# A monotonicity result for the first
Steklov-Dirichlet Laplacian eigenvalue

Nunzia Gavitone, Gianpaolo Piscitelli  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 consider the first Steklov-Dirichlet eigenvalue of the
Laplace operator in annular domains with a spherical hole. We prove a
monotoni-
city result with respect to the hole, when the outer region is centrally
symmetric.

MSC 2020: Primary: 35J25, 35P15; Secondary: 28A75, 49Q10.

Keywords: Laplacian eigenvalue, Steklov-Dirichlet boundary conditions, Shape
derivative.

AbstractAbstract

##  1\. Introduction and main result

In recent years, the study of eigenvalue problems in holed domains has been
the object of much interest. These kind of problems are usually defined in
annular domains with the outer region and the hole satisfying suitable
assumptions. Specifically, different boundary conditions can be imposed on the
outer and inner boundary and hence several optimization problems can be
studied (e.g. Robin-Neumann [PPT], Neumann-Robin [DP], Dirichlet-Neumann [AA,
AAK], Steklov-Dirichlet [PPS, GPPS, HLS], Steklov-Robin [GS]). At the mean
time, the optimal placement of an obstacle has been studied, so as to maximize
or minimize a prescribed functional (e.g. the Dirichlet heat content [L], the
first Steklov eigenvalue [F]). In this paper, we consider an eigenvalue
problem for the Laplace operator in a suitable annular domain with outer
Steklov and inner Dirichlet boundary conditions. More precisely, let
$\Omega_{0}\subset\mathbb{R}^{n}$, $n\geq 2$, be an open bounded set with
Lipschitz boundary and $B_{r}(y)$ be the ball of radius $r>0$, centered at
$y$, such that $B_{r}(y)\subset\abxring{\Omega}_{0}$. We study the following
Steklov-Dirichlet eigenvalue problem

$\sigma(\Omega)=\min_{\begin{subarray}{c}v\in H^{1}_{\partial
B_{r}(y)}(\Omega)\\\ v\not\equiv
0\end{subarray}}\left\\{\int_{\Omega}|Dv|^{2}\;dx,\,\|v\|_{L^{2}(\partial\Omega_{0})}=1\right\\},$
(1.1)

where $H^{1}_{\partial B_{r}(y)}(\Omega)$ denotes the set of Sobolev functions
which vanish on the boundary of $B_{r}(y)$ (see Section 2 for the precise
definition). If $u\in H^{1}_{\partial B_{r}(y)}(\Omega)$ is a minimizer of
(1.1), then it satisfies:

$\begin{cases}\Delta u=0&\mbox{in}\ \Omega\vspace{0.2cm}\\\ \dfrac{\partial
u}{\partial\nu}=\sigma\,u&\mbox{on}\ \partial\Omega_{0}\vspace{0.2cm}\\\
u=0&\mbox{on}\ \partial B_{r}(y),\end{cases}$ (1.2)

where $\Omega$ is the annular domain
$\Omega=\Omega_{0}\setminus\overline{B_{r}(y)}$, $\sigma\in\mathbb{R}$ and
$\nu$ is the outer unit normal to $\partial\Omega_{0}$. In [PPS] (see also [D,
HLS]), the authors prove that the minimum in (1.1) is achieved by a function
$u\in H^{1}_{\partial B_{r}(y)}(\Omega)$, which is a weak solution to problem
(1.2) with constant sign in $\Omega$, and that $\sigma(\Omega)$ is simple.
Furthermore, the authors prove that, keeping the measure of $\Omega$ and the
radius of the inner ball $r$ fixed, $\sigma(\Omega)$ is maximized, among
quasi-spherical sets, when $\Omega$ is a spherical shell, that is when
$\Omega_{0}$ is a ball with the same center of the hole. On the other hand, in
[GPPS] the authors extend this result to a class of annular sets with a
suitable convex outer domain $\Omega_{0}$. More properties are known when
$\Omega$ is an eccentric spherical shell, that is, when the outer domain
$\Omega_{0}$ is a ball not necessarly centered at the same point of the
spherical hole. In [VS], the authors study the optimal placement of the hole
in eccentric spherical shell $\Omega$ so that $\sigma(\Omega)$ is maximized
when the outer ball and the inner radius are fixed. If $n\geq 3$, they prove
that $\sigma(\Omega)$ achieves the maximum when the two balls are concentric.
Subsequently, this result has been also proved for any dimension $n\geq 2$ in
[F], by using different proofs (see also [S] for an analogous result in two-
points homogeneous spaces). Moreover, by performing numerical experiments, the
authors in [HLS] exhibit that $\sigma(\Omega)$ is monotone decreasing with
respect to the distance between the centers of the two disks. Our aim is to
prove that this monotonicity property holds in any dimension and in a more
general setting. Through this paper, we assume that the outer domain
$\Omega_{0}$ verifies the following hypotheses.

######  Definition 1.1.

Let $\Omega_{0}\subset\mathbb{R}^{n}$, $n\geq 2$, be an open, bounded set with
Lipschitz boundary and centrally symmetric with respect to
$x_{0}\in\abxring{\Omega}_{0}$, that is there exists
$x^{\prime}\in\overline{\Omega_{0}}$ such that
$\frac{1}{2}\left(x+x^{\prime}\right)=x_{0}$ for any
$x\in\overline{\Omega_{0}}$.

Let $r>0$ be fixed, our first question is

  * $(\mathcal{Q}_{1})$

Where we have to place the center of the spherical hole with fixed radius $r$
in order to maximize $\sigma(\Omega)$?

