# Rigidity results for the $p$-Laplacian Poisson problem with Robin boundary
conditions

Alba Lia Masiello*, Gloria Paoli

###### Abstract

Let $\Omega\subset\mathbb{R}^{n}$ be an open, bounded and Lipschitz set. We
consider the Poisson problem for the $p-$Laplace operator associated to
$\Omega$ with Robin boundary conditions. In this setting, we study the
equality case in the Talenti-type comparison stated in [6]. We prove that the
equality is achieved only if $\Omega$ is a ball and both the function $u$ and
the right hand side $f$ of the Poisson equation are radial.

Keywords: Robin boundary conditions, $p$-Laplace operator, rigidity result,
Talenti comparison.
MSC 2020: 35J92, 35J25, 46E30.

E-mail address, A.L. Masiello (corresponding author):
<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.

E-mail address, G. Paoli<EMAIL_ADDRESS>
Department of Data Science (DDS) Chair in Dynamics, Control and Numerics
(Alexander von Humboldt-Professorship), Cauerstr. 11, 91058 Erlangen, Germany.

##  1 Introduction

Symmetrization techniques in the context of qualitative properties of
solutions to second-order elliptic boundary value problems are introduced by
Talenti in [25]. In this seminal paper, the author considers an open, bounded
and Lipschitz set $\Omega\subset\mathbb{R}^{n}$, the ball $\Omega^{\sharp}$
with the same measure as $\Omega$ and the solutions $u$ and $v$ to the
following problems

$\begin{cases}-\Delta u_{D}=f\,&\text{in }\operatorname{\Omega},\\\
u_{D}=0&\text{on
}\partial\Omega,\end{cases}\quad\qquad\quad\begin{cases}-\Delta
v_{D}=f^{\sharp}\,&\text{in }\operatorname{\Omega}^{\sharp},\\\
v_{D}=0&\text{on }\partial\Omega^{\sharp},\end{cases}$ (1)

where $f\in L^{2}(\Omega)$ is a positive function and $f^{\sharp}$ is its
Schwarz rearrangement of $f$ (see Definition 2.3). In this setting, Talenti
proves the following point-wise estimate:

$u_{D}^{\sharp}(x)\leq v_{D}(x),\quad\text{ for all }x\in\Omega^{\sharp}.$ (2)

For sake of completeness, we observe that this result is proved more generally
for a uniformly elliptic linear operator in divergence form.

A version of this result for nonlinear operators in divergence form is
contained in [26], which includes as a special instance the case of the
$p$-Laplace operator. Moreover, these results are later extended, for
instance, to the anisotropic elliptic operators in [2], to the parabolic case
in [4], and to higher order operators in [9, 28].

Once a comparison result holds, it is natural to ask whether the equality
cases can be characterized and, so, if a rigidity result is in force. In [3],
the rigidity result linked to problem (1) is proved. Indeed, the authors prove
that if equality holds in (2), then $\Omega$ is a ball, $u$ is radially
symmetric and decreasing and $f=f^{\sharp}$. Rigidity results for a generic
linear, elliptic second order operator can be found in [17] and [18]. To the
best of our knowledge, rigidity results for nonlinear operators with Dirichlet
boundary conditions are not present in the literature. In this paper, we
obtain, as a corollary of our results, the rigidity for the $p-$Laplace
operator with Dirichlet boundary conditions in any dimension (see Corollary
3.3).

For a long time, it was believed that comparison results could not be proved
by means of spherical rearrangement argument when dealing with Robin boundary
conditions, until the recent paper [5]. The authors consider the following
problems

$\begin{cases}-\Delta u=f\,&\text{in }\operatorname{\Omega},\\\\[4.30554pt]
\displaystyle{\frac{\partial u}{\partial\nu}+\beta u=0}&\text{on
}\partial\Omega,\end{cases}\quad\qquad\quad\begin{cases}-\Delta
v=f^{\sharp}\,&\text{in }\operatorname{\Omega}^{\sharp},\\\\[4.30554pt]
\displaystyle{\frac{\partial v}{\partial\nu}+\beta v=0}&\text{on
}\partial\Omega^{\sharp},\end{cases}$

and they prove a comparison result involving Lorentz norms of $u$ and $v$
whenever $f$ is a non negative function in $L^{2}(\Omega)$ and $\beta$ is a
positive parameter. In particular, in the case $f\equiv 1$, they prove

${\left\|u\right\|}_{L^{p}(\Omega)}\leq{\left\|v\right\|}_{L^{p}(\Omega^{\sharp})},\quad
p=1,2,$

and, if $n=2$, the pointwise comparison holds:

$u^{\sharp}(x)\leq v(x),\quad\text{ for all }x\in\Omega^{\sharp}.$ (3)

In [21], it is proved that (3) is rigid, i.e. the equality case is possible if
and only if $\Omega$ is a ball and $u$ is a radial and decreasing function.

Generalizations of the results contained in [5] can be found for the
anisotropic case in [24], for mixed boundary conditions in [1], in the case of
the Hermite operator in [13].

In the present paper, we focus our study on the rigidity for $p-$Laplace
operator. In this case, the comparison results are obtained in [6] and the
setting is the following.

Let $\Omega$ be a bounded, open and Lipschitz set in
$\operatorname{\mathbb{R}}^{n}$, with $n\geq 2$. Let $p\in(1,+\infty)$ and let
$f\in L^{p^{\prime}}(\Omega)$ be a positive function, where
$p^{\prime}=p/(p-1)$. The Poisson problem for the $p-$Laplace operator with
Robin boundary conditions is

$\begin{cases}-\Delta_{p}u:=-\text{div}({\left|\nabla u\right|}^{p-2}\nabla
u)=f&\text{ in }\Omega\\\\[4.30554pt] {\left|\nabla
u\right|}^{p-2}\displaystyle{\frac{\partial
u}{\partial\nu}}+\beta{\left|u\right|}^{p-2}u=0&\text{ on
}\partial\Omega,\end{cases}$ (4)

where $\nu$ is the unit exterior normal to $\partial\Omega$ and $\beta>0$. A
function $u\in W^{1,p}(\Omega)$ is a weak solution to (4) if

$\int_{\Omega}{\left|\nabla u\right|}^{p-2}\nabla
u\nabla\varphi\,dx+\beta\int_{\partial\Omega}{\left|u\right|}^{p-2}u\varphi\,d\mathcal{H}^{n-1}(x)=\int_{\Omega}f\varphi\,dx,\qquad\forall\varphi\in
W^{1,p}(\Omega).$ (5)

The symmetrized problem associated to (4) is the following

$\begin{cases}-\Delta_{p}v=f^{\sharp}&\text{ in
}\Omega^{\sharp}\\\\[4.30554pt] {\left|\nabla
v\right|}^{p-2}\displaystyle{\frac{\partial
v}{\partial\nu}}+\beta{\left|v\right|}^{p-2}v=0&\text{ on
}\partial\Omega^{\sharp}.\end{cases}$ (6)

