta\neq K_{\Omega_{0}}$.
Indeed, if

$d_{\mathcal{H}}(\Theta,K_{\Omega_{0}})<t_{\Omega_{0}}-(R_{2}-R_{1}),$

then $\tilde{\mathcal{A}}(\Theta;\Omega_{0})=0$. Nevertheless, in that case,
$\Omega_{0}\neq B_{R_{2}}$, thus the global asymmetry satisfies

$\alpha(\Omega)=\max\left\\{g(\mathcal{A}_{\mathcal{H}}(\Omega_{0})),\tilde{\mathcal{A}}(\Theta;\Omega_{0})\right\\}=g(\mathcal{A}_{\mathcal{H}}(\Omega_{0}))>0.$

In other words, if $\Theta$ has a low Hausdorff distance from
$K_{\Omega_{0}}$, then the larger term in the computation of $\alpha(\Omega)$
is $g(\mathcal{A}(\Omega_{0}))$. In particular we point out that, provided
that $\Theta\neq K_{\Omega_{0}}$, $\tilde{\mathcal{A}}(\Theta;\Omega_{0})$
could be zero if and only if $\Omega_{0}\neq B_{R_{2}}$.

On the other hand, we stress the fact that $t_{\Omega_{0}}=R_{2}-R_{1}$ if and
only if $\Omega_{0}=B_{R_{2}}$. So, in this case, $K_{\Omega_{0}}=B_{R_{1}}$
and any admissible hole $\Theta\neq B_{R_{1}}$ satisfies
$\tilde{\mathcal{A}}(\Theta;B_{R_{2}})>0$, since $w_{\Omega_{0}}=z\gneq z_{m}$
on $\Theta\setminus B_{R_{1}}$. Consequently, we have

$\alpha(\Omega)=\max\left\\{g(\mathcal{A}_{\mathcal{H}}(B_{R_{2}})),\tilde{\mathcal{A}}(\Theta;B_{R_{2}})\right\\}=\tilde{\mathcal{A}}(\Theta;B_{R_{2}})>0.$

So, it actually holds $\alpha(\Omega)=0$ if and only if
$\Omega=A_{R_{1},R_{2}}$.

Now, we justify the fact that $\alpha(\Omega)$ is actually an asymmetry in the
sense that the set $\Omega$ tends to the spherical shell $A_{R_{1},R_{2}}$ in
the Hausdorff sense if and only if the hydrid asymmetry $\alpha(\Omega)$ tends
to zero.

######  Proposition 2.11.

Let $0<R_{1}<R_{2}<+\infty$ and $\Omega_{j}\subset\mathbb{R}^{n}$ in the class
$\mathcal{T}_{R_{1},R_{2}}$. Then

$\lim_{j\to+\infty}d_{\mathcal{H}}(\Omega_{j},A_{R_{1},R_{2}})=0\iff\lim_{j\to+\infty}\alpha(\Omega_{j})=0.$

###### Proof.

The necessary implication directly follows from Definition 1.1. Therefore, it
only remains to prove the converse condition.

Let $\\{\Omega_{j}\\}_{j\in\mathbb{N}}$ be a sequence such that
$\alpha(\Omega_{j})\to 0$. Since $\Omega_{j}\in\mathcal{T}_{R_{1},R_{2}}$ for
any $j\in\mathbb{N}$, then $\Omega_{j}=(\Omega_{0})_{j}\setminus\Theta_{j}$.
Consequently, from Definition 1.1, it is clear that
$d_{\mathcal{H}}({\Omega_{0}}_{j},B_{R_{2}})\to 0$.

By contradiction, let us assume that $d_{\mathcal{H}}(\Theta j,B_{R_{1}})>c$.
We have two possibilities: the difference between $\Theta$ and $B_{R_{1}}$ is
more concentrated either inside or outside $B_{R_{1}}$. Specifically

  1. (i)

there exists $x_{j}\in\Theta_{j}\cap B_{R_{1}}$ such that
$d(x_{j},B_{R_{1}})>c$ and $d_{\mathcal{H}}(\partial\Theta_{j}\setminus
B_{R_{1}},B_{R_{1}})\to 0$;

  2. (ii)

there exists $x_{j}\in\Theta_{j}\setminus B_{R_{1}}$ such that
$d(x_{j},B_{R_{1}})>c$.

In the case (i), the volume constraint gives the contradiction. Indeed, there
exists $\bar{j}$ such that $|\Theta_{j}|<|B_{R_{1}}|-c$, for any $j>\bar{j}$.
From the volume constraint, we have an absurd, since it would be
$|\Omega_{j}|\geq|A_{R_{1},R_{2}}|+c$.

Now, we consider the case (ii), where a key role is played by the set
$({\Omega_{0}}_{j})_{R_{2}-R_{1}}$, in which, by the definition (8), the test
function $w_{\Omega_{0}}$ is constant.

We point out that, in view of the behavior of the outer asymmetry, we have
$d_{\mathcal{H}}(B_{R_{1}},({\Omega_{0}}_{j})_{R_{2}-R_{1}})\to 0$; in
particular, there exists $c>0$ such that
$d_{\mathcal{H}}(({\Omega_{0}}_{j})_{R_{2}-R_{1}},B_{R_{1}})<c/2$. This
inequality, together with the assumption (ii), implies that there exists two
positive constants $c$ and $c^{\prime}$ such that
$d_{\mathcal{H}}(({\Omega_{0}}_{j})_{R_{2}-R_{1}},\Theta_{j})>c/2$ and, in
view of the inradius constraint in the class $\mathcal{T}_{R_{1},R_{2}}$, it
holds $|\Theta_{j}\setminus({\Omega_{0}}_{j})_{R_{2}-R_{1}}|>c^{\prime}/2$.

Since, by construction and by the volume constraint, we have that
$K_{(\Omega_{0})_{j}}\subseteq({\Omega_{0}}_{j})_{R_{2}-R_{1}}$, then
$d(x)<R_{2}-R_{1}-c/2$ and hence $w_{(\Omega_{0})_{j}}(x)>G(R_{2}-R_{1}-c/2)$
on $\Theta_{j}\setminus K_{(\Omega_{0})_{j}}$. Consequently, there exists
$c^{\prime\prime}>0$ such that

$\tilde{\mathcal{A}}(\Theta_{j},{\Omega_{0}}_{j})=\int_{\Theta_{j}\setminus
K_{(\Omega_{0})_{j}}}w_{(\Omega_{0})_{j}}^{2}(x)-z_{m}^{2}\
dx>c^{\prime\prime}>0,$

that is absurd.

Therefore, in both cases we contradict the assumption $d_{\mathcal{H}}(\Theta
j,B_{R_{1}})>c$. Hence we conclude. ∎

Let us observe that, if we denote by $\chi_{E}$ the characteristic function of
a set $E$, then the weak weighted Fraenkel asymmetry (10) can be also written
as

$\tilde{\mathcal{A}}(\Theta,{\Omega_{0}})=\int_{\mathbb{R}^{n}}\chi_{\Theta\setminus
K_{\Omega_{0}}}(w_{\Omega_{0}}^{2}(x)-z_{m}^{2})dx.$

Finally, it is worth noticing that

$\tilde{\mathcal{A}}(\Theta,{\Omega_{0}})\geq||w_{\Omega_{0}}-z_{m}||^{2}_{L^{2}(\Theta\setminus
K_{\Omega_{0}})}.$

##  3\. A quantitative result for a Steklov-type problem

The quantitative results for the Robin eigenvalue problem with negative
boundary parameter are often solved through Steklov type problems (see [6, 21,
24]). This strategy is suggested by a similar behavior of the functionals in
terms of monotonicity and of eigenfunctions. Indeed, as a first trivial link
between the two functionals, one can observe that the Rayleigh quotients of
both the Robin and the Steklov first eigenvalues are monotonically decreasing
with respect to the boundary integral.

The strategy of the proof is inspired by [21, 24], where the authors relate
the Raylegh quotient of a suitable Steklov-type eigenvalue problem with a
ratio involving only the boundary integral terms. Subsequently, the suitable
representations of the outer and inner boundary are useful to estimate the
difference between the eigenvalues on the radial and on the quasi-radial
shape.