To give an answer, let $w\in\mathbb{R}^{n}$ be a unit vector and let us
consider the holes $B_{r}(t)$ with the centers on the $w$-direction:

$B_{r}(t)=\left\\{x(t)=x+tw,\,x\in B_{r}(x_{0})\right\\}\quad 0\leq
t<\rho_{w}(x_{0},r)-r,$ (1.3)

where

$\rho_{w}=\rho_{w}(x_{0},r)=\sup\\{t>0\colon
B_{r}(x_{0}+tw)\subset\abxring{\Omega}\\}.$ (1.4)

We stress that $\rho_{w}$ is the distance between $x_{0}$ and the center of
the farthest ball from $x_{0}$, well contained in $\Omega_{0}$, in the
direction $w$ passing at $x_{0}$. Then we consider the following type of
annular domains

$\Omega(t)=\Omega_{0}\setminus\overline{B_{r}(t)},$ (1.5)

where $\Omega_{0}$ verifies the assumption of Definition 1.1. We stress that
when $\Omega_{0}$ is a ball, then the sets $\Omega(t)$ are eccentric spherical
shells. Furthermore, for any $0\leq t<\rho_{w}(x_{0},r)-r$, we denote by
$\sigma(t)$ the first Steklov-Dirichlet eigenvalue of the Laplacian in
$\Omega(t)$, that is

$\sigma(t)=\min_{\begin{subarray}{c}v\in H^{1}_{\partial
B_{r}(t)}(\Omega(t))\\\ v\not\equiv
0\end{subarray}}\left\\{\int_{\Omega(t)}|Dv|^{2}\;dx,\
\,\|v\|_{L^{2}(\partial\Omega_{0})}=1\right\\}.$ (1.6)

Our second questions is

  * $(\mathcal{Q}_{2})$

Is $\sigma(t)$ decreasing with respect to $t$?

Our main result gives an answer to both questions $(\mathcal{Q}_{1})$ and
$(\mathcal{Q}_{2})$.

######  Theorem 1.2.

Let $\Omega_{0}$ be as in Definition 1.1, $w\in\mathbb{R}^{n}$ be a unit
vector, $B_{r}(t)$, $\rho_{w}$, $\Omega(t)$ and $\sigma(t)$ be defined as in
(1.3), (1.4), (1.5) and (1.6), respectively. Then, $\sigma(t)$ is strictly
monotone decreasing with respect to $t\in[0,\rho_{w}-r)$.

As an immediate consequence of our main result, we obtain that, in order to
maximize $\sigma(t)$, the hole has to be centered at the symmetry point of
$\Omega_{0}$, when the inner radius is fixed. To prove Theorem 1.2, we use a
shape derivative approach. In particular we compute the first and the second
domain derivatives of the first Steklov-Dirichlet eigenvalue. We emphasize
that our result implies that the monotonicity property holds also for
eccentric spherical shell in any dimension and in particular in two
dimensions, as suggested by the numerical computation contained in [HLS].
Finally, we describe the outline of the paper. In Section 2, we summarize some
results about the first Steklov-Dirichlet eigenvalue problems. In Section 3,
we compute the first shape derivative of $\sigma(t)$, observing that a
stationary set occurs when the center of the hole coincides with the center of
symmetry of the outer region. Finally, in Section 4, we compute the second
shape derivative of $\sigma(t)$ and prove the main result.

##  2\. The Steklov-Dirichlet Laplacian eigenvalue problem

In this Section, we study the Steklov-Dirichlet Laplacian eigenvalue problem
both on fixed domain $\Omega$ and on a parameter-dependent domain $\Omega(t)$,
respectively.

###  2.1. Foundation of the problem

Let $\Omega_{0}\subset\mathbb{R}^{n}$ be an open, bounded set with Lipschitz
boundary and such that $B_{r}(y)\subset\abxring{\Omega}_{0}$, where $B_{r}(y)$
is the ball of radius $r>0$ centered at $y$. Let us consider the annular
domain $\Omega:=\Omega_{0}\setminus\overline{B_{r}(y)}$. In what follows we
denote the set of Sobolev functions on $\Omega$ vanishing on $\partial
B_{r}(y)$ by $H^{1}_{\partial B_{r}(y)}(\Omega)$, that is (see [ET]) the
closure in $H^{1}(\Omega)$ of the following set

$C^{\infty}_{\partial B_{r}(y)}(\Omega):=\\{u|_{\Omega}\ |\ u\in
C_{0}^{\infty}(\mathbb{R}^{n}),\ \operatorname{spt}(u)\cap\partial
B_{r}(y)=\emptyset\\}.$

It is known (see for instance [A, D, P]) that the spectrum of the Steklov-
Dirichlet eigenvalue problem (1.2) for the Laplace operator in $\Omega$ is
discrete and that the sequence of eigenvalues can be ordered as follows

$0<\sigma(\Omega)\leq\sigma_{2}(\Omega)\leq\sigma_{3}(\Omega)\leq\ldots\nearrow+\infty.$

In particular, the first eigenvalue $\sigma(\Omega)$ has the variational
characterization (1.1), see for instance [D, PPS, HLS]. Moreover in [PPS] (see
also [HLS]), the authors prove the following result.

######  Proposition 2.1.

There exists a function $u\in H^{1}_{\partial B_{r}(y)}(\Omega)$ which
achieves the minimum in (1.1) and is a weak solution to the problem (1.2).
Moreover $\sigma(\Omega)$ is simple and the first eigenfunctions have constant
sign in $\Omega$.

In order to prove our main result, we need to compute the first and the second
shape derivative of $\sigma(\Omega)$ and, to study these derivatives, we need
to consider a family of domains approaching to $\Omega$. In our case, we get
the desired monotonicity result by studying the behavior of the involved
quantities on the family of domains obtained by moving the hole in a fixed
direction. In what follows we fix notation and recall some preliminary
results. Let $\Omega_{0}$ be as in Definition 1.1. For the reader convenience,
from now on, we assume that $x_{0}=0$. In the sequel, we denote by $B_{r}$ the
ball centered at the origin with radius $r$ and we set
$\Omega=\Omega_{0}\setminus\overline{B_{r}}$ (see the figure below).

