# On a reverse Kohler-Jobin inequality

Luca Briani  ,  Giuseppe Buttazzo  and  Serena Guarino Lo Bianco

###### Abstract.

We consider the shape optimization problems for the quantities
$\lambda(\Omega)T^{q}(\Omega)$, where $\Omega$ varies among open sets of
$\mathbb{R}^{d}$ with a prescribed Lebesgue measure. While the
characterization of the infimum is completely clear, the same does not happen
for the maximization in the case $q>1$. We prove that for $q$ large enough a
maximizing domain exists among quasi-open sets and that the ball is optimal
among nearly spherical domains.

Keywords: torsional rigidity; shape optimization; principal eigenvalue;
capacitary measures.

2020 Mathematics Subject Classification: 49Q10, 49J45, 49R05, 35P15, 35J25

##  1\. Introduction

In the present paper we consider two well-known quantities that occur in the
study of elliptic equations in the Euclidean space $\mathbb{R}^{d}$, $d\geq
2$. The first one is usually called torsional rigidity and is defined, for
every nonempty open set $\Omega\subset\mathbb{R}^{d}$ with finite Lebesgue
measure (in the following a domain), as

$T(\Omega)=\int w_{\Omega}\,dx,$

where $w_{\Omega}$ is the unique solution of the PDE

$-\Delta u=1\quad\hbox{in }\Omega,\qquad u\in H^{1}_{0}(\Omega).$

Equivalently, we may define $T(\Omega)$ as

$T(\Omega)=\max\left\\{\Big{[}\int u\,dx\Big{]}^{2}\Big{[}\int|\nabla
u|^{2}\,dx\Big{]}^{-1}\ :\ u\in H^{1}_{0}(\Omega)\setminus\\{0\\}\right\\}.$

In the integrals above and in the following we use the convention that
integrals without the indicated domain are intended over the entire space
$\mathbb{R}^{d}$. The quantity $T(\Omega)$ verifies the scaling property

$T(t\Omega)=t^{d+2}T(\Omega)\qquad\text{for every }t>0;$

in addition, the maximum of $T(\Omega)$ among domains with prescribed measure
is reached by the ball (Saint Venant inequality), which can be written in the
scaling free formulation as

$|\Omega|^{-(d+2)/d}T(\Omega)\geq|B|^{-(d+2)/d}T(B),$

for every domain $\Omega$ and for every ball $B\subset\mathbb{R}^{d}$.

The second quantity is the first eigenvalue $\lambda(\Omega)$ of the Dirichlet
Laplacian, defined as the smallest $\lambda$ such that the PDE

$-\Delta u=\lambda u\quad\hbox{in }\Omega,\qquad u\in H^{1}_{0}(\Omega)$

admits a nonzero solution. Equivalently, $\lambda(\Omega)$ can be defined
through the minimization of the Rayleigh quotient

$\lambda(\Omega)=\min\left\\{\Big{[}\int|\nabla u|^{2}\,dx\Big{]}\Big{[}\int
u\,dx\Big{]}^{-2}\ :\ u\in H^{1}_{0}(\Omega)\setminus\\{0\\}\right\\}.$

The quantity $\lambda(\Omega)$ verifies the scaling property

$\lambda(t\Omega)=t^{-2}\lambda(\Omega)\qquad\text{for every }t>0;$

in addition, the minimum of $\lambda(\Omega)$ among domains with prescribed
measure is reached by the ball (Faber-Krahn inequality), which can be written
in the scaling free formulation as

$|\Omega|^{2/d}\lambda(\Omega)\geq|B|^{2/d}\lambda(B),$

for every domain $\Omega$ and for every ball $B\subset\mathbb{R}^{d}$.

The study of relations between $T(\Omega)$ and $\lambda(\Omega)$ was performed
in several papers (see for instance [2], [3], [4], [5], [6], [13], [14], [19],
[22], [23], [24]), where some important inequalities were established. In
particular:

  * -

the Kohler-Jobin inequality

$\lambda(\Omega)T^{q}(\Omega)\geq\lambda(B)T^{q}(B),$

valid for every $q\in[0,2/(d+2)]$ and for every domain $\Omega$, where $B$ is
any ball in $\mathbb{R}^{d}$ with $|B|=|\Omega|$;

  * -

the Pólya inequality

$0<\frac{\lambda(\Omega)T(\Omega)}{|\Omega|}<1,$

valid for every domain $\Omega$ of $\mathbb{R}^{d}$.

In the present paper we consider the scaling free shape functional

$F_{q}(\Omega)=\frac{\lambda(\Omega)T^{q}(\Omega)}{|\Omega|^{\alpha_{q}}},\quad\quad\text{
with }\alpha_{q}=\frac{-2+q(d+2)}{d},$

and the two quantities

$\begin{cases}m_{q}=\inf\big{\\{}F_{q}(\Omega)\ :\ \Omega\text{
domain}\big{\\}};\\\ M_{q}=\sup\big{\\{}F_{q}(\Omega)\ :\ \Omega\text{
domain}\big{\\}}.\end{cases}$

While the situation for $m_{q}$ is fully clear, and by Kohler-Jobin
inequality, together with the Saint Venant inequality, we have

$m_{q}=\begin{cases}F_{q}(B)&\text{if }q\leq 2/(d+2)\\\ 0&\text{if
}q>2/(d+2),\end{cases}$

the characterization of $M_{q}$ is not yet complete. The results available up
to now are (see [2] and [4]):

  * $M_{q}=\infty$ for every $q<1$;

  * $M_{q}=1$ when $q=1$, with the upper bound $1$ not reached by any domain $\Omega$;

  * $M_{q}<\infty$ for every $q>1$.

We investigate here this last case. The maximal expectation would be having
the following result (reverse Kohler-Jobin inequality):

  * -

for every $q>1$ the supremum $M_{q}$ is reached on an optimal domain
$\Omega_{q}$;

  * -

there exists a threshold $q^{*}>1$ such that for every $q\geq q^{*}$ the
supremum $M_{q}$ is reached by a ball.

We are unable to prove the results in the strong form above, and we prove here
the weaker results below:

  * -

for every $q>1$ the supremum $M_{q}$ is reached on a capacitary measure
$\mu_{q}$ (Theorem 4.3);

  * -

there exists a threshold $q_{0}>1$ such that for every $q\geq q_{0}$ the
supremum $M_{q}$ is reached by a domain $\Omega_{q}$ (Theorem 5.3);

  * -

there exists another threshold $q_{1}$ such that for every $q\geq q_{1}$ the
ball is a maximizer for the shape functional $F_{q}$ among nearly spherical
domains (Theorem 6.2).

While finishing this paper we have been informed that similar problems are
considered in the work in progress [12].