###  3.1. A Steklov-type problem

The auxiliary problem we deal with is the following:

(28) $\sigma_{1}(\Omega)=\min_{\begin{subarray}{c}\psi\in H^{1}(\Omega)\\\
\psi\not\equiv
0\end{subarray}}\dfrac{\displaystyle\int_{\Omega}|\nabla\psi|^{2}+\int_{\Omega}\psi^{2}\;dx}{\displaystyle\int_{\partial\Omega_{0}}\psi^{2}\;d\mathcal{H}^{n-1}}.$

If $w\in H^{1}(\Omega)$ is a minimizer of (29), then it satisfies:

(29) $\begin{cases}-\Delta w+w=0&\mbox{in}\ \Omega\vspace{0.2cm}\\\
\dfrac{\partial w}{\partial\nu}=\sigma_{1}(\Omega)w&\mbox{on}\
\partial\Omega_{0}\vspace{0.2cm}\\\ \dfrac{\partial w}{\partial\nu}=0&\text{on
}\partial\Theta.\end{cases}$

If $\Omega$ is the spherical shell
$A_{R_{1},R_{2}}=B_{R_{2}}\setminus\overline{B}_{R_{1}}$, the unique solution
(up to multiplicative constant) of problem (28) is radial and it is given by

$z(x)=\Psi(|x|),$

where

(30)
$\Psi(r)=r^{1-\frac{n}{2}}K_{\frac{n}{2}}(R_{1})I_{\frac{n}{2}-1}(r)+r^{1-\frac{n}{2}}I_{\frac{n}{2}}(R_{1})K_{\frac{n}{2}-1}(r)\quad\forall
r\in[R_{1},R_{2}].$

The starting point of our analysis is the following upper bound for
$\sigma_{1}(\Omega)$, only involving outer boundary integral terms.

######  Lemma 3.1.

It holds that

(31)
$\sigma_{1}(\Omega)\leq\dfrac{\displaystyle\int_{\partial\Omega_{0}}z\dfrac{\partial
z}{\partial\nu}\;d\mathcal{H}^{n-1}}{\displaystyle\int_{\partial\Omega_{0}}z^{2}\;d\mathcal{H}^{n-1}}=:\dfrac{N(\Omega)}{D(\Omega)},$

where $z(x)=\Psi(|x|)$ is defined in (30). Moreover, equality in (31) holds
iff $\Omega=A_{R_{1},R_{2}}$, that is the spherical shell with the same volume
of $\Omega$ and such that $P(\Omega_{0})=P(B_{R_{2}})$.

###### Proof.

Let $w\in H^{1}(\Omega)$ be a solution of (28). For every $\varphi\in
H^{1}(\Omega)$, we have that

$\displaystyle 0=\displaystyle\int_{\Omega}\varphi
w\;dx-\displaystyle\int_{\Omega}\varphi\Delta
w\;dx=\displaystyle\int_{\Omega}\varphi
w\;dx+\displaystyle\int_{\Omega}\nabla\varphi\nabla
w\;dx-\displaystyle\int_{\partial\Omega_{0}}\varphi\dfrac{\partial
w}{\partial\nu}\;d\mathcal{H}^{n-1}=$
$\displaystyle=\displaystyle\int_{\Omega}\varphi
w\;dx+\displaystyle\int_{\Omega}\nabla\varphi\nabla
w\;dx-\sigma_{1}(\Omega)\displaystyle\int_{\partial\Omega_{0}}\varphi
w\;d\mathcal{H}^{n-1}.$

Therefore, we have

$\sigma_{1}(\Omega)\displaystyle\int_{\partial\Omega_{0}}\varphi
w\;d\mathcal{H}^{n-1}=\displaystyle\int_{\Omega}\varphi
w\;dx+\displaystyle\int_{\Omega}\nabla\varphi\nabla w\;dx.$

If we choose $\varphi=w$, the divergence theorem and the fact that $w$ is a
solution of (29) lead to

$\displaystyle\sigma_{1}(\Omega)$
$\displaystyle=\dfrac{\displaystyle\int_{\Omega}w^{2}\;dx+\displaystyle\int_{\Omega}|\nabla
w|^{2}\;dx}{\displaystyle\int_{\partial\Omega_{0}}w^{2}\;d\mathcal{H}^{n-1}}=$
$\displaystyle=\dfrac{\displaystyle\int_{\Omega}w^{2}\;dx-\displaystyle\int_{\Omega}w\Delta
w\;dx+\displaystyle\int_{\partial\Omega_{0}}w\dfrac{\partial
w}{\partial\nu}\;d\mathcal{H}^{n-1}}{\displaystyle\int_{\partial\Omega_{0}}w^{2}\;d\mathcal{H}^{n-1}}=\dfrac{\displaystyle\int_{\partial\Omega_{0}}w\dfrac{\partial
w}{\partial\nu}\;d\mathcal{H}^{n-1}}{\displaystyle\int_{\partial\Omega_{0}}w^{2}\;d\mathcal{H}^{n-1}}.$

In this way, we have obtained the variational characterization of
$\sigma_{1}(\Omega)$:

(32) $\sigma_{1}(\Omega)=\min_{\begin{subarray}{c}\psi\in H^{1}(\Omega)\\\
\psi\not\equiv
0\end{subarray}}\dfrac{\displaystyle\int_{\partial\Omega_{0}}\psi\dfrac{\partial\psi}{\partial\nu}\;d\mathcal{H}^{n-1}}{\displaystyle\int_{\partial\Omega_{0}}\psi^{2}\;d\mathcal{H}^{n-1}}.$

Finally, let $z$ be the solution of (28) when $\Omega$ is the spherical shell;
the claimed result follows by testing the problem (32) with $z$. ∎

###  3.2. The weighted quotient

Now, our aim is to write the ratio in the right-hand side of (31) in the case
when $\Omega_{0}$ is a $R_{2}$-nearly spherical set. It is worth noticing
that, since in (31) an interior boundary term is not appearing, the same
variational characterization holds when $\Omega$ is a $(R_{1},R_{2})-$nearly
annular set.

Keeping in mind the radial solution (30), we define

(33)
$\eta(r)=\left[K_{\frac{n}{2}}(R_{1})I_{\frac{n}{2}-1}(r)+I_{\frac{n}{2}}(R_{1})K_{\frac{n}{2}-1}(r)\right]\quad\forall
r\in[R_{1},R_{2}],$ (34)
$\mu(r)=\left[K_{\frac{n}{2}}(R_{1})I_{\frac{n}{2}}(r)-I_{\frac{n}{2}}(R_{1})K_{\frac{n}{2}}(r)\right]\quad\forall
r\in[R_{1},R_{2}],$

where the second relation is obtained by deriving the first one with the use
of the derivative rules (16) and (17).

We observe that both the quantities $\eta(r)$ and $\mu(r)$ are strictly
positive. Indeed, $\eta>0$ is trivially verified and, as far as $\mu$ is
concerned, for any $\nu>-1$, we have that

$\dfrac{I_{\nu+1}}{K_{\nu+1}}(R_{1})<\dfrac{I_{\nu+1}}{K_{\nu+1}}(r)$

for any $r>R_{1}$, since

$\frac{d}{dt}\left[\dfrac{I_{\nu}}{K_{\nu}}(r)\right]=\dfrac{\left(I_{\nu-1}(r)+I_{\nu+1}(r)\right)K_{\nu}(r)+I_{\nu}(r)\left(K_{\nu-1}(r)+K_{\nu+1}(r)\right)}{2K_{\nu}(x)^{2}}>0.$

Furthermore, the monotonicity of $I_{\nu}$ and $K_{\nu}$ with respect to the
index (see as a reference [22]) implies that $\eta(r)>\mu(r)$ . Let us observe
that, by (16) and (17), the following derivation rules also hold:

(35)
$\left(r^{1-\frac{n}{2}}\eta(r)\right)^{\prime}=r^{1-\frac{n}{2}}\mu(r);\quad\left(r^{\frac{n}{2}}\mu(r)\right)^{\prime}=r^{\frac{n}{2}}\eta(r);$
(36)
$\eta^{\prime}(r)=\left(\frac{n}{2}-1\right)\frac{\eta(r)}{r}+\mu(r);\qquad\mu^{\prime}(r)=-\frac{n}{2}\frac{\mu(r)}{r}+\eta(r).$

Hence, the functions $\eta$ and $\mu$, defined in (33) and (34) respectively,
lead to the definitions of the integrand functions in the quotient (31), that
are

(37) $h(r)=r^{2-n}\eta^{2}(r)\quad\forall r\in[R_{1},R_{2}],$ (38)
$f(r)=\frac{h^{\prime}(r)}{2}=r^{2-n}\eta(r)\mu(r)\quad\forall
r\in[R_{1},R_{2}].$

Finally, using the parametrization of $R_{2}-$nearly spherical sets (see
Definition (2.4)), we get:

(39)
$N(\Omega)=\displaystyle\int_{\mathbb{S}^{n-1}}f(R_{2}+u)\left(R_{2}+u\right)^{n-1}\;d\mathcal{H}^{n-1},$
(40)
$D(\Omega)=\displaystyle\int_{\mathbb{S}^{n-1}}h(R_{2}+u)\left(R_{2}+u\right)^{n-1}\sqrt{1+\dfrac{|\nabla_{\tau}u|^{2}}{\left(R_{2}+u\right)^{2}}}\;d\mathcal{H}^{n-1}.$

Let us notice that, in the case $\Omega=A_{R_{1},R_{2}}$, we have

$\sigma_{1}(A_{R_{1},R_{2}})=\dfrac{f(R_{2})}{h(R_{2})}.$

The following Lemma is a consequence of the fact that the functions $h$ and
$f$ defined in (37) and (38) are analytic.