$\partial\Omega_{0}$$\Omega=\Omega_{0}\setminus{\overline{B_{r}}}$$r$$B_{r}$

In order to study the Steklov-Dirichlet eigenvalue problem on the annular
domain $\Omega(t)$ defined in (1.5), we define a suitable smooth vector field
which move the hole $B_{r}$ in a given direction $w$ keeping the boundary of
$\Omega_{0}$ and the inner radius $r$ fixed. Hence, the perturbed holes have
the form described in (1.3). We consider the following variational field in
$\mathbb{R}^{n}$:

$V(x)=w\varphi(x),$ (2.1)

where $\varphi\in C_{0}^{\infty}(\Omega_{0})$ is a cut-off function such that
$\varphi(x)=1$ on $\overline{B_{r}}$. Consequently, the perturbed annular
domains $\Omega(t)$, defined in (1.5), can be seen as

$\Omega(t)=\\{x(t)=x+tV(x),\,\,x\in\Omega\\}\quad t\in[0,\rho_{w}-r),$

where $\rho_{w}$ is defined in (1.4). Let us observe that
$\partial\Omega(t)=\partial\Omega_{0}\cup\partial B_{r}(t)$ and that
$\Omega(t)$ is centrally symmetric if and only if $t=0$ and the center of
symmetry is the origin. Also refer to the figure below.

$tw$$\partial\Omega_{0}$$w$$B_{r}(t)$$\Omega(t)=\Omega_{0}\setminus\overline{B_{r}(t)}$

For any $t\in[0,\rho_{w}-r)$, let $\sigma(t)$ be the first Steklov-Dirichlet
eigenvalue (1.6) of the Laplacian in $\Omega(t)$ and $u^{t}$ be the
corresponding normalized and positive eigenfunction. The first eigenvalue
admit the variational characterization (1.6) and the eigenfunction $u^{t}\in
H^{1}_{\partial B_{r}(t)}(\Omega(t))$ is a solution to the following
eigenvalue problem

$\begin{cases}\Delta u^{t}=0&\mbox{in}\ \Omega(t)\vspace{0.2cm}\\\
\dfrac{\partial u^{t}}{\partial\nu}=\sigma(t)u^{t}&\mbox{on}\
\partial\Omega_{0}\vspace{0.2cm}\\\ u^{t}=0&\mbox{on}\ \partial
B_{r}(t).\end{cases}$ (2.2)

By a little abuse of notation, we also indicate by $\nu$ the outer unit normal
to the boundary of the annular domain $\partial\Omega(t)$. Then, the
variational characterization assures that $u^{t}\in H^{1}_{\partial B_{r}(t)}$
is a positive function such that

$\sigma(t)=\displaystyle\int_{\Omega_{t}}|\nabla u^{t}|^{2}\,dx,$

and that verifies the following equality

$\displaystyle\int_{\partial\Omega_{0}}(u^{t})^{2}\,d\mathcal{H}^{n-1}=1.$
(2.3)

###  2.2. Differentiability of $\sigma(t)$

Standard arguments on shape derivatives assure that $u^{t}$ and $\sigma(t)$
are differentiable with respect to $t$. For the sake of completeness, we
sketchily give the proof, that follows exactly the same arguments contained in
[HP]. The main tool is a general version of the implicit function theorem (see
[HP, Sc] and also [Bo, Lem.2.1]) applied to the equation transferred onto the
fixed domain $\Omega_{0}$ (for the details, we refer to [HP] and also to [Bo,
Lem. 2.7] and [HLS, Th.1]).

######  Proposition 2.2.

Let $\Omega_{0}$ be as in Definition 1.1, $w\in\mathbb{R}^{n}$ be a unit
vector, $B_{r}(t)$, $\rho_{w}$, $\Omega(t)$ and $\sigma(t)$ be defined as in
(1.3), (1.4), (1.5) and (1.6), respectively. Let $u^{t}$ be the first
normalized eigenfunction of $\sigma(t)$, then the functions

$t\in[0,\rho_{w}-r)\to\sigma(t),\qquad t\in[0,\rho_{w}-r)\to u^{t}$

are differentiable for any direction $w\in\mathbb{S}^{n-1}$ and for any
$t\in[0,\rho_{w}-r)$.

###### Proof.

Let us fix $t\in[0,\rho_{w}-r)$ and let $s>0$ be such that $t+s<\rho_{w}-r$.
Therefore, we are able to consider $\sigma(t+s)$ and $u^{t+s}$ and the
following weak formulation for problem (1.6) holds:

$\int_{\Omega_{0}}\nabla
u^{t+s}\,\nabla\varphi\,dx=\sigma(t+s)\int_{\partial\Omega_{0}}u^{t+s}\varphi\,d\mathcal{H}^{n-1}\quad\forall\varphi\in
H^{1}_{\partial B_{r}(t+s)}(\Omega).$ (2.4)

Let $V$ be as in (2.1) and let us define the following map

$\Phi:(s,x)\in(-\rho_{w}+r-t,\rho_{w}-r-t)\times\Omega_{0}\to
x+sV(x)\in\mathbb{R}^{n}.$

It is easily seen that, for any $s$, $\Phi(s,\Omega_{0})=\Omega_{0}$,
$u^{t+s}(\Phi(s,\cdot))\in H^{1}_{\partial B_{r}(t)}(\Omega)$ and
$D\Phi(0,\cdot)=I$, where $D\Phi$ denotes the Jacobian matrix of $\Phi$ and
$I$ the identity matrix of order $n$. Therefore, there exists a neighborhood
$U$ of $0$ such that $\Phi(s,\cdot)$ is a diffeomorphism of $\Omega_{0}$ and
hence, by making a change of variables, (2.4) becomes