##  2\. Capacitary measures

The concept of capacitary measure and the related properties is a very useful
tool for our purposes. When dealing with sequences of PDEs of the form

$-\Delta u=f\quad\hbox{in }\Omega_{n},\qquad u\in H^{1}_{0}(\Omega_{n}),$

a natural question is to establish if the sequence $u_{n,f}$ of solutions, or
a subsequence of it, converges in $L^{2}$ to some function $u_{f}$ and to
determine in this case the PDE that the function $u_{f}$ solves. Starting from
the pioneering papers [16], [17] is now well understood that the right
framework to treat such a kind of questions is that of capacitary measures.
Below we recall the main results and definitions following [11] and [25]. For
further information we refer the reader to the monographs [9], [21] and
references therein.

######  Definition 2.1.

We say that a nonnegative Borel regular measure $\mu$, possibly taking the
value $\infty$, is a capacitary measure if

$\mu(E)=0\text{ whenever $E$ is a Borel set with }\mathrm{cap}(E)=0,$

being $\mathrm{cap}(E)$ the capacity

$\mathrm{cap}(E)=\inf\Big{\\{}\int_{\mathbb{R}^{d}}|\nabla u|^{2}+u^{2}\,dx\
:\ u\in H^{1}_{0}(\mathbb{R}^{d}),\ u=1\text{ in a neighborhood of
}E\Big{\\}}.$

A property $P(x)$ is said to hold quasi-everywhere (briefly q.e.) if the set
where $P(x)$ does not hold has zero capacity. A Borel set
$\Omega\subset\mathbb{R}^{d}$ is said to be quasi-open if there exists a
function $u\in H^{1}(\mathbb{R}^{d})$ such that $\Omega=\\{u>0\\}$ up to a set
of capacity zero. A function $f:\mathbb{R}^{d}\to\mathbb{R}$ is said to be
quasi-continuous if there is a sequence of open sets
$\omega_{n}\subset\mathbb{R}^{d}$ such that
$\lim_{n\to\infty}\mathrm{cap}(\omega_{n})=0$ and $f$ is continuous when
restricted to $\mathbb{R}^{d}\setminus\omega_{n}$. It is well known (see for
instance [20]) that every Sobolev function has a quasi-continuous
representative, and that two quasi-continuous representatives coincide quasi-
everywhere. We then identify the space $H^{1}(\mathbb{R}^{d})$ with the space
of quasi-continuous representatives. We recall that a sequence $u_{n}\in
H^{1}(\mathbb{R}^{d})$ that converges in norm to some $u\in
H^{1}(\mathbb{R}^{d})$, converges quasi-everywhere (up to a subsequence) to
$u$.

Given $\mu$ a capacitary measure we denote by $H^{1}_{\mu}$ the following
space

$H^{1}_{\mu}=H^{1}(\mathbb{R}^{d})\cap
L^{2}_{\mu}(\mathbb{R}^{d})=\left\\{u\in H^{1}(\mathbb{R}^{d})\,:\,\int
u^{2}\,d\mu<\infty\right\\}.$

The space $H^{1}_{\mu}$ is an Hilbert space when endowed with
$\|u\|_{H^{1}_{\mu}}=\|u\|_{H^{1}(\mathbb{R}^{d})}+\|u\|_{L^{2}_{\mu}(\mathbb{R}^{d})}$,
where the quantity $\|u\|_{L^{2}_{\mu}(\mathbb{R}^{d})}$ is well defined,
being Sobolev functions defined up to a set of zero capacity. We always
identify two capacitary measures $\mu,\nu$ for which

$\int u^{2}d\mu=\int u^{2}d\nu,\hbox{ for every }u\in H^{1}(\mathbb{R}^{d}).$
(2.1)

If instead (2.1) holds with “$\leq$” we say that $\mu\leq\nu$, and in this
case we have $H^{1}_{\nu}\subseteq H^{1}_{\mu}$. We can associate to any open
set (or more generally to any quasi-open set) $\Omega\subset\mathbb{R}^{d}$
the capacitary measure $I_{\Omega}$ defined as follows

$I_{\Omega}(E):=\begin{cases}0&\hbox{ if }\mathrm{cap}(E\setminus\Omega)=0,\\\
\infty&\hbox{ if }\mathrm{cap}(E\setminus\Omega)>0.\end{cases}$

Notice that, if $\mu=I_{\Omega}$ for some open set
$\Omega\subset\mathbb{R}^{d}$, then $H^{1}_{\mu}=H^{1}_{0}(\Omega)$.

To extend the notion of torsional rigidity to a capacitary measure $\mu$ we
need to carefully deal with the fact that the embedding
$H^{1}_{\mu}\hookrightarrow L^{1}(\mathbb{R}^{d})$ can be noncompact and even
noncontinuous. Nevertheless we can follow an approximation argument: for every
$R>0$, let $w_{R}$ be the solution to the following minimization problem

$\min\left\\{\int|\nabla u|^{2}\,dx+\int u^{2}\,d\mu-\int u\,dx\ :\ u\in
H^{1}_{\mu}\cap H^{1}_{0}(B_{R})\right\\}$

The torsion function $w_{\mu}$ and the torsional rigidity $T(\mu)$ of the
capacitary measure $\mu$ are defined as:

$w_{\mu}:=\sup_{R>0}w_{R},\quad T(\mu):=\int w_{\mu}dx.$

The Dirichlet eigenvalue of $\mu$ can be defined through the following
Rayleigh-type quotient:

$\lambda_{1}(\mu)=\inf_{u\subset H^{1}_{\mu}\setminus\\{0\\}}\frac{\int|\nabla
u|^{2}\,dx+\int u^{2}\,d\mu}{\int u^{2}\,dx}.$

Clearly, if $\mu=I_{\Omega}$ for some domain $\Omega\subset\mathbb{R}^{d}$, we
have $T(\mu)=T(\Omega)$ and $\lambda(\mu)=\lambda(\Omega)$ (we adopt this
notation also if $\Omega$ is a quasi-open set). For a general capacitary
measure $\mu$, neither $\lambda(\mu)$ is necessarily attained by some function
$u\in H^{1}_{\mu}$ nor $T(\mu)$ is necessarily finite. However, as shown in
[10], it holds the following:

$w_{\mu}\in L^{1}(\mathbb{R}^{d})\Longleftrightarrow
T(\mu)<\infty\Longrightarrow\lambda_{1}(\mu)\hbox{ is attained by some }u\in
H^{1}_{\mu}.$

For every capacitary measure $\mu$ with $T(\mu)<\infty$ we define the set of
finiteness $A_{\mu}$ as the quasi-open set

$A_{\mu}:=\\{w_{\mu}>0\\}.$

In the case when $\mu=I_{\Omega}$, for some domain
$\Omega\subset\mathbb{R}^{d}$, we have $A_{\mu}=\Omega$. The set of capacitary
measures with finite torsion can be endowed with the following notion of
distance.

######  Definition 2.2.

Given two capacitary measures $\mu,\nu$ such that $w_{\mu},w_{\nu}\in
L^{1}(\mathbb{R}^{d})$ we define the $\gamma-$distance between them as
$d_{\gamma}(\mu,\nu)=\|w_{\mu}-w_{\nu}\|_{L^{1}(\mathbb{R}^{d})}$. We say that
a sequence $\mu_{n}$ $\gamma-$converges to $\mu$ if
$d_{\gamma}(\mu_{n},\mu)\to 0$ as $n\to\infty$. When
$I_{\Omega_{n}}\overset{\gamma}{\to}\mu$ we simply write
$\Omega_{n}\overset{\gamma}{\to}\mu$.