In [6] the authors establish a comparison result between suitable Lorentz
norms (see Definition 2.4) of the solutions $u$ and $v$ to problems (4) and
(6) respectively. In particular, they prove

${\left\|u\right\|}_{L^{pk,p}(\Omega)}\,\leq{\left\|v\right\|}_{L^{pk,p}(\Omega^{\sharp})},\quad\,\;\forall\,0<k\leq\frac{n(p-1)}{(n-2)p+n},$
(7)

and in the case $f\equiv 1$, they prove

$u^{\sharp}(x)\leq v(x),\quad 1\leq p\leq\frac{n}{n-1}$ (8)

and

${\left\|u\right\|}_{L^{pk,p}(\Omega)}\,\leq{\left\|v\right\|}_{L^{pk,p}(\Omega^{\sharp})},\quad\,\;\forall\,0<k\leq\frac{n(p-1)}{(n-2)p+n},\quad\forall
p>1.$ (9)

In the present paper, we want to characterize the equality case in (7) and
(9), answering to the open problem contained in [21]. For simplicity, we state
the main Theorem only in the case $f\in L^{p^{\prime}}(\Omega)$ positive,
since in the case $f\equiv 1$ the proof is analogous, as we observe in Remark
4.1.

######  Theorem 1.1.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, open and Lipschitz set and let
$\Omega^{\sharp}$ be the ball centered at the origin with the same measure as
$\Omega$. Let $u$ be the solution to (4) and let $v$ be a solution to (6). If

${\left\|u\right\|}_{L^{pk,p}(\Omega)}={\left\|v\right\|}_{L^{pk,p}(\Omega^{\sharp})},\quad\text{for
some }k\in\left]0,\frac{n(p-1)}{(n-2)p+n}\right]$ (10)

then, there exists $x_{0}\in\operatorname{\mathbb{R}}^{n}$ such that

$\Omega=\Omega^{\sharp}+x_{0},\qquad u(\cdot+x_{0})=v(\cdot),\qquad
f(\cdot+x_{0})=f^{\sharp}(\cdot).$

The idea of the proof of Theorem 1.1 is the following. First of all, we prove
that hypothesis (10) implies that the superlevel sets of $u$ are balls. The
main difficulty is to prove that these balls are concentric.

Differently from the case of the Laplace operator with Dirichlet boundary
conditions studied in [4, 16], we can’t apply directly the steepest descent
method introduced in [8], because it strongly relays on the continuity of both
the solution and of its gradient. In the case of the $p-$Laplace equation, the
continuity of the solution up to the boundary depends on the regularity of the
given datum $f$. To overcome this regularity issue we show that $u$ is a
solution to a suitable Dirichlet problem and it satisfies the Pólya-Szegő
inequality with equality sign. Then, we can conclude that $u$ is radially
symmetric and decreasing, using the classical result contained in [10]. We
make use of Lemma 3.2, where the rigidity of the Poisson problem for the
$p$-Laplace operator with Dirichlet boundary condition is proved under the
assumption $f\in L^{p^{\prime}}(\Omega)$ and positive. Up to our knowledge,
Corollary 3.3 seems to be new in the literature.

The paper is organized as follows. In Section $2$ we recall some definitions
about rearrangement of functions and we state some lemmas that we will need in
the proof of the main theorem. Section $3$ is dedicated to the proof of the
main result and we conclude with a list of open problems.

##  2 Notation and Preliminaries

Throughout this article we will denote by $|\Omega|$ the Lebesgue measure of
an open and bounded Lipschitz set of $\mathbb{R}^{n}$, with $n\geq 2$, and by
$P(\Omega)$ the perimeter of $\Omega$. Since we are assuming that
$\partial\Omega$ is Lipschitz, we have that
$P(\Omega)=\mathcal{H}^{n-1}(\partial\Omega)$, where $\mathcal{H}^{n-1}$
denotes the $(n-1)-$dimensional Hausdorff measure.

We recall the classical isoperimetric inequality and and we refer the reader,
for example, to [22, 11, 12, 27] and to the original paper by De Giorgi [15].

######  Theorem 2.1 (Isoperimetric Inequality).

Let $E\subset\operatorname{\mathbb{R}}^{n}$ be a set of finite perimeter.
Then,

$n\omega_{n}^{\frac{1}{n}}{\left|E\right|}^{\frac{n-1}{n}}\leq P(E),$ (11)

where $\omega_{n}$ is the measure of the unit ball in
$\operatorname{\mathbb{R}}^{n}$. Equality occurs if and only if $E$ is
(equivalent to) a Ball.

For the following theorem, we refer to [7].

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

Let $\Omega\subset\mathbb{R}^{n}$ be an open set with Lipschitz boundary. Let
$f\in W^{1,1}_{\text{loc}}(\Omega)$ and let
$u:\Omega\to\operatorname{\mathbb{R}}$ be a measurable function. Then,

${\displaystyle\int_{\Omega}u(x)|\nabla
f(x)|dx=\int_{\mathbb{R}}dt\int_{\Omega\cap
f^{-1}(t)}u(y)\,d\mathcal{H}^{n-1}(y)}.$ (12)

Let us recall some basic notions about rearrangements. For a general overview,
see, for instance, [19].

######  Definition 2.1.

Let $u:\Omega\to\operatorname{\mathbb{R}}$ be a measurable function, the
_distribution function_ of $u$ is the function
$\mu:[0,+\infty[\,\to[0,+\infty[$ defined as the measure of the superlevel
sets of $u$, i.e.

$\mu(t)={\left|\Set{x\in\Omega\,:\,{\left|u(x)\right|}>t}\right|}.$

######  Definition 2.2.

Let $u:\Omega\to\operatorname{\mathbb{R}}$ be a measurable function, the
_decreasing rearrangement_ of $u$ is the distribution function of $\mu$. We
will denote it by $u^{\ast}(\cdot)$.

######  Remark 2.1.

Let us notice that the function $\mu(\cdot)$ is decreasing and right
continuous and the function $u^{\ast}(\cdot)$ is its generalized inverse.

######  Definition 2.3.

The _Schwartz rearrangement_ of $u$ is the function $u^{\sharp}$ whose
superlevel sets are balls with the same measure as the superlevel sets of $u$.