$\begin{split}\int_{\Omega_{0}}\nabla(u^{t+s}\left(\Phi(s,\cdot))\right)(D\Phi(s,\cdot))^{-1})(\nabla(\varphi\left(\Phi(s,\cdot))\right)(D\Phi(s,\cdot))^{-1})|D\Phi(s,\cdot)|\,dx\\\
=\sigma(t+s)\int_{\partial\Omega_{0}}(u^{t+s}\left(\Phi(s,\cdot))\right)(\varphi\left(\Phi(s,\cdot))\right)\,d\mathcal{H}^{n-1}\quad\forall\varphi\in
H^{1}_{\partial B_{r}(t+s)}(\Omega),\end{split}$

and the normalization becomes

$\int_{\partial\Omega_{0}}(u^{t+s}\left(\Phi(s,\cdot))\right)^{2}d\mathcal{H}^{n-1}=1.$

Now, let us denote by $(H^{1}_{\partial B_{r}(t)}(\Omega))^{\prime}$ the dual
space of $H^{1}_{\partial B_{r}(t)}(\Omega)$ and let us define

$f:(s,v,\sigma)\in\mathbb{R}\times H^{1}_{\partial
B_{r}(t)}(\Omega)\times\mathbb{R}\to(f_{1},f_{2})\in(H^{1}_{\partial
B_{r}(t)}(\Omega))^{\prime}\times\mathbb{R},$

where

$\begin{cases}\langle f_{1}(s,v,\sigma),\Psi\rangle=\int_{\Omega_{0}}((\nabla
v)(D\Phi(s,\cdot))^{-1})((\nabla\Psi)(D\Phi(s,\cdot))^{-1})|D\Psi(s,\cdot)|dx-\sigma\int_{\Omega_{0}}v\Psi
d\mathcal{H}^{n-1}\\\ \langle
f_{2}(s,v,\sigma),\Psi\rangle=\int_{\partial\Omega_{0}}v^{2}d\mathcal{H}^{n-1},\end{cases}$

for any $\Psi\in H^{1}_{\partial B_{r}(t)}(\Omega_{0})$. If we consider the
function

$g\colon s\in U\to\left(u^{t+s}(\Phi(s,\cdot),\sigma(t+s)\right)\in
H^{1}_{\partial B_{r}(t)}(\Omega)\times\mathbb{R},$

then $f(s,g(s))=0$ for any $s\in U$ and $g(0)=\left(u^{t},\sigma(t)\right)$.
In order to obtain the claim, we have to prove that $g$ is differentiable in
$s=0$, that follows by applying the implicit function Theorem. To do this, we
have to prove that

$\frac{\partial
f}{\partial(v,\sigma)}\Big{|}_{0,u^{t},\sigma(t)}:H^{1}_{\partial
B_{r}(t)}(\Omega)\times\mathbb{R}\to(H^{1}_{\partial
B_{r}(t)}(\Omega))^{\prime}\times\mathbb{R}$

is an isomorphism. This can be proved following line by line the same
arguments of [Bo, Lem 2.7] and [HLS, Th. 1] (see also chapter 5 in [HP]). ∎

######  Remark 2.3.

We observe that being $u^{t}$ harmonic for any $t\in[0,\rho_{w}-r)$, then
$u^{t}\in C^{\infty}\left(\overline{\Omega(t)}\right)$. Then by the general
theory of the shape derivatives (see Chapter 5 of [HP]), follows that $u^{t}$
is $C^{\infty}$ in a neighborhood of $t$.

Since, to reach our aims, we need to consider the total and partial derivative
of $u^{t}$ with respect to the parameter $t$, we observe that, by (1.3), we
have

$\frac{d}{dt}[u^{t}(t,x(t))]=(u^{t})^{\prime}+\langle\nabla u^{t},w\rangle,$
(2.5)

where $(u^{t})^{\prime}=\displaystyle\frac{\partial u^{t}}{\partial t}$ and
$\langle\cdot,\cdot\rangle$ denotes the usual scalar product in
$\mathbb{R}^{n}$. Recalling that the perturbed hole $B_{r}(t)$ is the zero-
level set of the function $u^{t}$, then (2.5) implies that

$(u^{t})^{\prime}=-\frac{\partial
u^{t}}{\partial\nu}\,\langle\nu,w\rangle,\quad\text{ on }\partial B_{r}(t),$
(2.6)

for any $t\in[0,\rho_{w}-r)$. Moreover, we observe that it holds

$\nu=-\frac{\nabla u^{t}}{|\nabla u^{t}|}\quad\text{ on }\partial B_{r}(t)$
(2.7)

and

$(n-1)H=\text{div}\left(\frac{\nabla u^{t}}{|\nabla u^{t}|}\right)\quad\text{
on }\partial B_{r}(t).$

Therefore, we have

$\begin{split}\frac{(n-1)}{r}=(n-1)H&=-\frac{1}{|\nabla
u^{t}|^{2}}\langle\nabla\left(|\nabla u^{t}|\right),\nabla u^{t}\rangle\\\
&=-\frac{1}{|\nabla u^{t}|^{3}}\langle\nabla u^{t}\cdot D^{2}(u^{t}),\nabla
u^{t}\rangle\quad\text{ on }\partial B_{r}(t),\end{split}$ (2.8)

where $D^{2}(u^{t})$ denotes the Hessian matrix of $u^{t}$. Furthermore, by
using (2.6) and the fact that $u^{t}$ is a solution to (2.2), we get that
$(u^{t})^{\prime}$ is a weak solution to the following problem

$\begin{cases}\Delta(u^{t})^{\prime}=0&\mbox{in}\ \Omega(t)\vspace{0.2cm}\\\
\dfrac{\partial(u^{t})^{\prime}}{\partial\nu}=\sigma^{\prime}(t)u^{t}+\sigma(t)(u^{t})^{\prime}&\mbox{on}\
\partial\Omega_{0}\vspace{0.2cm}\\\ (u^{t})^{\prime}=-\dfrac{\partial
u^{t}}{\partial\nu}\,\langle\nu,w\rangle&\mbox{on}\ \partial
B_{r}(t).\end{cases}$ (2.9)