We summarize the main properties of the $\gamma-$distance below:

  * •

The space $(\\{\mu:\ \mu\hbox{ capacitary measure with }w_{\mu}\in
L^{1}(\mathbb{R}^{d})\\},d_{\gamma})$ is a complete metric space and the set
$\\{I_{\Omega}:\ \Omega\subset\mathbb{R}^{d}\hbox{ open set with
}w_{\Omega}\in L^{1}(\mathbb{R}^{d})\\}$ is a dense subset of it.

  * •

The functionals $\mu\mapsto\lambda(\mu)$ and $\mu\mapsto T(\mu)$ are
$\gamma-$continuous.

  * •

The map $\mu\mapsto|A_{\mu}|$, or more generally integral functionals as
$\int_{A_{\mu}}f(x)\,dx$ with $f\geq 0$ and measurable, are lower
semicontinuous with respect to the $\gamma$-convergence.

  * •

The $\gamma$-convergence of $\mu_{n}$ to $\mu$ implies the
$\Gamma$-convergence in $L^{2}(\mathbb{R}^{d})$ of the functionals
$\|\cdot\|_{H^{1}_{\mu_{n}}}:L^{2}(\mathbb{R}^{d})\to L^{2}(\mathbb{R}^{d})$
defined by

$\|u\|_{H^{1}_{\mu}}=\begin{cases}\|u\|_{H^{1}(\mathbb{R}^{d})}+\int
u^{2}\,d\mu_{n}\ &\hbox{if }u\in H^{1}_{\mu_{n}}\\\ \infty&\hbox{if }u\not\in
H^{1}\end{cases}$

to the functional $\|\cdot\|_{H^{1}_{\mu}}:L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})$ ,

$\|u\|_{H^{1}_{\mu}}=\begin{cases}\|u\|_{H^{1}(\mathbb{R}^{d})}+\int
u^{2}\,d\mu\ &\hbox{if }u\in H^{1}_{\mu}\\\ \infty&\hbox{if }u\not\in
H^{1}.\end{cases}$

  * •

For a given capacitary measures $\mu$ with finite torsion we call resolvent of
$\mu$ the linear compact and self-adjoint operator

$R_{\mu}:L^{2}(\mathbb{R}^{d})\to L^{2}(\mathbb{R}^{d}),\quad
R_{\mu}(f)=w_{\mu,f},$

where $w_{\mu,f}$ is the solution of the problem

$w_{\mu,f}\in H^{1}_{\mu},\quad-\Delta w_{\mu,f}+w_{\mu,f}\mu=f,$

in the sense that

$w_{\mu,f}\in H^{1}_{\mu},\quad\int\nabla w_{\mu,f}\cdot\nabla\phi dx+\int
w_{\mu,f}\phi d\mu=\int f\phi dx\ \hbox{ for every }\phi\in H^{1}_{\mu}.$

The $\gamma$-convergence of $\mu_{n}$ to $\mu$ implies the norm convergence of
$R_{\mu_{n}}$ to $R_{\mu}$, i.e.

$\lim_{n\to\infty}\|R_{\mu_{n}}-R_{\mu}\|_{\mathcal{L}(L^{2}(\mathbb{R}^{d}),L^{2}(\mathbb{R}^{d}))}=0.$

  * •

If $\mu_{n}$ is a sequence of capacitary measures whose set of finiteness have
uniformly bounded measures $|A_{\mu_{n}}|$, then

$\mu_{n}\overset{\gamma}{\to}\mu\Longleftrightarrow\|R_{\mu_{n}}-R_{\mu}\|_{\mathcal{L}(L^{2}(\mathbb{R}^{d}),L^{2}(\mathbb{R}^{d}))}\to
0\Longleftrightarrow\|u\|_{H^{1}_{\mu_{n}}}\overset{\Gamma}{\longrightarrow}\|u\|_{H^{1}_{\mu}}\hbox{
on }L^{2}(\mathbb{R}^{d}).$

The classical concentration-compactness principle of P.L. Lions was extended
to sequences of open sets in [8]. Notably, the following result holds.

######  Theorem 2.3.

Let $\Omega_{n}$ be a sequence of open sets with uniformly bounded measures.
Then there exists a subsequence (still denoted with the same indices $n$) such
that one of the following situations occurs.

  * -

Compactness: there exists a sequence $x_{n}\subset\mathbb{R}^{d}$ such that
the sequence of capacitary measures $I_{\Omega_{n}}(x_{n}+\cdot)$
$\gamma-$converges.

  * -

Vanishing: the sequence $R_{I_{\Omega_{n}}}$ converges in norm to $0$.
Moreover we have $\|w_{\Omega_{n}}\|_{L^{\infty}}\to 0$ and
$\lambda(\Omega_{n})\to\infty$, as $n\to\infty$.

  * -

Dichotomy: there exist two sequences of quasi-open sets
$\Omega^{1}_{n},\Omega_{n}^{2}\subset\Omega_{n}$ such that

    * -

$\mathrm{dist}(\Omega^{1}_{n},\Omega^{2}_{n})\to\infty$, as $n\to\infty$;

    * -

$d_{\gamma}(I_{\Omega_{m}},I_{\Omega^{1}_{n}\cup\Omega^{2}_{n}})\to 0$, as
$n\to\infty$;

    * -

$\liminf_{n\to\infty}T(\Omega^{1}_{n})>0$ and
$\liminf_{n\to\infty}T(\Omega^{2}_{n})>0$.

The proof of the theorem above can be deduced by combining Theorem 2.2 of [8]
and Theorem 3.5 of [11].

##  3\. Relaxation of $F_{q}$

In this section we characterize the relaxation of the functional $F_{q}$ to
the set of capacitary measures. We define the set $\mathcal{M}_{ad}$ of
admissible capacitary measures as

$\mathcal{M}_{ad}=\\{\mu\ :\ \mu\hbox{ capacitary measure with }\
0<|A_{\mu}|<\infty\\}.$

For $\mu\in\mathcal{M}_{ad}$ we define the relaxed form of our functional
$F_{q}$ as

$F_{q}(\mu)=\sup\Big{\\{}\limsup_{n}F_{q}(\Omega_{n})\ :\
\Omega_{n}\subset\mathbb{R}^{d}\hbox{ open set such that
}\Omega_{n}\overset{\gamma}{\to}\mu\Big{\\}},$

so that

$M_{q}=\sup\\{F_{q}(\mu)\,:\,\mu\in\mathcal{M}_{ad}\\}.$

######  Lemma 3.1.

Let $\mu\in\mathcal{M}_{ad}$ and $\Omega_{n}$ a sequence of domains such that
$\Omega_{n}\overset{\gamma}{\to}\mu$. If $|A_{\mu}|<\infty$ then
$\Omega_{n}\cap A_{\mu}\overset{\gamma}{\to}\mu$.

