Rigidity Results for the p-Laplacian Poisson Problem with Robin Boundary Conditions

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: we prove that the equality is achieved only if $\Omega$ is a ball and both the solution u and the right-hand side f of the Poisson equation are radial and decreasing.

1 Introduction

Symmetrization techniques in the context of qualitative properties of solutions to second-order elliptic boundary value problems are introduced by Talenti in [28]. In this seminal paper, the author considers an open and bounded 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

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_D=f \, &{}\text {in } {{\,\mathrm{\Omega }\,}}, \\ u_D=0 &{} \text {on } \partial \Omega , \end{array}\right. } \quad \qquad \quad {\left\{ \begin{array}{ll} -\Delta v_D=f^\sharp \, &{} \text {in } {{\,\mathrm{\Omega }\,}}^\sharp , \\ v_D=0 &{} \text {on } \partial \Omega ^\sharp , \end{array}\right. } \end{aligned}$$

(1)

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

$$\begin{aligned} u_D^\sharp (x) \le v_D(x), \quad \text { for all } x\in \Omega ^\sharp . \end{aligned}$$

(2)

For the 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 [29], which includes as a special instance the case of the p-Laplace operator. Moreover, these results are later extended, for instance, to anisotropic elliptic operators in [2], to the parabolic case in [4, 24], and to higher order operators in [9, 31].

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 $f=f^\sharp$ up to a translation. Rigidity results for a generic linear, elliptic second-order operator can be found in [18] and [20]. 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

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=f \, &{}\text {in } {{\,\mathrm{\Omega }\,}}, \\ \displaystyle {\frac{\partial u}{\partial \nu }+\beta u=0} &{} \text {on } \partial \Omega , \end{array}\right. } \quad \qquad \quad {\left\{ \begin{array}{ll} -\Delta v=f^\sharp \, &{} \text {in } {{\,\mathrm{\Omega }\,}}^\sharp , \\ \displaystyle {\frac{\partial v}{\partial \nu }+\beta v=0} &{} \text {on } \partial \Omega ^\sharp , \end{array}\right. } \end{aligned}$$

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

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

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

$$\begin{aligned} u^\sharp (x) \le v(x), \quad \text { for all } x\in \Omega ^\sharp . \end{aligned}$$

(3)

Generalizations of the results contained in [5] can be found for the anisotropic case in [27], 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 of the 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 ${{\,\mathrm{\mathbb {R}}\,}}^n$, with $n\ge 2$. Let $p\in (1,+\infty )$ and let $f\in L^{p'}(\Omega )$ be a positive function, where $p'=p/(p-1)$. The Poisson problem for the p-Laplace operator with Robin boundary conditions is

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p u:= -\text {div}({\left| \nabla u\right| }^{p-2} \nabla u)=f &{} \text { in } \Omega \\ {\left| \nabla u\right| }^{p-2} \displaystyle {\frac{\partial u}{\partial \nu }} + \beta {\left| u\right| }^{p-2}u =0 &{} \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(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

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

(5)

The symmetrized problem associated to (4) is the following

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p v= f^{\sharp } &{} \text { in } \Omega ^\sharp \\ {\left| \nabla v\right| }^{p-2} \displaystyle {\frac{\partial v}{\partial \nu }} + \beta {\left| v\right| }^{p-2} v =0 &{} \text { on } \partial \Omega ^\sharp . \end{array}\right. } \end{aligned}$$

(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

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

(7)

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

$$\begin{aligned} u^\sharp (x)\le v(x), \quad 1 \le p \le \frac{n}{n-1} \end{aligned}$$

(8)

and

$$\begin{aligned} {\left\| u\right\| }_{L^{pk,p}(\Omega )} \, \le {\left\| v\right\| }_{L^{pk,p}(\Omega ^\sharp )}, \quad \, \; \forall \, 0 < k \le \frac{n(p-1)}{n(p-1)-p}, \quad \forall p>1, \end{aligned}$$

(9)

that is an improvement of (7) in the case $p\le 2n$.

In the present paper, we want to characterize the equality case in (7) and (9), answering the open problem contained in [23].

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 $p\in (1,+\infty )$ and let f be a positive function in $L^{p'}(\Omega )$. Let u be the solution to (4) and v be the solution to (6). If

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

(10)

then, there exists $x_0\in {{\,\mathrm{\mathbb {R}}\,}}^n$ such that

$$\begin{aligned} \Omega =\Omega ^\sharp +x_0, \qquad u(\cdot +x_0)=u^\sharp (\cdot ), \qquad f(\cdot + x_0)=f^\sharp (\cdot ). \end{aligned}$$

For simplicity, we state the rigidity result relative to (7) and we refer to Remark 3.1 for the rigidity result of (9), since the proof is analogous.

The idea of the proof of Theorem 1.1 is the following. We first prove that hypothesis (10) implies that the superlevel sets of u are balls. The main challenge lies in proving that these balls are concentric. Our problem is different from the case of the Laplace operator with Dirichlet boundary conditions studied in [4, 17], and from the case of the Laplace operator in the plane with Robin boundary conditions and $f\equiv 1$ studied in [23]. Indeed, we can’t apply directly the steepest descent method introduced in [8], because it strongly relies on the continuity of both the solution and its gradient. In particular, in [23] the constant right-hand side $f\equiv 1$ implies the regularity of the solutions that allows the authors to apply the steepest descent method. On the other hand, 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 $f\in L^{p'}(\Omega )$ and positive. Up to our knowledge, Corollary 3.3 seems to be novel in the literature.