We observe that, if we derive the normalized condition (2.3), we get

$\int_{\partial\Omega_{0}}(u^{t})^{\prime}\,u^{t}\,\,d\mathcal{H}^{n-1}=0.$
(2.10)

Finally, for every $t\in[0,\rho_{w}-r)$, it will be useful for the sequel to
consider the harmonic extension $H(|\nabla u^{t}|\langle\nu,w\rangle)$ of the
function with the same boundary value of (2.9) on $\partial B_{r}(t)$; that is

$\Delta H(|\nabla u^{t}|\langle\nu,w\rangle)=0\quad\text{in }B_{r}(t),\qquad
H(|\nabla u^{t}|\langle\nu,w\rangle)=|\nabla
u^{t}|\langle\nu,w\rangle\quad\text{on }\partial B_{r}(t).$ (2.11)

##  3\. The first order Shape Derivative of $\sigma(t)$

In this Section, we compute the first oder derivative of the eigenvalue
$\sigma(t)$ on $\Omega(t)$. Before doing this, we recall an Hadamard’s formula
in the framework of the domain derivative (see for instance [SZ, B, HP]). Let
$E\subset\mathbb{R}^{n}$ be an open bounded set with Lipschitz boundary and
let $V(x)$ a vector field such that $V\in
W^{1,\infty}(\mathbb{R}^{n};\mathbb{R}^{n})$. For any $t>0$, let
$E(t)=\\{x(t)=x+tV(x),\,x\in E\\}$, and $f(t,x(t))$ be such that
$f(t,\cdot)\in W^{1,1}(\mathbb{R}^{n})$ and differentiable at $t$. Then it
holds:

$\displaystyle\begin{split}\frac{d}{dt}\int_{E(t)}f(t,x)\,d\mathcal{H}^{n-1}=&\int_{E(t)}\frac{\partial}{\partial
t}f(t,x)\,d\mathcal{H}^{n-1}+\\\ &+\int_{+\partial
E(t)}f(t,x)\langle\nu,V(x)\rangle\,d\mathcal{H}^{n-1},\end{split}$ (3.1)

where $\nu$ is the outer unit normal to the boundary of $E(t)$ and
$\mathcal{H}^{n-1}$ denotes the Hausdorff measure (see for instance the
Chapter 5 of [HP]). We use formula (3.1) to prove the following.

######  Theorem 3.1.

Let $\Omega_{0}$ be as in Definition 1.1, $w\in\mathbb{R}^{n}$ be a unit
vector, $B_{r}(t)$, $\rho_{w}$, $\Omega(t)$ and $\sigma(t)$ be defined as in
(1.3), (1.4), (1.5) and (1.6), respectively. Then, for any
$t\in[0,\rho_{w}-r)$, it holds

$\displaystyle\frac{d}{dt}\sigma(t)=-\displaystyle\int_{\partial
B_{r}(t)}|\nabla u^{t}|^{2}\langle w,\nu\rangle\,d\mathcal{H}^{n-1},$

where $u^{t}$ the is the normalized, positive eigenfunction corresponding to
$\sigma(t)$.

###### Proof.

By using the Hadamard’s formula (3.1), by observing that the unit outer
normals $\nu(x)$ to $\partial B_{r}$ and $\nu(x(t))$ to $\partial B_{r}(t)$
coincide and by the fact that $u^{t}$ is a solution of (2.2), we get

$\displaystyle\begin{split}\displaystyle\frac{d}{dt}\sigma(t)=&\frac{d}{dt}\displaystyle\left(\int_{\Omega(t)}|\nabla
u^{t}|^{2}\,dx\right)\\\ =&2\displaystyle\int_{\Omega(t)}\langle\nabla
u^{t},\nabla(u^{t})^{\prime}\rangle\,dx+\displaystyle\int_{\partial
B_{r}(t)}|\nabla u^{t}|^{2}\langle w,\nu\rangle\,d\mathcal{H}^{n-1}\\\
=&2\left(\int_{\partial\Omega_{0}}(u^{t})^{\prime}\,\frac{\partial
u}{\partial\nu}\,d\mathcal{H}^{n-1}+\int_{\partial
B_{r}(t)}(u^{t})^{\prime}\,\frac{\partial
u^{t}}{\partial\nu}\,d\mathcal{H}^{n-1}\right)+\\\ &+\int_{\partial
B_{r}(t)}|\nabla u^{t}|^{2}\langle w,\nu\rangle\,d\mathcal{H}^{n-1}\\\
=&2\left(\sigma(t)\int_{\partial\Omega_{0}}(u^{t})^{\prime}\
\,u\,d\mathcal{H}^{n-1}-\int_{\partial B_{r}(t)}\left(\frac{\partial
u^{t}}{\partial\nu}\right)^{2}\,\langle w,\nu\rangle
d\mathcal{H}^{n-1}\right)+\\\ &+\int_{\partial B_{r}(t)}|\nabla
u^{t}|^{2}\langle w,\nu\rangle\,d\mathcal{H}^{n-1}.\end{split}$

By taking into account the relation (2.10), we get the conclusion. ∎

In order to obtain our main result, we study the behavior of the first order
derivative of $\sigma(t)$ for $t=0$. We stress that the symmetry of
$\Omega_{0}$ has a key role in the proofs of these results.