###### Proof.

Being the sequence $\Omega_{n}\cap A_{\mu}$ of uniformly bounded measure, by
the properties of $\gamma$-convergence seen above we have to show that

$\|u\|_{H^{1}_{\mu_{n}}}\overset{\Gamma}{\longrightarrow}\|u\|_{H^{1}_{\mu}}\hbox{
on }L^{2}(\mathbb{R}^{d}),$

where we set $\mu_{n}=I_{\Omega_{n}\cap A_{\mu}}$.

The “$\Gamma$-liminf” inequality readily follows by the fact that
$H^{1}_{\mu_{n}}=H^{1}_{0}(\Omega_{n}\cap A_{\mu})\subseteq
H^{1}_{0}(\Omega_{n})$ and by the $\Gamma$ convergence of
$\|\cdot\|_{H^{1}_{0}(\Omega_{n})}$ to $\|\cdot\|_{H^{1}_{\mu}}$ in
$L^{2}(\mathbb{R}^{d})$.

To prove the “$\Gamma$-limsup” inequality we can suppose without loss of
generality that $u\in H^{1}_{\mu}$. Since
$\Omega_{n}\overset{\gamma}{\to}\mu$, there exists a sequence $u_{n}\in
H^{1}_{0}(\Omega_{n})$ such that

$\begin{split}&u_{n}\longrightarrow u\hbox{ strongly
}L^{2}(\mathbb{R}^{d}),\\\ &\lim_{n\to\infty}\left(\int|\nabla
u_{n}|^{2}dx\right)=\int|\nabla u|^{2}dx+\int|u|^{2}d\mu.\end{split}$

We denote respectively by $u_{n}^{+}$ and $u_{n}^{-}$ the positive and
negative part of $u_{n}$. Since we have

$\int|\nabla(u_{n}^{+}-u_{n}^{-})|^{2}dx=\int|\nabla
u_{n}^{+}|^{2}dx+\int|\nabla u_{n}^{-}|^{2}dx,$

and $u_{n}=u_{n}^{+}-u_{n}^{-}$, by possibly passing to a subsequence (still
indexed by $n$) we can suppose that

$\begin{split}&\limsup_{n\to\infty}\left(\int|\nabla
u_{n}^{+}|^{2}dx\right)+\limsup_{n\to\infty}\left(\int|\nabla
u_{n}^{-}|^{2}dx\right)=\lim_{n\to\infty}\left(\int|\nabla(u_{n}^{+}-u_{n}^{-})|^{2}dx\right)\\\
&=\int|\nabla u|^{2}dx+\int u^{2}d\mu.\end{split}$ (3.1)

We define

$v_{n}^{+}=u_{n}^{+}\wedge u^{+}\in H^{1}(\mathbb{R}^{d}),\qquad
v_{n}^{-}=u_{n}^{-}\wedge u^{-}\in H^{1}(\mathbb{R}^{d}).$

Since $u\in H^{1}_{\mu}$ and $u_{n}\in H^{1}_{0}(\Omega_{n})$ we have $u=0$
q.e. on $A_{\mu}^{c}$ and $u_{n}=0$ q.e. on $\Omega_{n}^{c}$. This implies
that both $v_{n}^{+}$ and $v_{n}^{-}$ vanish q.e. on $(\Omega_{n}\cap
A_{\mu})^{c}$ and consequently that $v_{n}^{+},v_{n}^{-}\in
H^{1}_{0}(\Omega_{n}\cap A_{\mu})$. Moreover it is easy to show that

$v_{n}^{+}-v_{n}^{-}\longrightarrow u,\quad\hbox{strongly
}L^{2}(\mathbb{R}^{d}).$

Therefore the thesis is achieved if we show that

$\limsup_{n\to\infty}\left(\int|\nabla(v_{n}^{+}-v_{n}^{-})|^{2}\,dx\right)\leq\lim_{n\to\infty}\left(\int|\nabla(u_{n}^{+}-u_{n}^{-})|^{2}\,dx\right).$
(3.2)

We have

$\begin{split}\int|\nabla v^{+}_{n}|^{2}\,dx&=\int_{\\{u^{+}_{n}\leq
u^{+}\\}}|\nabla u^{+}_{n}|^{2}\,dx+\int_{\\{u^{+}_{n}>u^{+}\\}}|\nabla
u^{+}|^{2}\,dx\\\ &=\int|\nabla u^{+}_{n}|^{2}\,dx-\int\Big{(}|\nabla
u^{+}_{n}|^{2}-|\nabla u|^{2}\Big{)}1_{\\{u^{+}_{n}>u^{+}\\}}\,dx.\end{split}$
(3.3)

By lower semicontinuity we have

$\liminf_{n}\int\Big{(}|\nabla u^{+}_{n}|^{2}-|\nabla
u^{+}|^{2}\Big{)}1_{\\{u^{+}_{n}>u^{+}\\}}\,dx\geq 0.$ (3.4)

Indeed, to show the inequality above, it is enough to write

$\begin{split}\int\Big{(}|\nabla u^{+}_{n}|^{2}-|\nabla
u^{+}|^{2}\Big{)}1_{\\{u^{+}_{n}>u^{+}\\}}\,dx&=\int\Big{(}|\nabla
u^{+}_{n}|^{2}-|\nabla u^{+}|^{2}\Big{)}1_{\\{u^{+}_{n}\geq u^{+}\\}}\,dx\\\
&=\int|\nabla(u^{+}_{n}\vee u^{+})|^{2}-|\nabla u^{+}|^{2}\,dx\end{split}$

and to notice that $u^{+}_{n}\rightharpoonup u^{+}$ weakly in
$H^{1}(\mathbb{R}^{d})$ implies $u^{+}_{n}\vee u^{+}\rightharpoonup u^{+}$
weakly in $H^{1}(\mathbb{R}^{d})$ and so, by lower semicontinuity

$\liminf_{n}\int|\nabla(u^{+}_{n}\vee u^{+})|^{2}-|\nabla u^{+}|^{2}\,dx\geq
0.$

Combining (3.3) and (3.4) we deduce that

$\limsup_{n\to\infty}\left(\int|\nabla
v^{+}_{n}|^{2}\right)\leq\limsup_{n\to\infty}\left(\int|\nabla
u^{+}_{n}|^{2}\,dx\right).$ (3.5)

Similarly we have

$\limsup_{n\to\infty}\left(\int|\nabla
v^{-}_{n}|^{2}\right)\leq\limsup_{n\to\infty}\left(\int|\nabla
u^{-}_{n}|^{2}\,dx\right).$ (3.6)

Combining (3.1), (3.5) and (3.6) we finally deduce (3.2) and this concludes
the lemma. ∎

######  Remark 3.2.

By Lemma 3.1 for every measure $\mu\in\mathcal{M}_{ad}$ there exists a
sequence of quasi-open sets $\Omega_{n}$ (that can be taken open by a standard
approximation procedure) such that $I_{\Omega_{n}}$ $\gamma-$converges to
$\mu$ and for which

$|\Omega_{n}|\to|A_{\mu}|\quad\hbox{ as $n\to\infty$}.$

This in turns implies that the set

$\\{I_{\Omega}:\ \Omega\subset\mathbb{R}^{d}\hbox{ domain}\\}$

is $\gamma-$dense in $\mathcal{M}_{ad}$. Furthermore, we can extend both
Saint-Venant, Faber-Krahn and Pólya inequalities to any capacitary measure.
That is

$|A_{\mu}|^{-(d+2)/d}T(\mu)\leq|B|^{-(d+2)/d}T(B),\qquad|A_{\mu}|^{2/d}\lambda(\mu)\geq|B|^{2/d}\lambda(B),$
(3.7)

and

$0<|A_{\mu}|^{-1}\lambda(\mu)T(\mu)<1$ (3.8)

for every measure $\mu\in\mathcal{M}_{ad}$ and every ball
$B\subset\mathbb{R}^{d}$.