We have the following relation between $u^{\sharp}$ and $u^{*}$:

$u^{\sharp}(x)=u^{*}(\omega_{n}{\left|x\right|}^{n}),$

where $\omega_{n}$ is the measure of the unit ball in
$\operatorname{\mathbb{R}}^{n}$, and one can easily check that the functions
$u$, $u^{*}$ e $u^{\sharp}$ are equi-distributed, i.e. they have the same
distribution function, and it holds

$\displaystyle{{\left\|u\right\|}_{L^{p}(\Omega)}={\left\|u^{*}\right\|}_{L^{p}(0,{\left|\Omega\right|})}=\lVert{u^{\sharp}}\rVert_{L^{p}(\Omega^{\sharp})}},\quad\text{for
all }p\geq 1.$

We also recall the Hardy-Littlewood inequality, an important propriety of the
decreasing rearrangement,

$\int_{\Omega}{\left|h(x)g(x)\right|}\,dx\leq\int_{0}^{{\left|\Omega\right|}}h^{*}(s)g^{*}(s)\,ds.$

So, by choosing $h(\cdot)=\chi_{\left\\{{\left|u\right|}>t\right\\}}$, one has

$\int_{{\left|u\right|}>t}{\left|g(x)\right|}\,dx\leq\int_{0}^{\mu(t)}g^{*}(s)\,ds.$

We now introduce the Lorentz spaces (see [28] for more details on this topic).

######  Definition 2.4.

Let $0<p<+\infty$ and $0<q\leq+\infty$. The Lorentz space $L^{p,q}(\Omega)$ is
the space of those functions such that the quantity:

${\left\|u\right\|}_{L^{p,q}}=\begin{cases}\displaystyle{p^{\frac{1}{q}}\left(\int_{0}^{\infty}t^{q}\mu(t)^{\frac{q}{p}}\,\frac{dt}{t}\right)^{\frac{1}{q}}}&0<q<\infty\\\\[8.61108pt]
\displaystyle{\sup_{t>0}\,(t^{p}\mu(t))}&q=\infty\end{cases}$

is finite.

Let us observe that for $p=q$ the Lorentz space coincides with the $L^{p}$
space, as a consequence of the _Cavalieri’s Principle_

$\int_{\Omega}{\left|u\right|}^{p}=p\int_{0}^{+\infty}t^{p-1}\mu(t)\,dt.$

The solutions $u$ to problem (4) and $v$ to problem (6) are both
$p$-superharmonic and, as a consequence of the strong maximum principle and
the lower semicontinuity (see [29, 20]), they achieve their minima on the
boundary. If we set

$u_{m}=\min_{\Omega}u,\quad v_{m}=\min_{\Omega^{\sharp}}v$

the positiveness of $\beta$ and the Robin boundary conditions leads to
$u_{m}\geq 0$ and $v_{m}\geq 0$. Hence, $u$ and $v$ are strictly positive in
the interior of $\Omega$. Moreover, we can observe that

$u_{m}=\min_{\Omega}u\leq\min_{\Omega^{\sharp}}v=v_{m},$ (13)

indeed,

$\begin{split}v_{m}^{p-1}\text{P}(\Omega^{\sharp})&=\int_{\partial\Omega^{\sharp}}v(x)^{p-1}\,d\mathcal{H}^{n-1}(x)=\frac{1}{\beta}\int_{\Omega^{\sharp}}f^{\sharp}\,dx=\frac{1}{\beta}\int_{\Omega}f\,dx\\\
&=\int_{\partial\Omega}u(x)^{p-1}\,d\mathcal{H}^{n-1}(x)\\\ &\geq
u_{m}^{p-1}\text{P}(\Omega)\geq{u_{m}}^{p-1}\text{P}(\Omega^{\sharp}).\end{split}$

Moreover, it holds

$\mu(t)\leq\phi(t)={\left|\Omega\right|},\quad\forall t\leq v_{m}.$ (14)

Now, for $t\geq 0$, we introduce the following notations:

$U_{t}=\left\\{x\in\Omega:u(x)>t\right\\}\quad\partial U_{t}^{int}=\partial
U_{t}\cap\Omega,\quad\partial U_{t}^{ext}=\partial
U_{t}\cap\partial\Omega,\quad\mu(t)={\left|U_{t}\right|}$

and

$V_{t}=\left\\{x\in\Omega^{\sharp}:v(x)>t\right\\},\quad\partial
V_{t}^{int}=\partial V_{t}\cap\Omega,\quad\partial V_{t}^{ext}=\partial
V_{t}\cap\partial\Omega,\quad\phi(t)={\left|V_{t}\right|}.$

Because of the invariance of the $p-$Laplacian and of the Schwarz
rearrangement of $f$ by rotation, the solution $v$ to (6) is radial, so the
set $V_{t}$ are balls.

Now, we recall some technical Lemmas, proved in [6], that we need in what
follows. We recall the proof of Lemma 2.3 for reader’s convenience, while we
omit the proof of Lemma 2.4 and Lemma 2.5.

######  Lemma 2.3.

Let $u$ be the solution to (4) and let $v$ be the solution to (6). Then, for
almost every $t>0$, we have

$\gamma_{n}\mu(t)^{\left(1-\frac{1}{n}\right)\frac{p}{p-1}}\leq\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\mu^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{\text{ext}}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\right)$ (15)

and

$\gamma_{n}\phi(t)^{\left(1-\frac{1}{n}\right)\frac{p}{p-1}}=\left(\int_{0}^{\phi(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\phi^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
V_{t}^{\text{ext}}}\frac{1}{v}\,d\mathcal{H}^{n-1}(x)\right).$ (16)

where $\gamma_{n}=\left(n\omega_{n}^{1/n}\right)^{\frac{p}{p-1}}$.

###### Proof.

Let $t>0$ and $h>0$. In the weak formulation (5), we choose the following test
function

$\left.\varphi(x)=\right.\begin{cases}0&\text{ if }u<t\\\ u-t&\text{ if
}t<u<t+h\\\ h&\text{ if }u>t+h,\end{cases}$ (17)

obtaining

$\begin{split}\int_{U_{t}\setminus U_{t+h}}{\left|\nabla
u\right|}^{p}\,dx&+\beta h\int_{\partial
U_{t+h}^{ext}}u^{p-1}\,d\mathcal{H}^{n-1}(x)+\beta\int_{\partial
U_{t}^{ext}\setminus\partial
U_{t+h}^{ext}}u^{p-1}(u-t)\,d\mathcal{H}^{n-1}(x)\\\ &=\int_{U_{t}\setminus
U_{t+h}}f(u-t)\,dx+h\int_{U_{t+h}}f\,dx.\end{split}$ (18)

Dividing (18) by $h$, using coarea formula and letting $h$ go to $0$, we have
that for a.e. $t>0$

$\int_{\partial U_{t}}g(x)\,d\mathcal{H}^{n-1}=\int_{U_{t}}f\,dx,$

where

$\begin{cases}{\left|\nabla u\right|}^{p-1}&\text{ if }x\in\partial
U_{t}^{int},\\\ \beta u^{p-1}&\text{ if }x\in\partial U_{t}^{ext}.\end{cases}$
(19)