######  Corollary 3.2.

Let $\Omega_{0}$ be as in Definition 1.1, $w\in\mathbb{R}^{n}$ be a unit
vector, $B_{r}(t)$, $\rho_{w}$, $\Omega(t)$ and $\sigma(t)$ be defined as in
(1.3), (1.4), (1.5) and (1.6), respectively. Then it holds

$\displaystyle\frac{d}{dt}[\sigma(t)]_{t=0}=0.$

###### Proof.

Let $u^{t}$ be the normalized, positive eigenfunctions corresponding to
$\sigma(0)=\sigma(\Omega)$. We observe that, since $\Omega_{0}$ is centrally
symmetric with respect the origin (see Definition 1.1), then

$u^{t}(x)=u^{t}(-x)\quad x\in\partial B_{r}.$ (3.2)

If we denote by $x^{\prime}$ the symmetric point of $x$, then (3.2)
immediately follows by taking $v(x)=u^{t}(x^{\prime})$ as test function in
(1.1). Finally, by Theorem 3.1 and being $\nu=-\displaystyle\frac{x}{r}$, we
have

$\displaystyle\frac{d}{dt}[\sigma(t)]_{t=0}=-\displaystyle\int_{\partial
B_{r}}|\nabla u^{t}|^{2}\langle\nu,w\rangle\,d\mathcal{H}^{n-1}=0.$

∎

##  4\. The second order Shape Derivative of $\sigma(t)$

To prove the main result (Theorem 1.2), we need a stationary property of the
first order derivative (analyzed in the previous Section) and a sign of the
second order derivative. We compute the second order domain derivative of
$\sigma(t)$, by using the same notations of the previous Section and by
recalling some useful definitions from [BW]. For any bounded domain $E\in
C^{2,\alpha}$ and for any $f\in C^{1}(\partial E)$, the tangential derivative
of $v$ is given by

$\nabla^{\tau}f=\nabla f-\langle\nabla f,\nu\rangle\nu\quad\text{on }\partial
E,$ (4.1)

where $\nu$ is the unit outer normal. Furthermore, for any smooth vector field
$\Psi:\overline{E}\mapsto\mathbb{R}$, the tangential divergence is defined as

$\operatorname{div}_{\partial
E}(\Psi)=\operatorname{div}(\Psi)-\langle\nu,D(\Psi)\nu\rangle\quad\text{on
}\partial E,$ (4.2)

where $D(\Psi)$ is the Jacobian matrix of $\Psi$. Then, the Gauss theorem on
surfaces holds:

$\int_{\partial E}f\operatorname{div}_{\partial E}\Psi
d\mathcal{H}^{n-1}=-\int_{\partial E}\langle\Psi,\nabla^{\tau}f\rangle\
d\mathcal{H}^{n-1}+(n-1)\int_{\partial E}f\,H\,\langle\Psi,\nu\rangle\
d\mathcal{H}^{n-1}$ (4.3)

The following Theorem gives the expression of the second order domain
derivative of $\sigma(t)$.

######  Theorem 4.1.

Let $\Omega_{0}$ be as in Definition 1.1, $w\in\mathbb{R}^{n}$ be a unit
vector, $B_{r}(t)$, $\rho_{w}$, $\Omega(t)$ and $\sigma(t)$ be defined as in
(1.3), (1.4), (1.5) and (1.6), respectively. Then, for any
$t\in[0,\rho_{w}-r)$, it holds

$\begin{split}\displaystyle\frac{d^{2}}{dt^{2}}\sigma(t)=-2\int_{B_{r}(t)}|\nabla
H\left(|\nabla u^{t}|\langle
w,\nu\rangle\right)|^{2}dx-\dfrac{1}{r}\int_{\partial B_{r}(t)}|\nabla
u^{t}|^{2}d\mathcal{H}^{n-1}\\\ -\frac{3n-4}{r}\int_{\partial B_{r}(t)}|\nabla
u^{t}|^{2}\left(\langle
w,\nu\rangle\right)^{2}\,d\mathcal{H}^{n-1},\end{split}$

where $u^{t}$ the is the normalized positive eigenfunction corresponding to
$\sigma(t)$ and $H(|\nabla u^{t}|\langle w,\nu\rangle)$ is the harmonic
extension of $|\nabla u^{t}|\langle w,\nu\rangle$ in $\overline{B_{r}(t)}$,
defined in (2.11).

###### Proof.

In order to compute the second order derivative, we first observe that

$\displaystyle\int_{\partial B_{r}(t)}|\nabla u^{t}|^{2}\langle
w,\nu\rangle\,d\mathcal{H}^{n-1}=\int_{\partial\Omega(t)}|\nabla
u^{t}|^{2}\langle V,\nu\rangle\,d\mathcal{H}^{n-1},$

where $V$ is defined in (2.1). Therefore, by using the divergence Theorem, we
have

$\sigma^{\prime}(t)=-\int_{\Omega(t)}\operatorname{div}(|\nabla
u^{t}|^{2}V)\,dx.$

By using the Hadamard’s formula (3.1) in the right-hand side, we compute the
second order derivative of $\sigma(t)$:

$\displaystyle\begin{split}\sigma^{\prime\prime}(t)=-\frac{d}{dt}\int_{\Omega(t)}\operatorname{div}(|\nabla
u^{t}|^{2}V)\,dx=&-\int_{\Omega(t)}\operatorname{div}\left(\frac{\partial}{\partial
t}|\nabla u^{t}|^{2}V\right)\,dx\\\
&-\int_{\partial\Omega(t)}\operatorname{div}(|\nabla u^{t}|^{2}V)\ \langle
V,\nu\rangle\,d\mathcal{H}^{n-1}.\end{split}$ (4.4)

Let us consider the second term in (4.4), by using the tangential divergence
(4.2), we have

$\begin{split}&-\int_{\partial\Omega(t)}\operatorname{div}(|\nabla
u^{t}|^{2}V)\langle V,\nu\rangle\,d\mathcal{H}^{n-1}=-\int_{\partial
B_{r}(t)}\operatorname{div}(|\nabla u^{t}|^{2}w)\langle
w,\nu\rangle\,d\mathcal{H}^{n-1}\\\ &\quad=-\int_{\partial
B_{r}(t)}\operatorname{div}_{\partial B_{r}(t)}(|\nabla u^{t}|^{2}w)\langle
w,\nu\rangle\,d\mathcal{H}^{n-1}-\int_{\partial B_{r}(t)}\langle\nu,D(|\nabla
u^{t}|^{2}w)\nu\rangle\,d\mathcal{H}^{n-1}.\end{split}$ (4.5)