######  Lemma 3.2.

For any parametrization couple $u,v\in W^{1,\infty}(\mathbb{S}^{n-1})$ of a
$(R_{1},R_{2})-$nearly annular set and for every $0<\varepsilon<R_{1}/2$ such
that $||u||_{W^{1,\infty}},||v||_{W^{1,\infty}}\leq\varepsilon$, there exists
$C=C(n,R_{1},R_{2})>0$ such that, on $\mathbb{S}^{n-1}$, we have:

(41)
$\left|h(R_{2}+u)-h(R_{2})-h^{\prime}(R_{2})u-h^{\prime\prime}(R_{2})\frac{u^{2}}{2}\right|\leq
K\varepsilon u^{2},$ (42)
$\left|h(R_{1}+v)-h(R_{1})-h^{\prime}(R_{1})v-h^{\prime\prime}(R_{1})\frac{v^{2}}{2}\right|\leq
K\varepsilon v^{2},$ (43)
$\left|f(R_{2}+u)-f(R_{2})-f^{\prime}(R_{2})u-f^{\prime\prime}(R_{2})\frac{u^{2}}{2}\right|\leq
K\varepsilon u^{2}.$

The following monotonicity result will be useful in proving the stability
result for $\sigma_{1}(\Omega)$. The proof generalizes a result contained in
[3].

######  Lemma 3.3.

For any $R_{1}>0$, we have that

$\left(\frac{\eta(r)}{\mu(r)}\right)^{\prime}<0\quad\forall\ r\geq R_{1},$

where $\eta$ and $\mu$ are defined in (33) and (34).

###### Proof.

Since $\eta(r)$ is a linear combination of the modified Bessel functions
$I_{\nu}$ and $K_{\nu}$, it satisfies the following equation:

$\eta^{\prime\prime}+\frac{1}{r}\eta^{\prime}-\frac{r^{2}+(\frac{n}{2}-1)^{2}}{r^{2}}\eta=0,$

that is

$r^{2}\eta^{\prime}\eta^{\prime\prime}+r\eta^{\prime
2}-\left(r^{2}+\left(\frac{n}{2}-1\right)^{2}\right)\eta\eta^{\prime}=0,$

i.e.

$r\eta^{\prime}(r\eta^{\prime})^{\prime}-\left(r^{2}+\left(\frac{n}{2}-1\right)^{2}\right)\eta\eta^{\prime}=0.$

Therefore, dividing by $r\eta^{\prime}$, we obtain:

(44) $(r\eta^{\prime})^{\prime}=\frac{r^{2}+(\frac{n}{2}-1)^{2}}{r}\eta.$

In the same way, we also gain:

(45) $(r\mu^{\prime})^{\prime}=\frac{r^{2}+(\frac{n}{2})^{2}}{r}\mu.$

By using (44) and (45), we are able to write the following equality

$\int_{0}^{r}[\eta(s\mu^{\prime})^{\prime}-\mu(s\eta^{\prime})^{\prime}]ds=(n-1)\int_{0}^{r}\frac{\eta(s)\mu(s)}{s}ds>0.$

By applying the Green Theorem on the l.h.s, we have

$s[\eta\mu^{\prime}-\mu\eta^{\prime}]_{0}^{r}=(n-1)\int_{0}^{r}\frac{\eta(s)\mu(s)}{s}ds>0,$

that leads to

(46)
$\frac{\mu^{\prime}(r)}{\mu(r)}-\frac{\eta^{\prime}(r)}{\eta(r)}=\frac{n-1}{r\eta(r)\mu(r)}\int_{0}^{r}\frac{\eta(s)\mu(s)}{s}ds>0\quad\forall
r>0.$

By using the relation (46), the thesis follows by observing that

$\left(\dfrac{\eta(r)}{\mu(r)}\right)^{\prime}=\frac{\eta^{\prime}(r)\mu(r)-\eta(r)\mu^{\prime}(r)}{\mu^{2}(r)}=\frac{\eta^{\prime}(r)\mu(r)-\eta(r)\mu^{\prime}(r)}{\mu(r)\eta(r)}\frac{\eta(r)}{\mu(r)}=\left(\frac{\eta^{\prime}(r)}{\eta(r)}-\frac{\mu^{\prime}(r)}{\mu(r)}\right)\frac{\eta(r)}{\mu(r)}<0.$

∎

We end this section with a very technical Lemma establishing the sign of a
certain quantity, that will be useful to obtain the stability result. The
proof is based on the second part of [24, Prop. 2.5]. For sake of simplicity,
we set:

(47) $\begin{split}A_{0}=&R_{2}^{n-1}f(R_{2})h(R_{2}),\\\
A_{1}=&R_{2}^{n-1}[f(R_{2})h^{\prime}(R_{2})-f^{\prime}(R_{2})h(R_{2})],\\\
A_{2}=&R_{2}^{n-1}[f(R_{2})h^{\prime\prime}(R_{2})-f^{\prime\prime}(R_{2})h(R_{2})].\end{split}$

######  Lemma 3.4.

For any $R_{2}>0$, we have that

(48) $(n-1)R_{2}A_{1}+R_{2}^{2}A_{2}+2nA_{0}>0.$

###### Proof.

If we put in evidence $R_{2}^{2}$ in (48), we obtain
$R_{2}^{2}[\frac{(n-1)}{R_{2}}A_{1}+A_{2}+\frac{2n}{R_{2}^{2}}A_{0}]>0$ and,
so, we have to prove that

$\left[\frac{(n-1)}{R_{2}}A_{1}+A_{2}+\frac{2n}{R_{2}^{2}}A_{0}\right]=[r^{n-1}(f(r)h^{\prime}(r)-h(r)f^{\prime}(r)]^{\prime}+2nr^{n-3}h(r)f(r)>0$

for $r=R_{2}$. By using the definitions of $h$ and $f$ in (37)-(38) and the
derivation rules (35), the last relation becomes

$[r^{2-n}(r\eta^{2}(r)\mu^{2}(r)-r\eta^{4}(r)+(n-1)\eta^{3}(r)\mu(r))]^{\prime}+2nr^{1-n}\eta^{3}(r)\mu(r)>0$

and hence

$r^{1-n}\eta(r)\mu(r)[2r^{2}\mu^{2}(r)+2(n-1)r\eta(r)\mu(r)+(n+1-2r^{2})\eta^{2}(r)]>0.$

If we set

$\zeta(r):=2r^{2}\mu^{2}(r)+2(n-1)r\eta(r)\mu(r)+(n+1-2r^{2})\eta^{2}(r),$

we need to show that $\zeta(r)>0$. We observe that

$\zeta^{\prime}(r)=\frac{1}{r}\bigl{(}\zeta(r)+4r\eta(r)\mu(r)+(n^{2}-2n-3)\eta^{2}(r)\bigr{)}.$

If $n\geq 3$, since $n^{2}-2n-3\geq 0$ and $\eta(r),\mu(r)>0$ for any $r>0$,
we deduce that $r\zeta^{\prime}(r)\geq\zeta(r)$ and hence that $\zeta(r)>0$
for any $r>0$. Otherwise, if $n=2$, we observe that the function
$\xi(r)=r\zeta^{\prime}(r)$ is increasing for $r>0$ because, by using (36), we
have that

$\xi^{\prime}(r)=(r\zeta^{\prime}(r))^{\prime}=6r\mu^{2}(r)+2r\eta^{2}(r)>0\quad\forall
r>0.$

Moreover, it is immediate that $\xi^{\prime}(0)=\xi(0)=0$. This means that
$\xi(r)>0$ and hence $\zeta^{\prime}(r)>0$ for any $r>0$. Finally, since
$\zeta(0)=0$, this implies that $\zeta(r)>0$ for any $r>0$. ∎

###  3.3. The stability result

A crucial result to obtain the stability with respect to the outer boundary is
the following stability issue for the functional
$\Omega\mapsto\frac{N(\Omega)}{D(\Omega)}$. Up to some necessary technical
modification, the proof follows the scheme in [21, 24].