Using the isoperimetric inequality, for a.e. $t\in[0,+\infty)$ we have

$\displaystyle n\omega_{n}^{\frac{1}{n}}\mu(t)^{\frac{n-1}{n}}$
$\displaystyle\leq P(U_{t})=\int_{\partial U_{t}}\,d\mathcal{H}^{n-1}$ (20)
$\displaystyle\leq\left(\int_{\partial
U_{t}}g\,d\mathcal{H}^{n-1}(x)\right)^{\frac{1}{p}}\left(\int_{\partial
U_{t}}\frac{1}{g^{\frac{1}{p-1}}}\,d\mathcal{H}^{n-1}(x)\right)^{1-\frac{1}{p}}$
(21) $\displaystyle=\left(\int_{\partial
U_{t}}g\,d\mathcal{H}^{n-1}(x)\right)^{\frac{1}{p}}\left(\int_{\partial
U_{t}^{int}}\frac{1}{{\left|\nabla
u\right|}}\,d\mathcal{H}^{n-1}(x)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{ext}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\right)^{1-\frac{1}{p}}$ (22)
$\displaystyle\leq\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p}}\left(-\mu^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{ext}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\right)^{1-\frac{1}{p}},$ (23)

and, so, (15) follows. Finally, we notice that, if $v$ is the solution to (6),
then all the inequalities above are equalities, and, consequently, we have
(16). ∎

######  Lemma 2.4.

For all $\tau\geq v_{m}$, we have

$\int_{0}^{\tau}t^{p-1}\left(\int_{\partial
U_{t}^{ext}}\frac{1}{u(x)}\,d\mathcal{H}^{n-1}(x)\right)\,dt\leq\frac{1}{p\beta}\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds.$
(24)

Moreover,

$\int_{0}^{\tau}t^{p-1}\left(\int_{\partial
V_{t}\cap\partial\Omega^{\sharp}}\frac{1}{v(x)}\,d\mathcal{H}^{n-1}(x)\right)\,dt=\frac{1}{p\beta}\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds,$
(25)

######  Lemma 2.5 (Gronwall).

Let $\xi(\tau)$ be a continuously differentiable function, let $q>1$ and let
$C$ be a non negative constant $C$ such that the following differential
inequality holds

$\tau\xi^{\prime}(\tau)\leq(q-1)\xi(\tau)+C\quad\forall\tau\geq\tau_{0}>0.$

Then, we have

$\xi(\tau)\leq\left(\xi(\tau_{0})+\frac{C}{q-1}\right)\left(\frac{\tau}{\tau_{0}}\right)^{q-1}-\frac{C}{q-1}\quad\forall\tau\geq\tau_{0},$
(26)

and

$\xi^{\prime}(\tau)\leq\left(\frac{(q-1)\xi(\tau_{0})+C}{\tau_{0}}\right)\left(\frac{\tau}{\tau_{0}}\right)^{q-2}\quad\forall\tau\geq\tau_{0}.$
(27)

The following Lemma is contained in [4].

######  Lemma 2.6.

Let $f,g\in L^{2}(\Omega)$ be two positive functions. If

$\int_{\Omega}fg\;dx=\int_{\Omega^{\sharp}}f^{\sharp}g^{\sharp}\;dx,$ (28)

then, for every $\tau\geq 0$ there exists $t\geq 0$ such that we have, up to
zero measure set,

$\\{g>\tau\\}=\\{f>t\\}.$ (29)

We conclude this preliminary session, recalling the classical results
contained in [10] (see Theorem $1.1$ and Lemma $2.3$). In particular, the
result contained in (iii) of Lemma (2.7) gives the rigidity of the Pólya-Szegő
inequality (see [23]):

$\int_{\operatorname{\mathbb{R}}^{n}}|\nabla
u^{\sharp}|^{p}\,dx\leq\int_{\operatorname{\mathbb{R}}^{n}}{\left|\nabla
u\right|}^{p}\,dx,\quad\forall u\in W^{1,p}(\mathbb{R}^{n}).$ (30)

######  Theorem 2.7.

Let $w\in W^{1,p}(\mathbb{R}^{n})$, let $\sigma(t)$ be its distribution
function and let

$w_{M}:=\begin{cases}{\left\|w\right\|}_{\infty}&\text{if }w\in
L^{\infty}(\Omega)\\\ +\infty&\text{otherwise.}\end{cases}$

Then, the following are true:

  1. i.

For almost all $t\in(0,w_{M})$,

$\infty>-\sigma^{\prime}(t)\geq\displaystyle\int_{w^{-1}(t)}\dfrac{1}{|\nabla
w|}d\mathcal{H}^{n-1}$ (31)

  2. ii.

$\sigma$ is absolutely continuous if and only if

${\left|\Set{\lvert\nabla
w^{\sharp}\rvert=0}\cap\Set{0<w^{\sharp}<w_{M}}\right|}=0.$ (32)

  3. iii.

If

$\int_{\mathbb{R}^{n}}|\nabla w|^{p}=\int_{\mathbb{R}^{n}}|\nabla
w^{\sharp}|^{p},$ (33)

and (32) holds, then there exist a translate of $w^{\sharp}$ which is almost
everywhere equal to $w$.

######  Remark 2.2.

We observe that in [14], it is proved that the condition

${\left|\Set{{\left|\nabla w\right|}=0}\cap\Set{0<w<w_{M}}\right|}=0$ (34)

implies (32). So, by (iii) in Lemma 2.7, if we have (33) and (34), there
exists a translated of $w^{\sharp}$ which is almost everywhere equal to $w$.

######  Remark 2.3.

We observe that the Pólya -Szegő inequality (30) and the relative rigidity
result (iii) contained in Lemma 2.7 hold also if we assume $w\in
W^{1,p}_{0}(\Omega)$. Indeed, it is easily proved that for every $w\in
W^{1,p}_{0}(\Omega)$ one has $w^{\sharp}\in W^{1,p}_{0}(\Omega^{\sharp})$.

##  3 Proof of Theorem 1.1

In order to prove the main Theorem 1.1, we divide the proof into the following
steps. First of all, we prove that, under the assumptions of Theorem 1.1,
equality holds in (15) and this is the content of Proposition 3.1. Then, in
Proposition 3.4, we prove that equality in (15) implies the fact that $\Omega$
is a ball and $u$ and $f$ are radial functions. In order to prove this last
step, we need the key Lemma 3.2.

######  Proposition 3.1.

Let $u$ be the solution to (4) and let $v$ be the solution to (6). If there
exists $k$

$k\in\left]0,\frac{n(p-1)}{(n-2)p+n}\right]\quad\text{such that
}\quad{\left\|u\right\|}_{L^{pk,p}(\Omega)}={\left\|v\right\|}_{L^{pk,p}(\Omega^{\sharp})},$

then equality holds in (15) for almost every $t$.