######  Proposition 3.3.

Let $\mu\in\mathcal{M}_{ad}$. Then we have

$|A_{\mu}|=\inf\Big{\\{}\liminf_{n}|\Omega_{n}|\ :\ \Omega_{n}\text{ domain,
}\Omega_{n}\overset{\gamma}{\to}\mu\Big{\\}}.$ (3.9)

The quantity $|A_{\mu}|$ is then the relaxation, in the $\gamma$-convergence,
of the Lebesgue measure $|\Omega|$. As a consequence, we have

$F_{q}(\mu)=\frac{\lambda(\mu)T^{q}(\mu)}{|A_{\mu}|^{\alpha_{q}}}.$ (3.10)

###### Proof.

The inequality $\leq$ in (3.9) follows from the $\gamma$-lower semicontinuity
of the map $\mu\mapsto|A_{\mu}|$ seen above. The opposite inequality follows
at once by Remark 3.2. Since $T(\mu)$ and $\lambda(\mu)$ are
$\gamma$-continuous, the proof of (3.10) is achieved by a similar argument. ∎

The scaling properties of the shape functionals $|\Omega|$, $\lambda(\Omega)$,
$T(\Omega)$ and $F_{q}(\Omega)$ extend to their relaxations $|A_{\mu}|$,
$\lambda(\mu)$, $T(\mu)$ and $F_{q}(\mu)$ in $\mathcal{M}_{ad}$. More
precisely, setting for $t>0$

$\mu_{t}(E)=t^{d-2}\mu(E/t),$

we have

$|A_{\mu_{t}}|=t^{d}|A_{\mu}|,\quad\lambda(\mu_{t})=t^{-2}\lambda(\mu),\quad
T(\mu_{t})=t^{d+2}T(\mu),\quad F_{q}(\mu_{t})=F_{q}(\mu).$

##  4\. Existence of an optimal measure for $q>1$

In [4] it is proved that the supremum $M_{1}=1$ is not attained in the class
of domains. In the next proposition we point out that the same occurs even in
the class $\mathcal{M}_{ad}$.

######  Proposition 4.1 (Nonexistence for $q=1$ of an optimal measure).

Given a capacitary measure $\mu\in\mathcal{M}_{ad}$ the problem
$\sup\\{F_{1}(\mu):\ \mu\in\mathcal{M}_{ad}\\}$ does not have a maximizer.

###### Proof.

The proof follows at once by exploiting Theorem 1.1. of [4] which asserts that
there exists a dimensional constant $c_{d}>0$ for which

$F_{1}(\Omega)\leq 1-\frac{c_{d}T(\Omega)}{|\Omega|^{1+\frac{2}{d}}},$ (4.1)

for every domain $\Omega$. Then, for every $\mu\in\mathcal{M}_{ad}$, by Remark
3.2 we can select a sequence $\Omega_{n}\overset{\gamma}{\to}A_{\mu}$ for
which

$F_{1}(\Omega_{n})\to F(\mu),\quad T(\Omega_{n})\to
T(\mu),\quad|\Omega_{n}|\leq|A_{\mu}|\quad\text{as }n\to\infty.$

Thus, using (4.1) with $\Omega=\Omega_{n}$ and passing to the limit as
$n\to\infty$, we get $F_{1}(\mu)<1=M_{1}$. ∎

To prove the main result of this section we need the following elementary
lemma.

######  Lemma 4.2.

Let $0<c_{1}<c_{2}<\infty$, $1<\alpha_{1}<\alpha_{2}<\infty$. Then, there
exists $\beta<1$ such that, for every $a,b,c,d\in(c_{1},c_{2})$ it holds

$\frac{(a+b)^{\alpha_{1}}}{(c+d)^{\alpha_{2}}}\leq\beta\max\left\\{\frac{a^{\alpha_{1}}}{c^{\alpha_{2}}},\frac{b^{\alpha_{1}}}{d^{\alpha_{2}}}\right\\}.$

###### Proof.

Letting $x=b/a$ and $y=d/c$, is enough to prove that

$\frac{(1+x)^{\alpha_{1}}}{(1+y)^{\alpha_{2}}}\leq\beta\max\left\\{1,\frac{x^{\alpha_{1}}}{y^{\alpha_{2}}}\right\\}.$

Suppose that $x\leq y$. Since $x\geq\frac{c_{1}}{c_{2}}$, it holds

$(1+x)^{\alpha_{1}}=(1+x)^{\alpha_{2}}(1+x)^{\alpha_{1}-\alpha_{2}}\leq(1+y)^{\alpha_{2}}\left(1+\frac{c_{1}}{c_{2}}\right)^{\alpha_{1}-\alpha_{2}}.$
(4.2)

Similarly, if $x>y$, since $x\leq\frac{c_{2}}{c_{1}}$, it holds

$\left(1+\frac{1}{x}\right)^{\alpha_{1}}\leq\left(1+\frac{1}{y}\right)^{\alpha_{2}}\left(1+\frac{1}{x}\right)^{\alpha_{1}-\alpha_{2}}\leq\left(1+\frac{1}{y}\right)^{\alpha_{2}}\left(1+\frac{c_{2}}{c_{1}}\right)^{\alpha_{1}-\alpha_{2}}.$
(4.3)

Eventually we achieve the thesis by letting

$\beta=\left(1+\frac{c_{1}}{c_{2}}\right)^{\alpha_{1}-\alpha_{2}}$

and combining (4.2) and (4.3). ∎

######  Theorem 4.3 (Existence for $q>1$ of an optimal measure).

For every $q>1$ there exists a measure $\mu^{\star}\in\mathcal{M}_{ad}$ such
that

$F_{q}(\mu^{\star})=\sup\left\\{F_{q}(\mu)\,:\,\mu\in\mathcal{M}_{ad}\right\\}.$

###### Proof.

We select a sequence $\mu_{n}\in\mathcal{M}_{ad}$ such that $F_{q}(\mu_{n})\to
M_{q}$, as $n\to\infty$. By density, we can suppose that
$\mu_{n}=I_{\Omega_{n}}$, for some sequence of open sets $\Omega_{n}$.
Further, being $F_{q}$ scaling free, we can also assume $|\Omega_{n}|=1$.
Hence, we can apply Theorem 2.3.