######  Proposition 3.5.

Let $0<R_{1}<R_{2}<+\infty$. There exists $C_{1}=C_{1}(n,R_{1},R_{2})>0$ such
that, for any $(R_{1},R_{2})-$nearly annular set $\Omega$ with the outer
boundary parametrized by $u\in W^{1,\infty}(\mathbb{S}^{n-1})$, and for every
$0<\varepsilon<R_{1}/2$ such that $||u||_{W^{1,\infty}}\leq\varepsilon$, the
following stability inequality holds true:

$\frac{N(\Omega^{\\#})D(\Omega)-D(\Omega^{\\#})N(\Omega)}{n\omega_{n}}\geq
C_{1}(n,R_{1},R_{2})||\nabla_{\tau}u||_{L^{2}}^{2}.$

###### Proof.

We start by performing a suitable “Taylor expansion” in terms of the
deformations of $\partial\Omega_{0}$ compared to $\partial B_{R_{2}}$. By
using the characterization of $N(\Omega)$ and $D(\Omega)$ in (39) and (40), we
obtain

$\begin{split}\frac{N(\Omega^{\\#})D(\Omega)-D(\Omega^{\\#})N(\Omega)}{n\omega_{n}}&=f(R_{2})\int_{\mathbb{S}^{n-1}}h(R_{2}+u)(R_{2}+u)^{n-1}\sqrt{1+\frac{|\nabla_{\tau}u|^{2}}{(R_{2}+u)^{2}}}\;d\mathcal{H}^{n-1}\\\
&\quad-h(R_{2})\int_{\mathbb{S}^{n-1}}f(R_{2}+u)(R_{2}+u)^{n-1}\;d\mathcal{H}^{n-1}.\end{split}$

Then, by using the Lemmata 2.6 and 3.2, we get

(49)
$\begin{split}\frac{N(\Omega^{\\#})D(\Omega)-D(\Omega^{\\#})N(\Omega)}{n\omega_{n}}&\geq\int_{\mathbb{S}^{n-1}}R_{2}^{2n-2}[f(R_{2})h^{\prime}(R_{2})-f^{\prime}(R_{2})h(R_{2})]u\;d\mathcal{H}^{n-1}\\\
&\quad+\int_{\mathbb{S}^{n-1}}R_{2}^{2n-3}[2(n-1)(f(R_{2})h^{\prime}(R_{2})-f^{\prime}(R_{2})h(R_{2}))]\frac{u^{2}}{2}\;d\mathcal{H}^{n-1}\\\
&\quad+\int_{\mathbb{S}^{n-1}}R_{2}^{2n-2}[f(R_{2})h^{\prime\prime}(R_{2})-f^{\prime\prime}(R_{2})h(R_{2})]\frac{u^{2}}{2}\;d\mathcal{H}^{n-1}\\\
&\quad+\int_{\mathbb{S}^{n-1}}R_{2}^{2n-4}f(R_{2})h(R_{2})\frac{|\nabla_{\tau}u|^{2}}{2}\;d\mathcal{H}^{n-1}-\varepsilon
C_{1}\|\nabla_{\tau}u\|^{2}_{L^{2}}.\\\ \end{split}$

Now, using the notation in (47), inequality (49) becomes

(50)
$\begin{split}\frac{N(\Omega^{\\#})D(\Omega)-D(\Omega^{\\#})N(\Omega)}{n\omega_{n}}&\geq
R_{2}^{n-1}A_{1}\int_{\mathbb{S}^{n-1}}u\;d\mathcal{H}^{n-1}+R_{2}^{n-2}[2(n-1)A_{1}+R_{2}A_{2}]\int_{\mathbb{S}^{n-1}}\frac{u^{2}}{2}\;d\mathcal{H}^{n-1}\\\
&\quad+R_{2}^{n-3}A_{0}\int_{\mathbb{S}^{n-1}}\frac{|\nabla_{\tau}u|^{2}}{2}\;d\mathcal{H}^{n-1}-\varepsilon
C_{1}\|\nabla_{\tau}u\|^{2}_{L^{2}}.\end{split}$

Moreover, by the perimeter constraint given by (24), then (50) becomes

(51)
$\begin{split}\frac{N(\Omega^{\\#})D(\Omega)-D(\Omega^{\\#})N(\Omega)}{n\omega_{n}}&\geq
R_{2}^{n-2}\frac{nA_{1}+R_{2}A_{2}}{2}\int_{\mathbb{S}^{n-1}}u^{2}\;d\mathcal{H}^{n-1}\\\
&\quad+R_{2}^{n-3}\left(\frac{A_{0}}{2}-\frac{R_{2}A_{1}}{2(n-1)}\right)\int_{\mathbb{S}^{n-1}}|\nabla_{\tau}u|^{2}\;d\mathcal{H}^{n-1}-\varepsilon
C_{1}\|\nabla_{\tau}u\|^{2}_{L^{2}}.\end{split}$

Firstly, let us note that the term in round parenthesis is positive because it
is the sum of two positive terms. Indeed, $A_{0}$, defined in (47), is
positive, since $h>0$ and $f>0$ (defined respectevely in (37) and (38));
meanwhile

(52)
$A_{1}=R_{2}^{n-1}[f(R_{2})h^{\prime}(R_{2})-f^{\prime}(R_{2})h(R_{2})]=f^{2}(R_{2})\left[\frac{h(r)}{f(r)}\right]^{\prime}_{r=R_{2}}=f^{2}(R_{2})\left[\frac{\eta(r)}{\mu(r)}\right]^{\prime}_{r=R_{2}}<0,$

where the last inequality follows from Lemma 3.3.

Therefore, since it is possible to take $\varepsilon C_{1}$ arbitrarily small,
where the quantity $C_{1}$ does not depend on $\varepsilon$, the proof is
concluded if we verify that

(53) $\frac{nA_{1}+R_{2}A_{2}}{2}\geq 0.$

Otherwise, if (53) does not hold (i.e. if $\frac{nA_{1}+R_{2}A_{2}}{2}<0$), we
go on with the estimate from below. More precisely, by using the Poicarè
inequality (27), from (51), we get

$\begin{split}\frac{N(\Omega^{\\#})D(\Omega)-D(\Omega^{\\#})N(\Omega)}{n\omega_{n}}&\geq\frac{R_{2}^{n-3}}{4n}\left(\frac{n^{2}-3n}{n-1}R_{2}A_{1}+R_{2}^{2}A_{2}+2nA_{0}\right)\int_{\mathbb{S}^{n-1}}|\nabla_{\tau}u|^{2}\;d\mathcal{H}^{n-1}\\\
&\quad-\varepsilon C_{1}\|\nabla_{\tau}u\|^{2}_{L^{2}}.\end{split}$

Finally, in order to conclude the proof, it remains to show that

$\left(\frac{n^{2}-3n}{n-1}R_{2}A_{1}+R_{2}^{2}A_{2}+2nA_{0}\right)>0.$

This term can be written as the sum of two positive quantities in the
following way:

$\frac{n^{2}-3n}{n-1}R_{2}A_{1}+R_{2}^{2}A_{2}+2nA_{0}=\left[(n-1)R_{2}A_{1}+R_{2}^{2}A_{2}+2nA_{0}\right]-\frac{n+1}{n-1}R_{2}A_{1}.$

Indeed, the positive sign of the first addendum of the right hand side of the
last equality is proved in Lemma 3.4 and the sign of the second one follows
from (52). ∎

An immediate consequence of Proposition 3.5 is the following stability result
for $\sigma_{1}(\Omega)$.

######  Proposition 3.6.

Let $0<R_{1}<R_{2}<+\infty$. There exists $C_{1}=C_{1}(n,R_{1},R_{2})>0$ such
that, for any $(R_{1},R_{2})-$nearly annular set $\Omega$ with the outer
boundary parametrized by $u\in W^{1,\infty}(\mathbb{S}^{n-1})$, and for every
$0<\varepsilon<R_{1}/2$ such that $||u||_{W^{1,\infty}}\leq\varepsilon$, the
following stability inequality holds true:

$\sigma_{1}(\Omega^{\\#})-\sigma_{1}(\Omega)\geq
C_{1}(n,R_{1},R_{2})||\nabla_{\tau}u||_{L^{2}}^{2}.$

Moreover, the constant $C_{1}(n,R_{1},R_{2})$ depends continuously (actually
analytically) on $R_{1}$ and $R_{2}$.