Hence, by inserting (4.5) in (4.4), we have

$\begin{split}\sigma^{\prime\prime}(t)=&-\int_{\Omega(t)}\operatorname{div}\left(\frac{\partial}{\partial
t}|\nabla u^{t}|^{2}V\right)\,dx-\int_{\partial
B_{r}(t)}\operatorname{div}_{\partial B_{r}(t)}(|\nabla u^{t}|^{2}w)\langle
w,\nu\rangle\,d\mathcal{H}^{n-1}\\\ &-\int_{\partial
B_{r}(t)}\langle\nu,D(|\nabla
u^{t}|^{2}w)\nu\rangle\,d\mathcal{H}^{n-1}:=I+II+III.\end{split}$ (4.6)

For reader’s convenience, we separately study the terms $I$, $II$ and $III$ in
(4.6). Let us focus on $I$; by using the divergence Theorem, we have

$\begin{split}I=-\int_{\Omega(t)}\operatorname{div}\left(\frac{\partial}{\partial
t}|\nabla
u^{t}|^{2}V\right)\,dx&=-2\int_{\Omega(t)}\operatorname{div}\left(\langle\nabla
u^{t},\nabla(u^{t})^{\prime}\rangle V\right)\,dx\\\
&=-2\int_{\partial\Omega(t)}\langle\nabla u^{t},\nabla(u^{t})^{\prime}\rangle\
\langle V,\nu\rangle\,d\mathcal{H}^{n-1}\\\ &=-2\int_{\partial
B_{r}(t)}\langle\nabla u^{t},\nabla(u^{t})^{\prime}\rangle\ \langle
w,\nu\rangle\,d\mathcal{H}^{n-1}\\\ &=-2\int_{\partial
B_{r}(t)}\displaystyle\frac{\partial(u^{t})^{\prime}}{\partial\nu}(u^{t})^{\prime}d\mathcal{H}^{n-1}\\\
&=-2\int_{B_{r}(t)}|\nabla H\left(|\nabla u^{t}|\langle
w,\nu\rangle\right)|^{2}dx.\end{split}$ (4.7)

Let us remark that $\nu$ is also the inner unit normal to $\partial B_{r}(t)$.
Now let us consider $II$; by using the Gauss Theorem (4.3), we have

$\begin{split}II&=-\int_{\partial B_{r}(t)}\operatorname{div}_{\partial
B_{r}(t)}(|\nabla u^{t}|^{2}w)\langle w,\nu\rangle\,d\mathcal{H}^{n-1}\\\
&=\int_{\partial B_{r}(t)}\langle|\nabla u^{t}|^{2}w,\nabla^{\tau}(\langle
w,\nu\rangle)\rangle\,d\mathcal{H}^{n-1}-\frac{n-1}{r}\int_{\partial
B_{r}(t)}|\nabla u^{t}|^{2}(\langle w,\nu\rangle)^{2}\,d\mathcal{H}^{n-1}\\\
&=\int_{\partial B_{r}(t)}\langle|\nabla u^{t}|^{2}w,\nabla(\langle
w,\nu\rangle)-\langle\nabla(\langle
w,\nu\rangle),\nu\rangle\nu\rangle\,d\mathcal{H}^{n-1}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\
-\frac{n-1}{r}\int_{\partial B_{r}(t)}|\nabla u^{t}|^{2}(\langle
w,\nu\rangle)^{2}\,d\mathcal{H}^{n-1}\\\ &=\int_{\partial
B_{r}(t)}\left\langle|\nabla
u^{t}|^{2}w,-\frac{w}{r}+\left\langle\frac{w}{r},\nu\right\rangle\nu\right\rangle\,d\mathcal{H}^{n-1}-\frac{n-1}{r}\int_{\partial
B_{r}(t)}|\nabla u^{t}|^{2}(\langle w,\nu\rangle)^{2}\,d\mathcal{H}^{n-1}\\\
&=-\frac{1}{r}\int_{\partial B_{r}(t)}|\nabla
u^{t}|^{2}d\mathcal{H}^{n-1}+\frac{1}{r}\int_{\partial B_{r}(t)}|\nabla
u^{t}|^{2}(\langle w,\nu\rangle)^{2}\,d\mathcal{H}^{n-1}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\
-\frac{n-1}{r}\int_{\partial B_{r}(t)}|\nabla u^{t}|^{2}(\langle
w,\nu\rangle)^{2}\,d\mathcal{H}^{n-1}\\\ &=-\frac{1}{r}\int_{\partial
B_{r}(t)}|\nabla u^{t}|^{2}d\mathcal{H}^{n-1}-\frac{n-2}{r}\int_{\partial
B_{r}(t)}|\nabla u^{t}|^{2}(\langle
w,\nu\rangle)^{2}\,d\mathcal{H}^{n-1},\end{split}$ (4.8)

where in the third line we have used the tangential gradient (4.1). Let us
focus on $III$; we have

$\begin{split}III&=-\int_{\partial B_{r}(t)}\langle\nu,D(|\nabla
u^{t}|^{2}w)\nu\rangle\,d\mathcal{H}^{n-1}=-2\int_{\partial
B_{r}(t)}\langle\nu,D^{2}(u^{t})\nabla u^{t}\rangle\ \langle
w,\nu\rangle\,d\mathcal{H}^{n-1}\\\ &=2\int_{\partial
B_{r}(t)}\dfrac{1}{|\nabla u^{t}|^{3}}\langle\nabla u^{t},D^{2}(u^{t})\nabla
u^{t}\rangle|\nabla u^{t}|^{2}\langle w,\nu\rangle\,d\mathcal{H}^{n-1}\\\
&=-\dfrac{2(n-1)}{r}\int_{\partial B_{r}(t)}|\nabla u^{t}|^{2}\langle
w,\nu\rangle\,d\mathcal{H}^{n-1},\end{split}$ (4.9)

where we have used the relations (2.7) and (2.8). The conclusion follows by
using (4.7), (4.8) and (4.9) in (4.6). ∎

Finally, the proof of the main Theorem is a direct consequence of the computed
expressions of the first and second order shape derivative of the first
Steklov-Dirichlet eigenvalue.