If dichotomy occurs, then there exist two sequences of quasi-open sets
$\Omega_{n}^{1},\Omega_{n}^{2}\subset\Omega_{n}$ such that

$\Omega_{n}^{1}\cap\Omega_{n}^{2}=\emptyset,\quad
d_{\gamma}(I_{\Omega_{n}},I_{\Omega^{1}_{n}\cup\Omega^{2}_{n}})\to
0\quad\text{ as }n\to\infty.$

Taking into account the Saint-Venant inequality and the fact that
$|\Omega_{n}|=1$, there exist constants $c_{1},c_{2}>0$, which depend only on
the dimension, such that

$c_{1}<\inf_{n}|T(\Omega_{n}^{i})|\leq\sup_{n}|T(\Omega_{n}^{i})|<c_{2},\quad
c_{1}<\inf_{n}|\Omega_{n}^{i}|\leq\inf_{n}|\Omega_{n}^{i}|<c_{2},\hbox{ for
$i=1,2$}.$

Since $\lambda_{1}$ is increasing with respect to set inclusion, we have

$\lambda_{1}(\Omega_{n})\leq\min\\{\lambda_{1}(\Omega_{n}^{1}),\lambda(\Omega^{2}_{n})\\}.$
(4.4)

Lemma 4.2 together with (4.4) gives

$\frac{\lambda(\Omega_{n})\left(T(\Omega_{n}^{1}\cup\Omega_{n}^{2})\right)^{q}}{|\Omega_{n}|^{\alpha_{q}}}\leq\frac{\lambda(\Omega_{n})\left(T(\Omega_{n}^{1})+T(\Omega_{n}^{2})\right)^{q}}{(|\Omega_{n}^{1}|+|\Omega_{n}^{2}|)^{\alpha_{q}}}\leq\beta\max_{i=1,2}\frac{\lambda(\Omega_{n}^{i})T^{q}(\Omega_{n}^{i})}{|\Omega_{n}^{i}|^{\alpha_{q}}}<F_{q}(\Omega_{n}).$

By taking the limit for $n\to\infty$ in the latter inequality we obtain the
contradiction

$\sup_{\mu\in\mathcal{M}_{ad}}F(\mu)<\sup_{\mu\in\mathcal{M}_{ad}}F(\mu),$

and hence dichotomy cannot occur. Now, the maximality condition on the
sequence $\Omega_{n}$ together with Pólya inequality gives that for $n$ large
enough

$\lambda(B)T^{q}(B)/|B|^{\alpha_{q}}\leq\lambda(\Omega_{n})T^{q}(\Omega_{n})=\lambda(\Omega_{n})T(\Omega_{n})\cdot
T^{q-1}(\Omega_{n})\leq T^{q-1}(\Omega_{n}),$ (4.5)

where $B$ is any ball of $\mathbb{R}^{d}$. In particular it cannot be
$\lim_{n\to\infty}T(\Omega_{n})=0$, and this rules out the vanishing case.

Therefore compactness holds and there exists a capacitary measure
$\mu^{\star}$ and a sequence $x_{n}\in\mathbb{R}^{d}$ such that
$I_{x_{n}+\Omega_{n}}$ $\gamma-$converges to $\mu^{\star}$.

By (4.5) we deduce that $T(\mu^{\star})>0$ which by (3.7) implies
$|A_{\mu^{\star}}|>0$ and hence that $\mu^{\star}$ belongs to
$\mathcal{M}_{ad}$. Clearly the measure $\mu^{\star}$ maximizes the functional
$F_{q}$ on $\mathcal{M}_{ad}$ and this concludes the proof. ∎

##  5\. Optimal measures are quasi-open sets for large $q$

We are now interested to prove that, when $q$ is large enough, optimal
measures $\mu$ coming from Theorem 4.3 can be represented as quasi-open sets.
We begin by recalling the following result, see [18] and [25] Proposition
3.83.

######  Theorem 5.1.

Let $\mu$ be a capacitary measure with finite torsion. Then the eigenfunctions
$u\in L^{2}(\mathbb{R}^{d})$ of the operator $-\Delta+\mu$ with unitary
$L^{2}$ norm are in $L^{\infty}(\mathbb{R}^{d})$ and satisfy

$\|u\|_{\infty}\leq e^{1/(8\pi)}\lambda(\mu)^{d/4}.$

We also use the following lemma.

######  Lemma 5.2.

For every $q>1$ let $\mu_{q}\in\mathcal{M}_{ad}$ be a maximal measure for the
functional $F_{q}$, such that $|A_{\mu_{q}}|=1$. Then

$\liminf_{q\to\infty}T(\mu_{q})>0.$

###### Proof.

Let $q_{n}$ be a diverging sequence and $B\subset\mathbb{R}^{d}$ be a ball of
unitary measure. By a standard diagonal argument we can select a sequence
$\Omega_{n}\subset\mathbb{R}^{d}$ of open sets such that $|\Omega_{n}|=1$ for
every $n$ and

$|F_{q_{n}}(\Omega_{n})-F_{q_{n}}(\mu_{q_{n}})|=o(T^{q_{n}}(B))\quad\hbox{as
$n\to\infty$}.$ (5.1)

Then we can apply Theorem 2.3 to the sequence $\Omega_{n}$. Dichotomy can be
ruled out by the same argument as in the proof of Theorem 4.3 once noticed
that a combination of (3.7) and (3.8) implies

$F^{1/{q_{n}}}_{q_{n}}(\mu)\leq T^{(q_{n}-1)/q_{n}}(B)\rightarrow T(B)\,\text{
as }n\rightarrow\infty.$

The vanishing case can be excluded too by following again the proof of Theorem
4.3. Indeed, for $n$ large enough, Pólya inequality and (5.1) imply

$T^{q_{n}-1}(\Omega_{n})\geq F_{q_{n}}(\Omega_{n})\geq
F_{q_{n}}(\mu_{q_{n}})-|F_{q_{n}}(\Omega_{n})-F_{q_{n}}(\mu_{q_{n}})|\geq
F_{q_{n}}(B)+o(T^{q_{n}}(B)).$

Hence we deduce

$\liminf_{n\to\infty}T^{(1-1/q_{n})}(\Omega_{n})>0,$

which implies that it cannot be $T(\Omega_{n})\to 0$, as $n\to\infty$.
Therefore compactness holds true and the sequence $\Omega_{n}$ has a
subsequence (still denoted by the same indices) that $\gamma$-converges to
some $\mu\in\mathcal{M}_{ad}$ up to translations.

By the maximality of $\mu_{q_{n}}$ it holds

$F^{1/q_{n}}_{q_{n}}(B)\leq
F^{1/q_{n}}_{q_{n}}(\mu_{q_{n}})=T(\Omega_{n})(\lambda(\Omega_{n})+o(1))^{1/q_{n}}$

and we deduce, passing to the limit as $n\to\infty$

$T(B)\leq T(\mu)=\lim_{n\to\infty}T(\Omega_{n}).$

Since the sequence $q_{n}$ was arbitrary we obtain the conclusion. ∎

######  Theorem 5.3.

Let $\mu\in\mathcal{M}_{ad}$ be an optimal measure for $F_{q}$ with $q>1$.
There exists $q_{0}>1$ such that for $q>q_{0}$ we have $\mu=I_{A_{\mu}}$. In
particular the optimal measure can be represented by a quasi-open set.