###### Proof.

The proof follows by combining Lemma 3.1 and Proposition 3.5. ∎

##  4\. Proof of the Main Result

In this section we provide the proof of the main stability result. First of
all, we show that we can reduce our study to the nearly annular sets and then
we use the results of Section 3.3 related to the auxiliary problem (28) to
gain the outer stability result. The strategy follows an idea developed in
[21, 24], linking the stability of the Steklov-Neumann eigenvalue (28) to the
stability of the Robin-Neumann eigenvalue (1).

Furthermore, in the last part of the section, we develop the key idea of
considering an additional term to take into account the asymmetry of the inner
boundary.

###  4.1. Towards the nearly spherical sets

We want to show that the stability result gets meaningful if we reduce to
nearly annular sets. To do that, the first step is to provide a uniform bound
on the diameters of the sets whose eigenvalues are "not so far" from the
optimum. This kind of isodiametric control of the eigenvalues is common when
dealing with maximization problems in shape optimization and it often provides
extra compactness when working on existence problems without a bounded design
region (see for instance the isodiametric control of the Robin spectrum proved
in [16] or for the Steklov spectrum proved in [10], both valid also for higher
eigenvalues in a wider class of sets).

######  Lemma 4.1.

Let $0<R_{1}<R_{2}<+\infty$. There exists a positive constant
$C(R_{1},R_{2},n,\beta)$ such that, for every
$\Omega=\Omega_{0}\setminus\overline{\Theta}\subset\mathbb{R}^{n}$ with
$\Omega_{0}$ and $\Theta$ convex sets such that
$\overline{\Theta}\subset\Omega_{0}$, $P(\Omega_{0})=P(B_{R_{2}})$ and
$|\Omega|=|A_{R_{1},R_{2}}|$, if

$\lambda_{1}({\beta},\Omega)>2\lambda_{1}({\beta},A_{R_{1},R_{2}})$

then

$\text{diam}(\Omega)<C(R_{1},R_{2},n,\beta).$

###### Proof.

The proof is a straightforward adaptation to our context of [21, Lemma 3.5].

Let us argue by contradiction and suppose that there exists a sequence
$\\{\Omega_{j}\\}_{j\in\mathbb{N}}$ of domains of the form
$\Omega_{j}={\Omega_{0}}_{j}\setminus\overline{\Theta_{j}}$ such that

$\lambda_{1}({\beta},A_{R_{1},R_{2}})\geq\lambda_{1}({\beta},\Omega_{j})>2\lambda_{1}({\beta},A_{R_{1},R_{2}})\
\text{and}\ \text{diam}(\Omega_{j})=\text{diam}({\Omega_{0}}_{j})\to+\infty.$

In view of the convexity of ${\Omega_{0}}_{j}$ and of the constraint
$P({\Omega_{0}}_{j})=m$, the sequence $\\{\rho_{j}\\}_{j\in\mathbb{N}}$ of the
inradii of $\\{{\Omega_{0}}_{j}\\}_{j\in\mathbb{N}}$ is necessarily vanishing.
Recalling that, for any convex set $A$ with inradius $\rho$, it holds

$|A|\leq\rho P(A)$

(see, for instance, [15, Prop. 2.4.3]), we deduce that $|{\Omega_{0}}_{j}|$
vanishes as $j$ goes to $+\infty$. Now, using the charachteristic function
$\chi_{\Omega_{j}}$ as a test for $\lambda_{1}({\beta},\Omega_{j})$, we obtain

$\lambda_{1}({\beta},\Omega_{j})\leq-\beta\frac{P({\Omega_{0}}_{j})}{|\Omega_{j}|}\leq-\beta\frac{P({\Omega_{0}}_{j})}{{|{\Omega_{0}}_{j}}|}\to-\infty,$

in contradiction with the lower bound on $\lambda_{1}({\beta},\Omega_{j})$. ∎

We point out that the previous result is immediate if we restrict to the class
$\mathcal{T}_{R_{1},R_{2}}$, defined in (5). Now, we provide two useful
semicontinuity results: the first one gives the lower semicontinuity of the
boundary integral term and the second one give the upper semicontinuity of the
map $\Omega\mapsto\lambda_{1}(\beta,\Omega)$.

######  Lemma 4.2 ([20], Proposition 2.1).

Let $E_{j},E\subset\mathbb{R}^{n}$ be convex domains such that $E_{j}\to E$ in
the sense of Hausdorff and in measure. Let $w_{j},w\in H^{1}(\mathbb{R}^{n})$.
If $w_{j}\rightharpoonup w$ in $H^{1}(\mathbb{R}^{n})$, then it holds

$\int_{\partial E}w^{2}\>d\sigma\leq\liminf_{j}\int_{\partial
E_{j}}w_{j}^{2}\>d\sigma$

######  Lemma 4.3.

Let $\Omega_{j},\Omega\subset\mathbb{R}^{n}$ be in the class
$\mathcal{T}_{R_{1},R_{2}}$, with $\Omega_{j}\to\Omega$ in the sense of
Hausdorff and in measure. Then

$\limsup_{j\to+\infty}\lambda_{1}(\beta,\Omega_{j})\leq\lambda_{1}(\beta,\Omega).$

###### Proof.

We proceed similarly to [20, Prop. 3.1], being careful to the fact that the
boundary integral keeps into account only $\partial\Omega_{0}$. Let $u\in
H^{1}(\Omega)$ an eigenfunction for $\lambda_{1}(\beta,\Omega)$, and let
$\overline{u}\in H^{1}(\mathbb{R}^{n})$ an extension of $u$ to the whole of
$\mathbb{R}^{n}$. Let us notice that, in view of the hypotheses, the
convergence of the convex sets ${\Omega_{0}}_{j}\to\Omega_{0}$ in the sense of
Hausdorff and in measure holds. Thus we have

$\int_{\partial\Omega_{0}}u^{2}d\mathcal{H}^{n-1}=\int_{\partial\Omega_{0}}\overline{u}^{2}d\mathcal{H}^{n-1}\leq\liminf_{j\to+\infty}\int_{\partial{\Omega_{0}}_{j}}\overline{u}^{2}d\mathcal{H}^{n-1}$

as a consequence of Lemma 4.2. Moreover, both volume integrals are continuous
in view of the convergence in measure $\Omega_{j}\to\Omega$ and this implies
the upper semicontinuity of the Rayleigh quotients, defined in (2). Therefore,
we have

$\lambda_{1}({\beta},\Omega)=R_{\Omega}(u)=R_{\Omega}(\overline{u})\geq\limsup_{j\to+\infty}R_{\Omega_{j}}(\overline{u})\geq\limsup_{j\to+\infty}\lambda_{1}(\beta,\Omega_{j}).$

∎

Since $A_{R_{1},R_{2}}$ is the unique maximizer of $\lambda_{1}(\beta,\cdot)$
in our class of admissible sets, as a consequence of the upper semicontinuity
of $\lambda_{1}(\beta,\Omega)$ and of the isodiametric control in Lemma 4.1,
the following convergence result holds.

######  Lemma 4.4.

Let $\\{\Omega_{j}\\}_{j\in\mathbb{N}}\subset\mathcal{T}_{R_{1},R_{2}}$ be a
maximizing sequence for Problem (6) with $\Omega_{j}$ is barycentered at the
origin. Then

$d_{\mathcal{H}}(\Omega_{j},A_{R_{1},R_{2}})\to 0.$

###### Proof.

Thanks to Lemma 4.1, the diameters of the convex sets ${\Omega_{0}}_{j}$ and
$\Theta_{j}$ are uniformly bounded.

The uniform upper bound on the diameters and the uniform lower bound
$\rho(\Theta_{j})>\vartheta_{R_{1},R_{2}}$ imply that, up to subsequences,
there exist an open bounded convex set $\Omega_{0}$ and an open bounded convex
set $\Theta\subset\subset\Omega_{0}$ such that
$\Omega:=\Omega_{0}\setminus\overline{\Theta}\in\mathcal{T}_{R_{1},R_{2}}$ and
$d_{\mathcal{H}}(\Omega_{j},\Omega)\to 0$. Since $\lambda_{1}(\beta,\cdot)$ is
upper semicontinuous in view of Lemma 4.3, we obtain that

$\max_{\mathcal{A}_{R_{1},R_{2}}}\lambda_{1}(\beta,\cdot)=\limsup_{j\to+\infty}\lambda_{1}(\beta,\Omega_{j})\leq\lambda_{1}(\beta,\Omega).$

So, since the spherical shell $A_{R_{1},R_{2}}$ is the unique maximizer for
$\lambda_{1}(\beta,\cdot)$ in $\mathcal{T}_{R_{1},R_{2}}$, we get that
$\Omega=A_{R_{1},R_{2}}$ necessarily. ∎

As a consequence of the previous lemma, we can actually restrict our main
stability result to nearly spherical sets barycentered in the origin.