###### Proof of the Theorem 1.2.

Since $n\geq 2$, the claim is a direct consequence of the Theorems 3.1,
Corollary 3.2 and Theorem 4.1 ∎

######  Remark 4.2.

An immediate consequence of Theorem 1.2 is that $\sigma(t)$ is maximum when
the hole is centered at the symmetry point of $\Omega_{0}$, when $t=0$.

######  Remark 4.3.

We stress that, when $\Omega_{0}=B_{R}$, the authors in [HLS] prove the
following estimate in two dimensions for $R>r$:

$\liminf_{t\to(R-r)^{-}}\sigma(t)\geq\displaystyle\frac{r}{2R(R-r)},$ (4.10)

where $\sigma(t)=\sigma(\Omega(t))$ and $\Omega(t)=B_{R}\setminus B_{r}(t)$.
By Theorem 1.2, the estimate (4.10) can be written as the following lower
bound

$\frac{1}{R\log(\frac{R}{r})}=\sigma(B_{R}\setminus
B_{r})\geq\sigma(t)\geq\displaystyle\frac{r}{2R(R-r)},\quad\forall
t\in[0,\rho_{w}-r).$ (4.11)

Finally, we observe that the inequality (4.11) gives an upper and lower bound
for $\sigma(t)$ in terms of the two radius of the eccentric annulus.

## 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). We would like to thank the reviewer for his/her
suggestions to improve the paper.

## Compliance with Ethical Standards

This paper does not disclose of potential conflicts of interest.

## References

  * [A] M. S. Agranovich. On a mixed Poincaré-Steklov type spectral problem in a Lipschitz domain. Russ. J. Math. Phys. 13.3 (2006): 239-244.
  * [AA] T. V. Anoop, K. Ashok Kumar. Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality. ArXiv (2022): 1-21.
  * [AAK] T. V. Anoop, K. Ashok Kumar, S. Kesavan. A shape variation result via the geometry of eigenfunctions. J. Differential Equations 298 (2021): 430-462.
  * [BW] C. Bandle, A. Wagner. Second domain variation for problems with Robin boundary conditions. J. Optim. Theory Appl. 167.2 (2015): 430-463.
  * [B] G. Bellettini. Lecture notes on mean curvature flow: barriers and singular perturbations. Springer (2014): 327 pp.
  * [Bo] J. F. Bonder, P. Groisman, J. D. Rossi. Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach. Ann. Mat. Pura Appl. (4) 186.2 (2007): 341-358.
  * [DP] F. Della Pietra, G. Piscitelli. An optimal bound for nonlinear eigenvalues and torsional rigidity on domains with holes. Milan J. Math. 88.2 (2020): 373-384.
  * [D] B. Dittmar. Eigenvalue problems and conformal mapping, Handbook of complex analysis: geometric function theory. Vol. 2, Elsevier Sci. B. V., Amsterdam (2005): 669-686.
  * [ET] M. Egert, P. Tolksdorf. Characterizations of Sobolev functions that vanish on a part of the boundary. Discrete Contin. Dyn. Syst. Ser. S 10.4 (2017): 729-743.
  * [F] I. Ftouhi. Where to place a spherical obstacle so as to maximize the first Steklov eigenvalue. ESAIM: Control Optim. Calc. Var. 28.6 (2022): 1-21.
  * [GPPS] N. Gavitone, G. Paoli, G. Piscitelli, R. Sannipoli. An Isoperimetric inequality for the first Steklov-Dirichlet Laplacian eigenvalue of convex sets with a spherical hole. Pacific J. Math. 320.2 (2022): 241-259.
  * [GS] N. Gavitone, R. Sannipoli. On a Steklov-Robin eigenvalue problem. J. Math. Anal. Appl. (in press).
  * [HP] A. Henrot, M. Pierre. Shape variation and optimization. EMS Tracts in Mathematics 28 (2018): 401 pp.
  * [HLS] J. Hong, M. Lim, D.H. Seo. On the first Steklov?Dirichlet eigenvalue for eccentric annuli. Ann. Mat. Pur. Appl. (1923-) 201.2 (2022): 769-799.
  * [L] L. Li. On the placement of an obstacle so as to optimize the Dirichlet heat content. SIAM J. Math. Anal. 54.3 (2022): 3275-3291.
  * [P] B. V. E. Pal’tsev. Mixed problems with non-homogeneous boundary conditions in Lipschitz domains for second-order elliptic equations with a parameter. Sb. Math. 187.4 (1996): 525.
  * [PPS] G. Paoli, G. Piscitelli, R. Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Comm. Pure Appl. Anal. 20.1 (2021): 145-158.
  * [PPT] G. Paoli, G. Piscitelli, L. Trani, Sharp estimates for the first $p$-laplacian eigenvalue and for the $p$-torsional rigidity on convex sets with holes. ESAIM Control Calc. Var. 26.111 (2020): 1-15.
  * [S] D.H. Seo. _A shape optimization problem for the first mixed Steklov-Dirichlet eigenvalue._ Ann. Glob. Anal. Geom. 59.3: 345-365.
  * [Sc] L. Schwartz, _Cours d’analyse. 1 et 2_. Second edition. Hermann, Paris (1981).
  * [SZ] J. Sokolowski, P. Zolésio. Introduction to shape optimization. In: Introduction to shape optimization. Springer, Berlin, Heidelberg (1992): 5-12.
  * [VS] S. Verma, G. Santhanam. On eigenvalue problems related to the laplacian in a class of doubly connected domains. Monatsh. Math. 193 (2020): 879-899.