###### Proof.

Since $F_{q}$ is scaling free, we can suppose that $|A_{\mu}|=1$. Let
$\varepsilon>0$ be a small parameter and let $\mu_{\varepsilon}$ be the
capacitary measure defined by

$\mu_{\varepsilon}(E)=(1-\varepsilon)\mu(E).$

Being $A_{\mu}=\mathcal{A}_{\mu_{\varepsilon}}$ we have
$\mu_{\varepsilon}\in\mathcal{M}_{ad}$. We assume by contradiction that
$\mu\neq I_{A_{\mu}}$ (notice that this implies $\mu_{\varepsilon}\neq\mu$).
For the sake of brevity, we denote respectively by $w$ and $w_{\varepsilon}$
the torsion functions of $\mu$ and $\mu_{\varepsilon}$. It is easy to verify
that, as $\varepsilon\to 0$,

$\|\cdot\|_{H^{1}_{\mu_{\varepsilon}}}\overset{\Gamma}{\to}\|\cdot\|_{H^{1}_{\mu}},\quad\hbox{
on }L^{2}(\mathbb{R}^{d}),$

and therefore we have $\mu_{\varepsilon}\overset{\gamma}{\to}\mu$ and
$w_{\varepsilon}\to w$ in $L^{1}(\mathbb{R}^{d})$, as $\varepsilon\to 0$. Let
us denote by $t(\varepsilon)$, $l(\varepsilon)$ and $f_{q}(\varepsilon)$ the
real functions

$\varepsilon\mapsto
t(\varepsilon)=T(\mu_{\varepsilon}),\quad\varepsilon\mapsto
l(\epsilon)=\lambda(\mu_{\varepsilon}),\quad\varepsilon\mapsto
f_{q}(\varepsilon)=F_{q}(\mu_{\varepsilon}),$

and by $t^{\prime}_{+}(0)$, $l^{\prime}_{+}(0)$, $(f_{q})^{\prime}_{+}(0)$ the
limits for $\varepsilon\to 0$ of the respective different quotients.

By writing $w_{\varepsilon}=w+\varepsilon\xi_{\varepsilon}$ for some
$\xi_{\varepsilon}\in L^{1}(\mathbb{R}^{d})$ and using the fact that
$w,w_{\varepsilon}$ respectively weakly solve the PDEs:

$-\Delta w+w\mu=1,$ $-\Delta
w_{\varepsilon}+w_{\varepsilon}\mu_{\varepsilon}=1,$ (5.2)

we deduce that $\xi_{\varepsilon}$ weakly solves the PDE

$-\Delta\xi_{\varepsilon}+\xi_{\varepsilon}\mu_{\varepsilon}=w\mu.$ (5.3)

This allows us to compute the derivative

$\begin{split}t_{+}^{\prime}(0)&=\lim_{\varepsilon\to
0}\left(\int\xi_{\varepsilon}\,dx\right)=\lim_{\varepsilon\to
0}\left(\int\nabla w_{\varepsilon}\nabla\xi_{\varepsilon}dx+\int
w_{\varepsilon}\xi_{\varepsilon}d\mu_{\varepsilon}\right)\\\
&=\lim_{\varepsilon\to 0}\left(\int ww_{\varepsilon}d\mu\right),\end{split}$

where we test (5.2) with $\xi_{\varepsilon}$ and we use (5.3) tested with
$w_{\varepsilon}$. Since, as $\varepsilon\to 0$, $w_{\varepsilon}\to w$ in
$L^{1}(\mathbb{R}^{d})$ we obtain

$t_{+}^{\prime}(0)=\int w^{2}\,d\mu.$ (5.4)

We can treat with a similar argument the eigenvalue. Let $u,u_{\varepsilon}$
be the first eigenfunctions (with unitary $L^{2}$ norm) respectively of the
operator $-\Delta+\mu_{\varepsilon}$ and $-\Delta+\mu$ and let
$v_{\varepsilon}\in L^{2}(\mathbb{R}^{d})$ be such that
$u_{\varepsilon}=u+\varepsilon v_{\varepsilon}$. Since

$-\Delta u+u\mu=\lambda(\mu)u,\quad-\Delta
u_{\varepsilon}+u_{\varepsilon}\mu_{\varepsilon}=\lambda(\mu_{\varepsilon})u_{\varepsilon}$

we have

$-\Delta v_{\varepsilon}+v_{\varepsilon}\mu-u\mu-\varepsilon
v_{\varepsilon}\mu=\left(\frac{\lambda(\mu_{\varepsilon})-\lambda(\mu)}{\varepsilon}\right)u+\lambda(\mu_{\varepsilon})v_{\varepsilon}.$

By testing the PDE above with $u\in H^{1}_{\mu}$ and since $\int u^{2}dx=1$,
we obtain

$\left(\frac{\lambda(\mu_{\varepsilon})-\lambda(\mu)}{\varepsilon}\right)=\int\nabla
v_{\varepsilon}\nabla u\,dx+\int v_{\varepsilon}u\,d\mu-\int
u^{2}\,d\mu-\varepsilon\int
v_{\varepsilon}u\,d\mu-\lambda(\mu_{\varepsilon})\int v_{\varepsilon}u\,dx.$

By taking the limit as $\varepsilon\to 0$ and exploiting the fact that
$u_{\varepsilon}\to u$ weakly in $H^{1}_{\mu}$ and
$\lambda(\mu_{\varepsilon})\to\lambda(\mu)$ we get

$l_{+}^{\prime}(0)=-\int u^{2}\,d\mu.$ (5.5)

By combining (5.4) and (5.5) we get

$\begin{split}&(f_{q})_{+}^{\prime}(0)=l^{\prime}_{+}(0)T^{q}(\mu)+q\lambda(\mu)T^{q-1}(\mu)t^{\prime}_{+}(0)=F_{q}(\mu)\int\left(-\frac{u^{2}}{\lambda(\mu)}+q\frac{w^{2}}{T(\mu)}\right)\,d\mu.\end{split}$

Now, the optimality condition on $\mu$ implies $(f_{q})^{\prime}(0)\leq 0$ and
hence that

$\int\left(\frac{u^{2}}{\lambda(\mu)}-q\frac{w^{2}}{T(\mu)}\right)\,d\mu\geq
0.$ (5.6)

We claim that

$\frac{u^{2}}{\lambda(\mu)}-q\frac{w^{2}}{T(\mu)}<0\quad\hbox{q.e on
}\mathbb{R}^{d}$ (5.7)

for $q$ large enough. Indeed, by an application of Theorem 5.1 together with a
comparison principle, we have

$u\leq e^{1/(8\pi)}\lambda^{d/4+1}(\mu)w\quad\hbox{q.e on }\mathbb{R}^{d},$

and so by the Pólya inequality

$u^{2}\leq
e^{1/(4\pi)}\lambda^{d/2}(\mu)\frac{\lambda(\mu)}{T(\mu)}w^{2}\quad\hbox{q.e
on }\mathbb{R}^{d}.$

The latter implies that

$\frac{u^{2}}{\lambda(\mu)}-q\frac{w^{2}}{T(\mu)}\leq\frac{w^{2}}{T(\mu)}\left(e^{1/(4\pi)}\lambda^{d/2}(\mu)-q\right)\quad\text{q.e.
on }\mathbb{R}^{d}.$

Therefore, for every $q$ such that

$\sup_{\mu\in\mathcal{M}_{ad}}e^{1/(4\pi)}\lambda^{d/2}(\mu)<q,$

(5.7) is verified. Notice that the supremum in the inequality above is finite
as a consequence of Lemma 5.2 combined again with Pólya inequality.