######  Lemma 4.5.

Let $0<R_{1}<R_{2}<+\infty$. There exists a positive constant
$\delta_{0}=\delta_{0}(n,\beta,R_{1},R_{2})$ such that, if
$\Omega\in\mathcal{T}_{R_{1},R_{2}}$ and

$\lambda_{1}(\beta,A_{R_{1},R_{2}})-\lambda_{1}(\beta,\Omega)\leq\delta_{0}$

then, up to a translation, $\Omega$ is a $(R_{1},R_{2})-$nearly annular set.

###### Proof.

The proof is based on a straightforward contradiction argument. Indeed, let us
suppose that, for every $j\in\mathbb{N}$, there exists an admissible set
$\Omega_{j}={\Omega_{0}}_{j}\setminus\overline{\Theta_{j}}$ such that

(54)
$\lambda_{1}(\beta,A_{R_{1},R_{2}})-\lambda_{1}(\beta,\Omega_{j})\leq\frac{1}{j}$

and any translations of $\Omega_{j}$ is not a $(R_{1},R_{2})-$nearly annular
set, i.e.

(55) $\text{either}\quad
d_{\mathcal{H}}({\Omega_{0}}_{j},B_{R_{2}})>\frac{R_{2}}{2}\quad\text{or}\quad
d_{\mathcal{H}}(\Theta_{j},B_{R_{1}})>\frac{R_{1}}{2}.$

By (54), $\\{\Omega_{j}\\}_{j\in\mathbb{N}}$ is a maximizing sequence for
Problem (6); if we consider the sequence (still denoted by
$\\{\Omega_{j}\\}_{j\in\mathbb{N}}$) of translated sets with barycenter at the
origin, we are under the hypotheses of Lemma 4.4 and thus we get
$d_{\mathcal{H}}(\Omega_{j},A_{R_{1},R_{2}})\to 0$, that is in contradiction
with (55). ∎

###  4.2. Outer asymmetry

At this stage, we are able to restrict our study to the class of nearly
annular sets. We will use the quantitative estimate given through the Steklov-
Neumann eigenvalues in Section 3.3.

######  Proposition 4.6.

Let $\delta>0$. For any $(R_{1},R_{2})-$nearly annular set $\Omega$, with
$P(\Omega_{0})=n\omega_{n}R_{2}^{n-1}$ and $|\Omega|=|A_{R_{1},R_{2}}|$,
having the outer boundary parametrized by $u\in
W^{1,\infty}(\mathbb{S}^{n-1})$, and for every $0<\varepsilon<R_{1}/2$ such
that $||u||_{W^{1,\infty}}\leq\varepsilon$, if
$\lambda_{1}(\beta,A_{R_{1},R_{2}})-\lambda_{1}(\beta,\Omega)<\delta$, then

(56) $\lambda_{1}(\beta,A_{R_{1},R_{2}})-\lambda_{1}(\beta,\Omega)\geq
C(n,\beta,R_{1},R_{2},\delta)\|\nabla_{\tau}u\|_{L^{2}}^{2}.$

###### Proof.

Let $\delta>0$ and let $\Omega$ be chosen as in the statement. The map
$\beta\mapsto\lambda_{1}(\beta,\Omega)$ is continuous and monotonically
increasing from $(-\infty,0)$ onto $(-\infty,0)$. Then, there exists
$\bar{\beta}\leq\beta$ such that

$\lambda_{1}({\beta},\Omega)=\lambda_{1}({\bar{\beta}},A_{R_{1},R_{2}}).$

Hence,

$\lambda_{1}({\beta},A_{R_{1},R_{2}})-\lambda_{1}({\beta},\Omega)=\lambda_{1}({\beta},A_{R_{1},R_{2}})-\lambda_{1}({\bar{\beta}},A_{R_{1},R_{2}}).$

Let us consider the positive constant

$\kappa:=\sqrt{|\lambda_{1}({\beta},\Omega)|}=\sqrt{|\lambda_{1}({\bar{\beta}},A_{R_{1},R_{2}})|};$

for the rescaled sets $\kappa\Omega$ and $\kappa A_{R_{1},R_{2}}$ it holds
that

$\displaystyle-\kappa^{2}$ $\displaystyle=\lambda_{1}({\beta},\Omega)$
$\displaystyle=\kappa^{2}\lambda_{1}\left({\frac{\beta}{\kappa}},\kappa\Omega\right)\leq\kappa^{2}\frac{\displaystyle\int_{\kappa\Omega}|\nabla\psi|^{2}\>dx+\frac{\beta}{\kappa}\int_{\partial(\kappa\Omega_{0})}\psi^{2}\>d\mathcal{H}^{n-1}}{\displaystyle\int_{\kappa\Omega}\psi^{2}\>dx}\quad\forall\psi\in
H^{1}(\kappa\Omega)$

and

$\displaystyle-\kappa^{2}$
$\displaystyle=\lambda_{1}({\bar{\beta}},A_{R_{1},R_{2}})$
$\displaystyle=\kappa^{2}\lambda_{1}\left({\frac{\bar{\beta}}{\kappa}},\kappa
A_{R_{1},R_{2}}\right)\leq\kappa^{2}\frac{\displaystyle\int_{\kappa
A_{R_{1},R_{2}}}|\nabla\Psi|^{2}\>dx+\frac{\bar{\beta}}{\kappa}\int_{\partial(\kappa
B_{R_{2}})}\Psi^{2}\>d\mathcal{H}^{n-1}}{\displaystyle\int_{\kappa
A_{R_{1},R_{2}}}\Psi^{2}\>dx}\quad\forall\Psi\in H^{1}(\kappa
A_{R_{1},R_{2}}),$

with equality holding if $\psi$ and $\Psi$ are, respectively, the
eigenfunctions for the Robin-Neumann problem on $\kappa\Omega$ with parameter
$\beta$ and on $\kappa A_{R_{1},R_{2}}$ with boundary parameter $\bar{\beta}$.
Thus we get

$-\frac{\beta}{\kappa}\leq\frac{\displaystyle\int_{\kappa\Omega}|\nabla\psi|^{2}\>dx+\int_{\kappa\Omega}\psi^{2}\>dx}{\displaystyle\int_{\partial(\kappa\Omega_{0})}\psi^{2}\>d\mathcal{H}^{n-1}}\quad\forall\psi\in
H^{1}(\kappa\Omega)$

and

$-\frac{\bar{\beta}}{\kappa}\leq\frac{\displaystyle\int_{\kappa
A_{R_{1},R_{2}}}|\nabla\Psi|^{2}\>dx+\int_{\kappa
A_{R_{1},R_{2}}}\Psi^{2}\>dx}{\displaystyle\int_{\partial(\kappa
B_{R_{2}})}\Psi^{2}\>d\mathcal{H}^{n-1}}\quad\forall\Psi\in H^{1}(\kappa
A_{R_{1},R_{2}}).$

It follows that the infimum for the Steklov-Neumann problem (29) is achieved
on $\kappa\Omega$ and $\kappa A_{R_{1},R_{2}}$ if $\psi$ and $\Psi$ are,
respectively, the eigenfunctions for the Robin-Neumann problem on
$\kappa\Omega$ with parameter $\beta$ and on $\kappa A_{R_{1},R_{2}}$ with
boundary parameter $\bar{\beta}$. Therefore, we obtain

$\sigma_{1}(\kappa\Omega)=-\frac{\beta}{\kappa},\quad\sigma_{1}(\kappa
A_{R_{1},R_{2}})=-\frac{\bar{\beta}}{\kappa}.$

If we denote with $z_{\beta}$ the first eigenfunction of $A_{R_{1},R_{2}}$ for
the Robin-Neumann problem with parameter $\beta$, Using the variational
characterization of $\lambda_{1}(\beta,\Omega)$ and
$\lambda_{1}({\bar{\beta}},A_{R_{1},R_{2}})$ we have that

$\displaystyle\lambda_{1}(\beta,A_{R_{1},R_{2}})-\lambda_{1}({\beta},\Omega)$
$\displaystyle=\lambda_{1}({\beta},A_{R_{1},R_{2}})-\lambda_{1}({\bar{\beta}},A_{R_{1},R_{2}})\geq({\beta}-\bar{\beta})\frac{\displaystyle\int_{\partial
B_{R_{2}}}z_{\beta}^{2}\>d\mathcal{H}^{n-1}}{\displaystyle\int_{A_{R_{1},R_{2}}}z_{\beta}^{2}\>dx}$
$\displaystyle=C_{2}(n,\beta,R_{1},R_{2})(\beta-\bar{\beta})\geq\kappa
C_{2}(n,\beta,R_{1},R_{2})(\sigma_{1}(\kappa
A_{R_{1},R_{2}})-\sigma_{1}(\kappa\Omega))$ $\displaystyle\geq\kappa
C_{2}(n,\beta,R_{1},R_{2})C_{1}(n,\kappa^{\frac{1}{n}}R_{1},\kappa^{\frac{1}{n}}R_{2})||\nabla_{\tau}u||_{L^{2}}^{2},$

where $C_{1}$ is the constant found in Proposition 3.5. The conclusion follows
by setting $C:=\kappa C_{2}C_{1}$. ∎

Now we are in position to prove the main stability result regarding the outer
boundary.