###### Proof.

Since we are assuming that
${\left\|u\right\|}_{L^{pk,p}(\Omega)}={\left\|v\right\|}_{L^{pk,p}(\Omega^{\sharp})}$,
we have that

$\int_{0}^{+\infty}t^{p-1}\mu(t)^{\frac{1}{k}}\,dt=\int_{0}^{+\infty}t^{p-1}\phi(t)^{\frac{1}{k}}\,dt.$
(35)

Let us multiply (15) by $t^{p-1}\mu(t)^{\alpha}$, where
$\displaystyle\alpha={\frac{1}{k}-\left(1-\frac{1}{n}\right)\frac{p}{p-1}}$,
and let us integrate from 0 to $+\infty$:

$\displaystyle\gamma_{n}\int_{0}^{+\infty}t^{p-1}\mu^{\frac{1}{k}}(t)\,dt$
(36) $\displaystyle\leq$
$\displaystyle\int_{0}^{+\infty}\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\mu^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{ext}}\frac{1}{u}\,d\mathcal{H}^{n-1}\right)t^{p-1}\mu(t)^{\alpha}\,dt$
$\displaystyle\leq$
$\displaystyle\int_{0}^{+\infty}t^{p-1}\mu(t)^{\alpha}\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}(-\mu^{\prime}(t))\,dt+\frac{{\left|\Omega\right|}^{\alpha}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}},$

where in the last inequality we have used $\mu(t)\leq{\left|\Omega\right|}$
and (24) in Lemma 2.4. As far as $v$ is concerned, it holds

$\displaystyle\gamma_{n}\int_{0}^{+\infty}t^{p-1}\phi^{\frac{1}{k}}(t)\,dt$
(37)
$\displaystyle=\int_{0}^{+\infty}\left(\int_{0}^{\phi(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\phi^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
V_{t}^{ext}}\frac{1}{u}\,d\mathcal{H}^{n-1}\right)t^{p-1}\phi(t)^{\alpha}\,dt$
$\displaystyle=\int_{0}^{+\infty}t^{p-1}\phi(t)^{\alpha}\left(\int_{0}^{\phi(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}(-\phi^{\prime}(t))\,dt+\frac{{\left|\Omega\right|}^{\alpha}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}.$

We observe that the left-hand-side of (36) and the left-hand-side of (37) are
equal from (35). So, it follows

$\displaystyle\int_{0}^{+\infty}\\!\\!\\!\\!\\!\\!t^{p-1}\phi(t)^{\alpha}\left(\int_{0}^{\phi(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\\!\\!\\!\\!\\!\\!(-\phi^{\prime}(t))\,dt\leq$
$\displaystyle\int_{0}^{+\infty}\\!\\!\\!\\!\\!\\!t^{p-1}\mu(t)^{\alpha}\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\\!\\!\\!\\!\\!(-\mu^{\prime}(t))\,dt.$
(38)

Setting
$\displaystyle{F(l)=\int_{0}^{l}\omega^{\delta}\Biggl{(}\int_{0}^{\omega}f^{\ast}(s)\,ds\Biggr{)}^{\frac{1}{p-1}}\\!\\!\\!d\omega}$,
and integrating (38) by parts, we get

$\int_{0}^{\infty}t^{p-2}F(\phi(t))\,dt\leq\int_{0}^{\infty}t^{p-2}F(\mu(t))\,dt,$

being $\mu(t)=\phi(t)=0$ for $t>v_{M}$. In [6] (see the proof of Theorem
$1.1$), it is proved that

$\int_{0}^{\infty}t^{p-2}F(\mu(t))\,dt\leq\int_{0}^{\infty}t^{p-2}F(\phi(t))\,dt.$
(39)

and we recall here the proof for the reader’s convenience. In order to do
that, we multiply (15) by
$\displaystyle{t^{p-1}F(\mu(t))\mu(t)^{-\frac{(n-1)p}{n(p-1)}}}$ and we
integrate between $0$ and $\tau>v_{m}$. First, we observe that, by the
hypothesis $\displaystyle{k\leq\frac{n(p-1)}{(n-2)p+n}}$, it follows that the
function $\displaystyle{h(l)=F(l)l^{-\frac{(n-1)p}{n(p-1)}}}$ is non
decreasing. Hence, we obtain

$\begin{split}\int_{0}^{\tau}\gamma_{n}t^{p-1}F(\mu(t))\,dt&\leq\int_{0}^{\tau}\Bigl{(}-\mu^{\prime}(t)\Bigr{)}t^{p-1}\mu(t)^{-\frac{(n-1)p}{n(p-1)}}F(\mu(t))\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\,dt\\\
&+F({\left|\Omega\right|})\frac{{\left|\Omega\right|}^{-\frac{p(n-1)}{n(p-1)}}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}.\end{split}$

If we integrate by parts both sides of the last expression and we set

$\displaystyle{C=F({\left|\Omega\right|})\frac{{\left|\Omega\right|}^{-\frac{p(n-1)}{n(p-1)}}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}},$

we obtain

$\tau\int_{0}^{\tau}\gamma_{n}t^{p-2}F(\mu(t))\,dt+\tau
H_{\mu}(\tau)\leq\int_{0}^{\tau}\int_{0}^{t}r^{p-2}F(\mu(r))\,drdt+\int_{0}^{\tau}H_{\mu}(t)\,dt+C,$
(40)

where

$H_{\mu}(\tau)=-\int_{\tau}^{+\infty}t^{p-2}\mu(t)^{-\frac{p(n-1)}{n(p-1)}}F(\mu(t))\biggl{(}\int_{0}^{\mu(t)}f^{*}(s)\,ds\biggr{)}^{\frac{1}{p-1}}\,d\mu(t).$

Setting now

$\begin{multlined}\xi(\tau)=\int_{0}^{\tau}\int_{0}^{t}\gamma_{n}r^{p-2}F(\mu(r))\,dr+\int_{0}^{t}H_{\mu}(t)\,dt,\end{multlined}\xi(\tau)=\int_{0}^{\tau}\int_{0}^{t}\gamma_{n}r^{p-2}F(\mu(r))\,dr+\int_{0}^{t}H_{\mu}(t)\,dt,$

inequality (40) becomes

$\tau\xi^{\prime}(\tau)\leq\xi(\tau)+C.$

So, Lemma 2.5, with $\tau_{0}=v_{m}$ and q=2, gives

$\int_{0}^{\tau}\gamma_{n}t^{p-2}F(\mu(t))\,dt+H_{\mu}(\tau)\leq\left(\frac{\displaystyle{\int_{0}^{v_{m}}t^{p-2}F(\mu(t)\,dt+H_{\mu}(v_{m})+C}}{v_{m}}\right).$

Of course, the inequality holds as equality if we replace $\mu(t)$ with
$\phi(t)$, so we get:

$\int_{0}^{\tau}\gamma_{n}t^{p-2}F(\mu(t))\,dt+H_{\mu}(\tau)\leq\int_{0}^{\tau}\gamma_{n}F(\phi(t))\,dt+H_{\phi}(\tau),$

keeping in mind that $\mu(t)\leq\phi(t)={\left|\Omega\right|}$ for $t\leq
v_{m}$. Now, letting $\tau\to\infty$, one has

$\int_{0}^{\infty}t^{p-2}F(\mu(t))dt\leq\int_{0}^{\infty}t^{p-2}F(\phi(t))dt,$

since $H_{\mu}(\tau),H_{\phi}(\tau)\to 0$.