To conclude it is now enough to notice that (5.7) contradicts (5.6). ∎

##  6\. Optimality for nearly spherical domains

In the following we consider the classes $\mathcal{S}_{\delta,\gamma}$ of
nearly spherical domains. Let $B_{1}$ be the unitary ball of $\mathbb{R}^{d}$.
A domain $\Omega$ such that

$|\Omega|=|B_{1}|,\quad\int_{\Omega}xdx=0,$

belongs to the class $\mathcal{S}_{\delta,\gamma}$ if there exists $\phi\in
C^{2,\gamma}(\partial B_{1})$ with $\|\phi\|_{L^{\infty}(\partial B_{1})}\leq
1/2$ and such that

$\partial\Omega=\\{x\in\mathbb{R}^{d}:\ x=(1+\phi(y))y,\ y\in\partial
B_{1}\\},\quad\|\phi\|_{C^{2,\gamma}}(\partial B_{1})\leq\delta.$

We recall the following result.

######  Theorem 6.1.

Let $\gamma\in(0,1)$. There exists $\delta=\delta(d,\gamma)>0$ such that if
$\Omega\in\mathcal{S}_{\delta,\gamma}$ then

$\begin{split}&T(B_{1})-T(\Omega)\geq C_{1}\|\phi\|^{2}_{H^{1/2}(\partial
B_{1})}\\\ &\lambda(\Omega)-\lambda(B_{1})\leq
C_{2}\|\phi\|^{2}_{H^{1/2}(\partial B_{1})}\end{split}$

for suitable constants $C_{1}$ and $C_{2}$ depending only on the dimension
$d$.

###### Proof.

The inequality for the torsional rigidity follows from Theorem 3.3 in [7]
while the inequality for the eigenvalue follows by combining Theorem 1.2 and
Lemma 2.8 of [15]. ∎

######  Theorem 6.2.

Let $\gamma\in(0,1)$. There exists $\delta>0$ and $q_{1}>1$ such that for
every $q\geq q_{1}$ and every $\Omega\in\mathcal{S}_{\gamma,\delta}$ it holds

$\lambda(B_{1})T^{q}(B_{1})\geq\lambda(\Omega)T^{q}(\Omega).$

###### Proof.

For every domain $\Omega$ we have

$\lambda(B_{1})T^{q}(B_{1})-\lambda(\Omega)T^{q}(\Omega)=\lambda(B_{1})(T^{q}(B_{1})-T^{q}(\Omega))+T^{q}(\Omega)(\lambda(B_{1})-\lambda(\Omega)),$

which, by the elementary inequality

$x^{q}-y^{q}\geq qy^{q-1}(y-x),\quad\hbox{for every }x,y\geq 0,\ q>1,$

implies

$\lambda(B_{1})T^{q}(B_{1})-\lambda(\Omega)T^{q}(\Omega)\geq
T^{q-1}(\Omega)[q(T(B_{1})-T(\Omega))-T(\Omega)(\lambda(\Omega)-\lambda(B_{1}))].$
(6.1)

Let $\delta$ the constant determined by Theorem 6.1 and assume
$\Omega\in\mathcal{S}_{\gamma,\delta}$. Since $2^{-1}B_{1}\subset\Omega\subset
2B_{1}$, we get

$2^{-(2+d)}T(B_{1})\leq T(\Omega)\leq 2^{2+d}T(B_{1}).$

Combining Theorem 6.1 and inequality (6.1) we get

$\lambda(B_{1})T^{q}(B_{1})-\lambda(\Omega)T^{q}(\Omega)\geq(2^{-(2+d)}T(B_{1}))^{q-1}(qC_{1}-2^{2+d}C_{2}T(B_{1}))\|\phi\|^{2}_{H^{1/2}(\partial
B_{1})}.$

Hence, if $q$ is such that

$q\geq 2^{d+2}\frac{C_{2}}{C_{1}}T(B_{1}),$

we obtain

$\lambda(B_{1})T^{q}(B_{1})\geq\lambda(\Omega)T^{q}(\Omega)$

and this concludes the proof. ∎

######  Remark 6.3.

Although for large $q$ we expect the ball to be optimal for the functional
$F_{q}$, it is easy to see that this does not occur when $q$ approaches $1$.
Indeed, if the ball maximizes $F_{q}$ for every $q>1$, passing to the limit as
$q\to 1$, this would happen also for $q=1$, which is not true, even in the
class of convex domains. To see this it is enough to notice that

$F_{1}(B_{1})=\frac{\lambda(B_{1})}{d(d+2)}\leq\frac{d+4}{2(d+2)},$

where the last inequality follows simply by taking $u(x)=1-|x|^{2}$ as a test
function for $\lambda(B_{1})$. On the other hand, taking as
$\Omega_{\varepsilon}$ the thin slab $]0,1[^{d-1}\times]0,\varepsilon[$, gives

$\lim_{\varepsilon\to 0}F_{1}(\Omega_{\varepsilon})=\frac{\pi^{2}}{12}$

and

$\frac{\pi^{2}}{12}>\frac{d+4}{2(d+2)}\qquad\text{for every }d\geq 2.$

Acknowledgments. SG gratefully acknowledges Professor Dorin Bucur for the
stimulating discussions, and the hospitality, during a research period, at
Université de Savoie-Mont Blanc. The work of GB is part of the project
2017TEXA3H “Gradient flows, Optimal Transport and Metric Measure Structures”
funded by the Italian Ministry of Research and University. The authors are
members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le
loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica
(INdAM).

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Luca Briani:
Dipartimento di Matematica, Università di Pisa
Largo B. Pontecorvo 5, 56127 Pisa - ITALY
<EMAIL_ADDRESS>

Giuseppe Buttazzo:
Dipartimento di Matematica, Università di Pisa
Largo B. Pontecorvo 5, 56127 Pisa - ITALY
<EMAIL_ADDRESS>
http://www.dm.unipi.it/pages/buttazzo/

Serena Guarino Lo Bianco:
Dipartimento di Scienze F.I.M., Università degli Studi di Modena e Reggio
Emilia
Via Campi 213/A, 41125 Modena - ITALY
<EMAIL_ADDRESS>