######  Theorem 4.7.

Let $0<R_{1}<R_{2}<+\infty$. There exist two positive constants
$C(n,R_{1},R_{2},\beta)$ and $\delta_{0}(n,R_{1},R_{2},\beta)$, such that, for
every $\Omega\in\mathcal{T}_{R_{1},R_{2}}$, if
$\lambda_{1}(\beta,A_{R_{1},R_{2}})-\lambda_{1}(\beta,\Omega)\leq\delta_{0}$,
we have

$\lambda_{1}(\beta,A_{R_{1},R_{2}})-\lambda_{1}(\beta,\Omega)\geq
C(n,R_{1},R_{2},\beta)g(\mathcal{A}_{\mathcal{H}}(\Omega_{0}))$

where $\mathcal{A}_{\mathcal{H}}$ is the Hausdorff asymmetry defined in (7)
and $g$ is the positive increasing function defined by

$g(s)=\begin{cases}s^{2}&\text{if $n=2$}\\\ f^{-1}(s^{2})&\text{if $n=3$}\\\
s^{\frac{n+1}{2}}&\text{if $n\geq 4$},\end{cases}$

with $f(t)=\sqrt{t\log\left(\frac{1}{t}\right)}$ for $0<t<e^{-1}$.

###### Proof.

Lemma 4.5 ensures us that, if $\delta_{0}$ is small enough, then we can
suppose without loss of generality that $\Omega$ is a nearly
$(R_{1},R_{2})-$annular set with barycenter at the origin,
$P(\Omega_{0})=n\omega_{n}R_{2}^{n-1}=P(B_{R_{2}})$ and
$|\Omega|=A_{R_{1},R_{2}}$. Thus, as far as the outer boundary, we have that

$\partial\Omega_{0}=\\{\xi(R_{2}+u(\xi)),\,\,\xi\in\mathbb{S}^{n-1}\\}$

for some $u\in W^{1,\infty}(\mathbb{S}^{n-1})$ with
$\|u\|_{W^{1,\infty}}<R_{2}/2$. Let $\rho_{2}$ such that
$|B_{\rho_{2}}|=|\Omega_{0}|$; then, we have that

(57)
$\frac{1}{n}\int_{\mathbb{S}^{n-1}}(R_{2}+u)^{n}d\mathcal{H}^{n-1}=\omega_{n}\rho_{2}^{n}.$

Now, let us define the function $h:=(R_{2}+u)^{n}-\rho_{2}^{n}$. By a
straightforward expansion of the left hand side of (57), we get

$R_{2}^{n}-\rho_{2}^{n}=-\frac{1}{n\omega_{n}}\displaystyle\int_{\mathbb{S}^{n-1}}\sum_{k=1}^{n}\binom{n}{k}u^{k}R_{2}^{n-k}d\mathcal{H}^{n-1},$

which immediately implies $|R_{2}-\rho_{2}|<C(n)\|u\|_{W^{1,\infty}}$. Hence,
it holds

(58) $C_{2}\|h\|_{L^{\infty}}\leq\|u\|_{L^{\infty}}\leq
C_{3}\|h\|_{L^{\infty}},$

where $C_{2}$ and $C_{3}$ are constant depending only on the dimension.
Moreover, since

$\nabla_{\tau}h=n(R_{2}+u)^{n-1}\nabla_{\tau}u,$

then

$C_{4}\|\nabla_{\tau}u\|_{L^{2}}\leq\|\nabla_{\tau}h\|_{L^{2}}\leq
C_{5}\|\nabla_{\tau}u\|_{L^{2}},$

where $C_{4},C_{5}$ depend on the dimension and on $R_{2}$. Moreover, from
(57), we know that $h$ has zero integral, thus we can apply Lemma 2.8 to $h$
and use (58) to infer

(59)
$\displaystyle||u||_{L^{\infty}}^{n-1}\leq\begin{cases}\pi\|\nabla_{\tau}u\|_{L^{2}}&n=2\\\
4||\nabla_{\tau}u||^{2}_{L^{2}}\log\frac{8e||\nabla_{\tau}u||_{L^{\infty}}^{n-1}}{||\nabla_{\tau}u||_{L^{2}}^{2}}&n=3\\\\[5.69046pt]
C(n)||\nabla_{\tau}u||_{L^{2}}^{2}||\nabla_{\tau}u||_{L^{\infty}}^{n-3}\,\,&n\geq
4.\end{cases}$

For the sake of brevity, we show the conclusion of the proof only for $n\geq
4$, since for $n=2,3$ the argument is the same. Inequality (59) and Lemma 2.9
lead to

$\|u\|_{L^{\infty}(\mathbb{S}^{n-1})}^{n-1}\leq
C(n)\|D_{\tau}u\|_{L^{2}(\mathbb{S}^{n-1})}^{2}\|u\|_{L^{\infty}(\mathbb{S}^{n-1})}^{\frac{n-3}{2}},$

and hence we have

$\|u\|_{L^{\infty}(\mathbb{S}^{n-1})}^{\frac{n+1}{2}}\leq
C(n)\|D_{\tau}u\|_{L^{2}(\mathbb{S}^{n-1})}^{2}.$

Recalling now that
$R_{2}\|u\|_{L^{\infty}}=d_{\mathcal{H}}(\Omega_{0},B_{R_{2}})$, we get

$\|\nabla_{\tau}u\|_{L^{2}(\mathbb{S}^{n-1})}^{2}\geq
C_{6}(n,R_{2})g(d_{\mathcal{H}}(\Omega_{0},B_{R_{2}})).$

Plugging the previous estimate in (56) and recalling that $\delta_{0}$ depends
only on $n,\beta,R_{1},R_{2}$, we finally obtain

$\displaystyle\lambda_{1}(\beta,A_{R_{1},R_{2}})-\lambda_{1}(\beta,\Omega)$
$\displaystyle\geq
C_{7}(n,\beta,R_{1},R_{2},\delta_{0})\|\nabla_{\tau}u\|_{L^{2}(\mathbb{S}^{n-1})}^{2}\geq
C(n,\beta,R_{1},R_{2})g(\mathcal{A}_{\mathcal{H}}(\Omega_{0})).$

∎

###  4.3. Inner asymmetry and conclusion

In this section we explain the key idea to keep into account both inner and
outer perturbations of the optimal set $A_{R_{1},R_{2}}$. As previously
explained, the Neumann condition on the inner boundary and the measure
constraint on $\Omega_{0}\setminus\overline{\Theta}$ do not seem to allow a
Fuglede type approach, since we do not have any control on
$d_{\mathcal{H}}(\Theta,B_{R_{1}})$. Intuitively, this is due to lack of a
constraint on $P(\Theta)$ and to the fact that in the Rayleigh quotient (2),
there is no boundary integral on $\partial\Theta$. By using the weak asymmetry
relative to the outer box defined in (10), we prove the following result.

######  Theorem 4.8.

Let $0<R_{1}<R_{2}<+\infty$. Then, for every
$\Omega=\Omega_{0}\setminus\overline{\Theta}\subset\mathbb{R}^{n}$ with
$\Omega_{0}$ and $\Theta$ convex sets such that
$\overline{\Theta}\subset\Omega_{0}$, $P(\Omega_{0})=P(B_{R_{2}})$ and
$|\Omega|=|A_{R_{1},R_{2}}|$, it holds

$\lambda_{1}(\beta,A_{R_{1},R_{2}})-\lambda_{1}(\beta,\Omega)\geq|\lambda_{1}(\beta,A_{R_{1},R_{2}})|\tilde{\mathcal{A}}(\Theta;\Omega_{0}).$

where $\tilde{\mathcal{A}}(\Theta;\Omega_{0})$ is defined in (10).