So, we get equality in (36) and, consequently, in (15) for almost every $t$,
indeed

$\displaystyle\gamma_{n}\int_{0}^{+\infty}t^{p-1}\mu^{\frac{1}{k}}(t)\,dt$
$\displaystyle\leq\int_{0}^{+\infty}\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\mu^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{ext}}\frac{1}{u}\,d\mathcal{H}^{n-1}\right)t^{p-1}\mu(t)^{\alpha}\,dt$
$\displaystyle\leq\int_{0}^{+\infty}t^{p-2}F(\mu(t))\,dt+\frac{{\left|\Omega\right|}^{\alpha}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}$
$\displaystyle=\int_{0}^{+\infty}t^{p-2}F(\phi(t))\,dt+\frac{{\left|\Omega\right|}^{\alpha}}{p\beta^{\frac{p}{p-1}}}\left(\int_{0}^{{\left|\Omega\right|}}f^{\ast}(s)\,ds\right)^{\frac{p}{p-1}}$
$\displaystyle=\int_{0}^{+\infty}\left(\int_{0}^{\phi(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\phi^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
V_{t}^{ext}}\frac{1}{v}\,d\mathcal{H}^{n-1}\right)t^{p-1}\phi(t)^{\alpha}\,dt$
$\displaystyle=\gamma_{n}\int_{0}^{+\infty}t^{p-1}\phi^{\frac{1}{k}}(t)\,dt=\gamma_{n}\int_{0}^{+\infty}t^{p-1}\mu^{\frac{1}{k}}(t)\,dt.$

∎

In the following Lemma we prove that a solution to a Dirichlet problem, such
that its distribution function satisfies the differential equation (42), is
necessarily defined on a ball and it has to be radial and decreasing.

######  Lemma 3.2.

Let $\Omega\subset\operatorname{\mathbb{R}}^{n}$ be an open, bounded and
Lipschitz set. Let $f\in L^{p^{\prime}}(\Omega)$ be a positive function, let
$w$ be a weak solution to

$\begin{cases}-\Delta_{p}w=f&\text{ in }\Omega\\\ w=0&\text{ on
}\partial\Omega,\end{cases}$ (41)

and let $\sigma$ be the distribution function of $w$. If $\sigma$ satisfies
the following condition

$\gamma_{n}\sigma(t)^{\left(1-\frac{1}{n}\right)\frac{p}{p-1}}=\left(\int_{0}^{\sigma(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\sigma^{\prime}(t)\right),\quad\text{for
a.e. }t\in[0,w_{M}]$ (42)

then, there exists $x_{0}$ such that

$\Omega=\Omega^{\sharp}+x_{0},\quad w(\cdot+x_{0})=w^{\sharp}(\cdot),\quad
f(\cdot+x_{0})=f^{\sharp}(\cdot).$

###### Proof.

First of all, we recall that $w$ is a weak solution to (41) if and only if

$\int_{\Omega}{\left|\nabla w\right|}^{p-2}\nabla
w\nabla\varphi\,dx=\int_{\Omega}f\varphi\,dx,\qquad\forall\varphi\in
W_{0}^{1,p}(\Omega).$ (43)

Arguing as in the proof of (15) in Lemma 2.3, choosing the same test function
$\varphi$, defined in (17),

$\left.\varphi(x)=\right.\begin{cases}0&\text{ if }w<t\\\ w-t&\text{ if
}t<w<t+h\\\ h&\text{ if }w>t+h,\end{cases}$

one obtains

$\int_{\partial W_{t}}{\left|\nabla
w\right|}^{p-1}\,d\mathcal{H}^{n-1}=\int_{W_{t}}f(x)\,dx\leq\int_{0}^{\sigma(t)}f^{\star}(s)\,ds,$
(44)

where $W_{t}=\\{x\in\Omega\;:w(x)>t\\}$.

If we apply the isoperimetric inequality to the superlevel set $W_{t}$, the
Hölder inequality and the Hardy-Littlewood inequality, we get, for almost
every $t$,

$\displaystyle n\omega_{n}^{\frac{1}{n}}\sigma(t)^{\frac{n-1}{n}}$
$\displaystyle\leq P(W_{t})=\int_{\partial W_{t}}\,d\mathcal{H}^{n-1}$ (45)
$\displaystyle\leq\left(\int_{\partial W_{t}}{\left|\nabla
w\right|}^{p-1}\,d\mathcal{H}^{n-1}(x)\right)^{\frac{1}{p}}\left(\int_{\partial
W_{t}}\frac{1}{{\left|\nabla
w\right|}}\,d\mathcal{H}^{n-1}(x)\right)^{1-\frac{1}{p}}$ (46)
$\displaystyle\leq\left(\int_{0}^{\sigma(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p}}\left(-\sigma^{\prime}(t)\right)^{1-\frac{1}{p}}.$
(47)

So, hypothesis (42) ensures us that equality holds in the isoperimetric
inequality (45), in the Hölder inequality (46) and in the Hardy-Littlewood
inequality (47).
We now divide the proof into three steps.
Step 1. Let us prove that the superlevel set $\Set{w>t}$ is a ball for all
$t\in[0,w_{M})$. Equality in (45) implies that, for almost every $t$, $W_{t}$
is a ball. On the other hand, for all $t\in[0,w_{M})$, there exists a sequence
$\Set{t_{k}}$ such that

  1. 1.

$t_{k}\to t$;

  2. 2.

$t_{k}>t_{k+1}$;

  3. 3.

$\\{w>t_{k}\\}$ is a ball for all $k$.

Since $\displaystyle{\Set{w>t}=\cup_{k}\Set{w>t_{k}}}$ can be written as an
increasing union of balls, $\\{w>t\\}$ is a ball for all $t$ and, in
particular, $\operatorname{\Omega}=\\{w>0\\}$ is a ball too and we obtain that
$\operatorname{\Omega}=x_{0}+\operatorname{\Omega}^{\sharp}$.
From now on, we can assume without loss of generality that $x_{0}=0$.
Step 2. Let us prove that the superlevel sets are concentric balls.
Equality in (46) implies also equality in Hölder inequality, i.e.