###### Proof.

Let $z\in H^{1}(A_{R_{1},R_{2}})$ be the first positive eigenfunction of
$\lambda_{1}(\beta,A_{R_{1},R_{2}})$ with $\|z\|_{L^{2}(A_{R_{1},R_{2}})}=1$,
let $w_{\Omega_{0}}\in H^{1}(\Omega_{0})$ the test function defined in (8) and
$t_{\Omega_{0}},K_{\Omega_{0}}$ as in the Definition 1.1.

We observe that $t_{\Omega_{0}}\geq R_{2}-R_{1}$ in view of the isoperimetric
inequality; it follows $w_{\Omega_{0}}\geq z_{m}$ in
$\Omega_{0}\setminus\overline{K_{\Omega_{0}}}$ and $w_{\Omega_{0}}=z_{m}$ in
$K_{\Omega_{0}}$.

We point out that the choice of $w_{\Omega_{0}}$ depends only on $\Omega_{0}$
and thus it is a suitable web function for both
$\Omega_{0}\setminus\overline{\Theta}$ and
$\Omega_{0}\setminus\overline{K_{\Omega_{0}}}$. From [42, eq. (3.11)] applied
to the admissible set $\Omega_{0}\setminus\overline{K_{\Omega_{0}}}$, we have
that

(60) $\int_{\Omega_{0}\setminus
K_{\Omega_{0}}}w^{2}_{\Omega_{0}}\>dx\leq\int_{A_{R_{1},R_{2}}}z^{2}\>dx.$

Moreover, it holds the following

(61)
$\begin{split}\int_{\Omega_{0}}w_{\Omega_{0}}^{2}\>dx&=\int_{\Omega_{0}\setminus\Theta}w_{\Omega_{0}}^{2}\>dx+\int_{\Theta\setminus
K_{\Omega_{0}}}w_{\Omega_{0}}^{2}\>dx+\int_{K_{\Omega_{0}}\cap\Theta}w_{\Omega_{0}}^{2}\>dx\\\
\end{split}$

and, by using (60), it holds

(62)
$\begin{split}\int_{\Omega_{0}}w_{\Omega_{0}}^{2}\>dx&=\int_{\Omega_{0}\setminus
K_{\Omega_{0}}}w_{\Omega_{0}}^{2}\>dx+\int_{K_{\Omega_{0}}\setminus\Theta}w_{\Omega_{0}}^{2}\>dx+\int_{K_{\Omega_{0}}\cap\Theta}w_{\Omega_{0}}^{2}\>dx\\\
&\leq\int_{A_{R_{1},R_{2}}}z^{2}\>dx+z_{m}^{2}|\Theta\setminus
K_{\Omega_{0}}|+\int_{K_{\Omega_{0}}\cap\Theta}w_{\Omega_{0}}^{2}\>dx,\end{split}$

since $|K_{\Omega_{0}}\setminus\Theta|=|\Theta\setminus K_{\Omega_{0}}|$,
which follows from the fact that $|K_{\Omega_{0}}|=|\Theta|$. Therefore, by
combining (61) and (62), we get

$\int_{\Omega_{0}\setminus\Theta}w^{2}_{\Omega_{0}}\>dx\leq\int_{A_{R_{1},R_{2}}}z^{2}\>dx-\int_{\Theta\setminus
K_{\Omega_{0}}}(w_{\Omega_{0}}^{2}-z_{m}^{2})\>dx.$

Now, by using $w_{\Omega_{0}}$ as a test function for
$\lambda_{1}(\beta,\Omega)$, we have

$\displaystyle\lambda_{1}(\beta,\Omega)$
$\displaystyle\leq\frac{\displaystyle\int_{\Omega_{0}\setminus\Theta}|\nabla
w_{\Omega_{0}}|^{2}\>dx+\beta\int_{\partial\Omega_{0}}w^{2}_{\Omega_{0}}\>d\sigma}{\displaystyle\int_{\Omega_{0}\setminus\Theta}w^{2}_{\Omega_{0}}\>dx}\leq\frac{\displaystyle\int_{A_{R_{1},R_{2}}}|\nabla
z|^{2}\>dx+\beta\int_{\partial
A_{R_{1},R_{2}}}z^{2}\>d\sigma}{\displaystyle\int_{A_{R_{1},R_{2}}}z^{2}\>dx-\int_{\Theta\setminus
K_{\Omega_{0}}}(w_{\Omega_{0}}^{2}-z_{m}^{2})\>dx}$
$\displaystyle=\frac{\lambda_{1}(\beta,A_{R_{1},R_{2}})}{\displaystyle
1-\int_{\Theta\setminus K_{\Omega_{0}}}(w_{\Omega_{0}}^{2}-z_{m}^{2})\>dx},$

and thus, from
$-\lambda_{1}(\beta,\Omega)\geq-\lambda_{1}(\beta,A_{R_{1},R_{2}})=|\lambda_{1}(\beta,A_{R_{1},R_{2}})|$,
we finally obtain

$\displaystyle\lambda_{1}(\beta,A_{R_{1},R_{2}})-\lambda_{1}(\beta,\Omega)$
$\displaystyle\geq-\lambda_{1}(\beta,\Omega)\int_{\Theta\setminus
K_{\Omega_{0}}}(w_{\Omega_{0}}^{2}-z_{m}^{2})\>dx$
$\displaystyle\geq|\lambda_{1}(\beta,A_{R_{1},R_{2}})|\int_{\Theta\setminus
K_{\Omega_{0}}}(w_{\Omega_{0}}^{2}-z_{m}^{2})\>dx.$

∎

Now we are in position to prove the Main Theorem stated in the introduction.

###### Proof of the Main Theorem.

It is a straightforward combination of Theorem 4.7 and Theorem 4.8. ∎

######  Remark 4.9.

We point out that, as usual when dealing with negative Robin boundary
conditions, the final constant appearing in the stability inequality (11) does
not only depend on the dimension, but also on the boundary parameter $\beta$
and on $R_{1}$ and $R_{2}$, i.e. on the size of the admissible sets for the
maximization problem.

##  5\. Further remarks and open problems

In this Section, we collect some final remarks and open problems.

######  Remark 5.1.

It is possible to prove the result of the Main Theorem also in some larger
classes than $\mathcal{T}_{R_{1},R_{2}}$. For instance, one can consider
$\Theta$ as a uniformly finite union of nondegenerate convex sets such that
the connected components of $\partial\Omega$ lie at uniformly bounded distance
and the inradii are uniformly bounded from below. More generally, one can
replace the convexity constraint and the nondegeneracy assumptions on the
admissible sets $\Theta$ by a uniform cone condition.

Indeed, the isoperimetric inequality (4) has been proved in [42] in a larger
setting than $\mathcal{T}_{R_{1},R_{2}}$, and so the proof of the stability
(11) can be easily generalized to the above mentioned more general cases.

######  Open Problem 5.2.

By repeating the very same computations as in Theorem 4.8 for the $p$-Laplace
Robin-Neumann eigenvalue $\lambda_{1}^{(p)}(\beta,\Omega)$, we get

$\lambda^{(p)}_{1}(\beta,A_{R_{1},R_{2}})-\lambda_{1}^{(p)}(\beta,\Omega)\geq|\lambda^{(p)}_{1}(\beta,A_{R_{1},R_{2}})|\int_{\Theta\setminus
K}(w_{\Omega_{0}}^{p}-z_{m}^{p})\>dx.$

To get a complete stability result also for the nonlinear case we need an
estimate for the asymmetry of $\Omega_{0}$ which at the moment does not seem
available, since our argument based on a Fuglede type approach does not seem
to apply in the nonlinear framework due to the lack of an explicit expression
for $z$.

Moreover, we ask whether the method used in this paper may be applied in more
general frameworks, as e.g. the Minkowski spaces, in which the standard
Euclidean norm is replaced by a Finsler norm [45].

######  Open Problem 5.3.

Finally, it would be interesting to consider the case $\beta>0$. In [42], it
is proved that the spherical shell such that $|A|=|\Omega|$ and
$P(B_{R_{2}})=P(\Omega_{0})$ is maximal:

$\displaystyle\lambda_{1}(\beta,\Omega)\leq\lambda_{1}(\beta,A).$

A stability result in this sense requires different methods because, for
example, it is not possible to consider the auxiliary Steklov-type problem.

## Acknowledgment

Gloria Paoli and Gianpaolo Piscitelli were partially supported by Italian MUR
through research project PRIN 2017 “Direct and inverse problems for partial
differential equations: theoretical aspects and applications”. Gloria Paoli
was supported by the Alexander von Humboldt Foundation through an Alexander
von Humboldt research fellowship. The three authors were partially supported
by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro
Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM).

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