$\int_{\partial W_{t}}\,d\mathcal{H}^{n-1}=\left(\int_{\partial
W_{t}}{\left|\nabla
w\right|}^{p-1}\,d\mathcal{H}^{n-1}(x)\right)^{\frac{1}{p}}\left(\int_{\partial
W_{t}}\frac{1}{{\left|\nabla
w\right|}}\,d\mathcal{H}^{n-1}(x)\right)^{1-\frac{1}{p}}.$

This means that, for almost every $t$, ${\left|\nabla w\right|}$ is constant
$\mathcal{H}^{n-1}-$almost everywhere on $\partial W_{t}$ , and we denote by
$C_{t}$ the ($\mathcal{H}^{n-1}-$a.e.) constant value of ${\left|\nabla
w\right|}$ on $\partial W_{t}$. We claim that $C_{t}\neq 0$ for almost every
$t$. Indeed, (44) and the positivity of $f$ ensure us that

$P(W_{t})C_{t}^{p-1}=\int_{\partial W_{t}}{\left|\nabla
w\right|}^{p-1}\,d\mathcal{H}^{n-1}=\int_{W_{t}}f(x)\,dx>0.$

Integrating (42), we obtain $w^{\sharp}(x)=z(x)$, for all
$x\in\Omega^{\sharp}$, where $z$ is the solution to

$\begin{cases}-\Delta_{p}z=f^{\sharp}&\text{ in }\Omega^{\sharp}\\\ z=0&\text{
on }\partial\Omega^{\sharp},\end{cases}$ (48)

and it has the following explicit form:

$z(x)=\int_{\omega_{n}{\left|x\right|}^{n}}^{{\left|\Omega\right|}}\frac{1}{\gamma_{n}}\left(\int_{0}^{s}f^{\star}(r)\,dr\right)^{1/(p-1)}\frac{1}{s^{(1-1/n)(p/(p-1))}}\,ds,$

so it easily follows that

${\left|\Set{\lvert\nabla
w^{\sharp}\rvert=0}\cap\Set{0<w^{\sharp}<w_{M}}\right|}=0.$ (49)

Using (ii) in Lemma 2.7, we have that (49) implies the absolutely continuity
of $\sigma$.

Now, we denote by $C^{\sharp}_{t}$ the ($\mathcal{H}^{n-1}-$a.e.) constant
value of ${\left|\nabla w^{\sharp}\right|}$ on $\partial W^{\sharp}_{t}$. We
recall that it holds

$-\sigma^{\prime}(t)=\int_{\partial W^{\sharp}_{t}}\frac{1}{{\left|\nabla
w^{\sharp}\right|}}=\frac{P(\partial W^{\sharp}_{t})}{C^{\sharp}_{t}}.$

and, by the absolutely continuity of $\sigma$, we have

$-\sigma^{\prime}(t)=\int_{\partial W_{t}}\frac{1}{{\left|\nabla
w\right|}}=\frac{P(\partial W_{t})}{C_{t}}.$

Since $w$ and $w^{\sharp}$ are equi-distributed, we have,

$\frac{P(\partial W_{t})}{C_{t}}=\frac{P(\partial
W^{\sharp}_{t})}{C^{\sharp}_{t}}$

Moreover, since $P(\partial W_{t})=P(\partial W^{\sharp}_{t})$, we have that
$C_{t}=C^{\sharp}_{t}$. So, by the coarea formula, we get

$\displaystyle\int_{\Omega}{\left|\nabla w\right|}^{p}\,dx$
$\displaystyle=\int_{0}^{+\infty}\int_{\partial W_{t}}{\left|\nabla
w\right|}^{p-1}\,d\mathcal{H}^{n-1}=\int_{0}^{+\infty}C_{t}^{p-1}P(W_{t})\,dt\,d\mathcal{H}^{n-1}$
$\displaystyle=\int_{0}^{+\infty}\left(C_{t}^{\sharp}\right)^{p-1}P(W_{t})\,dt\,d\mathcal{H}^{n-1}=\int_{0}^{+\infty}\int_{\partial
W^{\sharp}_{t}}\lvert\nabla
w^{\sharp}\rvert^{p-1}\,d\mathcal{H}^{n-1}=\int_{\Omega^{\sharp}}\lvert\nabla
w^{\sharp}\rvert^{p}\,dx.$

By (iii) in Lemma 2.7, we conclude that $u=u^{\sharp}$.

Step 3. Let us prove that $f$ is radial and decreasing.

Equality in (47) reads, for almost every $t$,

$\int_{W_{t}}f(x)\,dx=\int_{0}^{\sigma(t)}f^{\ast}(s)\,ds.$

Moreover, for all $\tau\in[0,w_{M})$, there exists a sequence $\Set{\tau_{k}}$
such that

  1. 1.

$\tau_{k}\to\tau$;

  2. 2.

$\tau_{k}>\tau_{k+1}$;

  3. 3.

$\displaystyle\int_{W_{\tau_{k}}}f(x)\,dx=\displaystyle\int_{0}^{\sigma(\tau_{k})}f^{\ast}(s)\,ds$,

and, by the continuity of $\sigma(\cdot)$, we have

$\int_{0}^{\sigma(\tau)}f^{\ast}(s)\,ds=\lim_{k}\int_{0}^{\sigma(\tau_{k})}f^{\ast}(s)=\lim_{k}\int_{W_{\tau_{k}}}f(x)\,dx=\int_{W_{\tau}}f(x)\;dx.$

By Lemma 2.6, we have that for all $\tau$, there exists $\alpha_{\tau}$ such
that

$\\{w>\tau\\}=\\{f>\alpha_{\tau}\\}.$

Consequently, we have that also $f$ is radial and decreasing, so
$f=f^{\sharp}$. ∎

As a direct consequence of Lemma 3.2, we obtain the rigidity for the
$p-$Laplace operator with Dirichlet boundary conditions.

######  Corollary 3.3.

Let $\Omega\subset\operatorname{\mathbb{R}}^{n}$ be an open, bounded and
Lipschitz set. Let $f\in L^{p^{\prime}}(\Omega)$ be a positive function and
let $w$ and $z$ be weak solutions respectively to

$\begin{cases}-\Delta_{p}w=f&\text{ in }\Omega\\\ w=0&\text{ on
}\partial\Omega,\end{cases}\quad\begin{cases}-\Delta_{p}z=f^{\sharp}&\text{ in
}\Omega^{\sharp}\\\ z=0&\text{ on }\partial\Omega^{\sharp}.\end{cases}$ (50)

If $w^{\sharp}(x)=z(x)$, for all $x\in\Omega^{\sharp}$, then there exists
$x_{0}\in\operatorname{\mathbb{R}}^{n}$ such that

$\Omega=\Omega^{\sharp}+x_{0},\qquad w(\cdot+x_{0})=z(\cdot),\qquad
f(\cdot+x_{0})=f^{\sharp}(\cdot).$

###### Proof.

From the proof of Lemma 3.2, it follows that the distribution function of $w$,
denoted by $\sigma$, satisfies

$n\omega_{n}^{\frac{1}{n}}\sigma(t)^{\frac{n-1}{n}}\leq\left(\int_{0}^{\sigma(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p}}\left(-\sigma^{\prime}(t)\right)^{1-\frac{1}{p}}.$
(51)

Now, we integrate (51) from $0$ to $t$, obtaining

$u^{\ast}(t)=\int_{\sigma(t)}^{{\left|\Omega\right|}}\frac{1}{\gamma_{n}}\left(\int_{0}^{s}f^{\star}(r)\,dr\right)^{1/(p-1)}\frac{1}{s^{(1-1/n)(p/(p-1))}}\,ds=z^{\ast}(t).$

So, if $w^{\sharp}=z$, we have $w^{\ast}=z^{\ast}$, and consequently we obtain
equality in (51) for almost every $t\in[0,w_{M}]$. We can conclude by applying
Lemma 3.2.

∎

Now, using Lemma 3.2, we are in position to conclude the proof of the main
Theorem.

######  Proposition 3.4.

Let $\Omega\subset\mathbb{R}^{n}$ be an open, bounded and Lipschitz set and
let $\Omega^{\sharp}$ be the ball with the same measure as $\Omega$. Let $u$
be the solution to (4) and let $\mu$ be its distribution function. If equality
holds in (15), then there exists $x_{0}\in\operatorname{\mathbb{R}}^{n}$ such
that

$\Omega=\Omega^{\sharp}+x_{0},\qquad u(\cdot+x_{0})=v(\cdot),\qquad
f(\cdot+x_{0})=f^{\sharp}(\cdot).$

###### Proof.

Firstly, we claim that the superlevel sets $\Set{u>t}$ are balls for every
$t\in[0,u_{M})$. Equality in (15) implies the equality in (20), i.e.

$n\omega_{n}^{\frac{1}{n}}\mu(t)^{\frac{n-1}{n}}=P(U_{t}),\quad\text{for a. e.
}t\in[0,u_{M}]$

that means that almost every superlevel set is a ball. Arguing as in Step $1$
of Lemma 3.2, we can conclude that every superlevel set is a ball, so,
$\operatorname{\Omega}=\\{u>u_{m}\\}$ is a ball and we obtain that
$\operatorname{\Omega}=x_{0}+\operatorname{\Omega}^{\sharp}$.

Let us observe that for every $t,s\in[u_{m},u_{M}]$ with $t<s$, as both
$U_{t}$ and $U_{s}$ are balls, we have that $\partial U_{t}\cap\partial U_{s}$
contains at most one point. In particular, the function $w=u-u_{m}$ is a weak
solution to the Dirichlet problem (41) in $\Omega$.

We claim that $\sigma(t)={\left|\Set{w>t}\right|}$ satisfies (42). Since
$\Set{w>t}=\Set{u>t+u_{m}}$, we have $\sigma(t)=\mu(t+u_{m})$ for all
$t\in[0,u_{M}-u_{m}]$. Moreover, we have

$\int_{\partial U_{t}}\frac{1}{u}\,d\mathcal{H}^{n-1}=0,\quad\forall t>u_{m}$

So, using the fact that we have equality in (15) by hypothesis, we get

$\displaystyle\gamma_{n}\sigma(t)^{\left(1-\frac{1}{n}\right)\frac{p}{p-1}}$
$\displaystyle=\gamma_{n}\mu(t+u_{m})^{\left(1-\frac{1}{n}\right)\frac{p}{p-1}}$
$\displaystyle=\left(\int_{0}^{\mu(t+u_{m})}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\mu^{\prime}(t+u_{m})+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t+u_{m}}^{\text{ext}}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\right)$
$\displaystyle=\left(\int_{0}^{\sigma(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\sigma^{\prime}(t)\right),$

for all $t\in(0,u_{M}-u_{m})$. So, we can conclude by Lemma 3.2. ∎

We conclude now with the proof of the main Theorem.

###### Proof of Theorem 1.1.

From Proposition 3.1, we have that the hypothesis of Theorem 1.1

${\left\|u\right\|}_{L^{pk,p}(\Omega)}={\left\|v\right\|}_{L^{pk,p}(\Omega^{\sharp})},\quad\text{for
some }k\in\left]0,\frac{n(p-1)}{(n-2)p+n}\right]$

implies the following equality for almost every $t\in(0,u_{M})$

$\gamma_{n}\mu(t)^{\left(1-\frac{1}{n}\right)\frac{p}{p-1}}=\left(\int_{0}^{\mu(t)}f^{\ast}(s)\,ds\right)^{\frac{1}{p-1}}\left(-\mu^{\prime}(t)+\frac{1}{\beta^{\frac{1}{p-1}}}\int_{\partial
U_{t}^{\text{ext}}}\frac{1}{u}\,d\mathcal{H}^{n-1}(x)\right),$

where $\mu(t)$ is the distribution function of $u$.

Now, we are in position to apply Proposition 3.4, and, so, there exists
$x_{0}\in\operatorname{\mathbb{R}}^{n}$ such that

$\Omega=\Omega^{\sharp}+x_{0},\qquad u(\cdot+x_{0})=v(\cdot),\qquad
f(\cdot+x_{0})=f^{\sharp}(\cdot).$

∎

##  4 Remarks and open problems

######  Remark 4.1.

In [6] the authors also prove that in the case $f\equiv 1$, it holds

${\left\|u\right\|}_{L^{pk,p}(\Omega)}\leq{\left\|v\right\|}_{L^{pk,p}(\Omega^{\sharp})},\quad\quad{\rm
if}\>\;\displaystyle{0<k\leq\frac{n(p-1)}{n(p-1)-p}}.$ (52)

We stress that the proof of Theorem 1.1 can be adapted to case $f\equiv 1$,
regardless of the fact that now the admissible $k$ varies in a wider range.

######  Open problem 4.2.

  * Below we present a list of open problems and work in progress.

  * •

Generalize the rigidity results in the anisotropic setting, starting from the
comparison proved in [24].

  * •

Generalize the rigidity results to other problems, such as the ones
investigated in [1], [13].

## Acknowledgements

The authors Alba Lia Masiello and Gloria Paoli are supported by GNAMPA of
INdAM. The author Gloria Paoli is supported by the Alexander von Humboldt
Foundation with an Alexander von Humboldt research fellowship.

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