The paper is organized as follows. In Sect. 2 we recall some definitions about the 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 $n\ge 2$, 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 we refer the reader, for example, to [11, 12, 25, 30] and to the original paper by De Giorgi [16].

Theorem 2.1

(Isoperimetric Inequality) Let $E\subset {{\,\mathrm{\mathbb {R}}\,}}^n$ be a set of finite perimeter. Then,

$$\begin{aligned} n \omega _n^{\frac{1}{n}} {\left| E\right| }^{\frac{n-1}{n}}\le P(E), \end{aligned}$$

(11)

where $\omega _n$ is the measure of the unit ball in ${{\,\mathrm{\mathbb {R}}\,}}^n$. Equality occurs if and only if E is a ball up to a set of measure zero.

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 $u:\Omega \rightarrow {{\,\mathrm{\mathbb {R}}\,}}$ be a measurable function. Then,

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

(12)

Let us recall some basic notions about rearrangements. For a general overview, see, for instance, [21].

Definition 2.1

Let $u: \Omega \rightarrow {{\,\mathrm{\mathbb {R}}\,}}$ be a measurable function, the distribution function of u is the function $\mu : [0,+\infty [\, \rightarrow [0, +\infty [$ defined as the measure of the superlevel sets of ${\left| u\right| }$, i.e.

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

Definition 2.2

Let $u: \Omega \rightarrow {{\,\mathrm{\mathbb {R}}\,}}$ be a measurable function, the decreasing rearrangement of u, denoted by $u^*(\cdot )$, is defined as

$$\begin{aligned} u^*(s)=\inf \{t\ge 0:\mu (t)<s\}. \end{aligned}$$

Remark 2.1

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

Definition 2.3

The Schwartz rearrangement of u is the function $u^\sharp$ whose superlevel sets are concentric balls with the same measure as the superlevel sets of u.

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

$$\begin{aligned} u^\sharp (x)= u^*(\omega _n{\left| x\right| }^n), \end{aligned}$$

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

$$\begin{aligned} \displaystyle {{\left\| u\right\| }_{L^p(\Omega )}={\left\| u^*\right\| }_{L^p(0, {\left| \Omega \right| })}=\Vert {u^\sharp }\Vert _{L^p(\Omega ^\sharp )}}, \quad \text {for all } p\ge 1. \end{aligned}$$

We also recall the Hardy-Littlewood inequality, an important property of the decreasing rearrangement (see [19])

$$\begin{aligned} \int _{\Omega } {\left| h(x)g(x)\right| } \, dx \le \int _{0}^{{\left| \Omega \right| }} h^*(s) g^*(s) \, ds. \end{aligned}$$

Thus, choosing $h(\cdot )=\chi _{\left\{ {\left| u\right| }>t\right\} }$, one has

$$\begin{aligned} \int _{{\left| u\right| }>t} {\left| g(x)\right| } \, dx \le \int _{0}^{\mu (t)} g^*(s) \, ds. \end{aligned}$$

We now introduce the Lorentz spaces (see [31] for more details on this topic).

Definition 2.4

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

$$\begin{aligned} {\left\| u\right\| }_{L^{p,q}} = {\left\{ \begin{array}{ll} \displaystyle { p^{\frac{1}{q}} \left( \int _{0}^{\infty } t^{q} \mu (t)^{\frac{q}{p}}\, \frac{dt}{t}\right) ^{\frac{1}{q}}} &{} 0<q<\infty \\[2ex] \displaystyle {\sup _{t>0} \, (t^p \mu (t))} &{} q=\infty \end{array}\right. } \end{aligned}$$

is finite.

Let us observe that for $p=q$ the Lorentz space coincides with the $L^p$ space, as a consequence of the Cavalieri Principle

$$\begin{aligned} \int _\Omega {\left| u\right| }^p =p \int _0^{+\infty } t^{p-1} \mu (t) \, dt. \end{aligned}$$

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 of p-superharmonic functions (see [22, 32]), they achieve their minima on the boundary. If we set

$$\begin{aligned} u_m=\min _\Omega u, \quad v_m=\min _{\Omega ^\sharp } v \end{aligned}$$

the positivity of $\beta$ and the Robin boundary conditions lead to $u_m \ge 0$ and $v_m \ge 0$. Hence, u and v are strictly positive in the interior of $\Omega$. Moreover, we can observe that

$$\begin{aligned} u_m = \min _\Omega u \le \min _{\Omega ^\sharp } v= v_m, \end{aligned}$$

(13)

since the following holds

$$\begin{aligned} \begin{aligned} v_m^{p-1} \text {P}(\Omega ^\sharp )&= \int _{\partial \Omega ^\sharp } v(x)^{p-1} \, d\mathcal {H}^{n-1}(x)= \frac{1}{\beta }\int _{\Omega ^\sharp } f^\sharp \, dx=\frac{1}{\beta } \int _{\Omega } f \, dx \\&= \int _{\partial \Omega } u(x)^{p-1} \, d\mathcal {H}^{n-1}(x) \\&\ge u_m^{p-1} \text {P}(\Omega ) \ge { u}_m^{p-1} \text {P}(\Omega ^\sharp ), \end{aligned} \end{aligned}$$

and it leads to

$$\begin{aligned} \mu (t) \le \phi (t) = {\left| \Omega \right| }, \quad \forall t \le v_m. \end{aligned}$$

(14)

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

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

and

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

Because of the invariance of the p-Laplacian and of the Schwarz rearrangement of f by rotation, the solution v to (6) is radial and, consequently, the superlevel sets $V_t$ 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 the 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

$$\begin{aligned} \gamma _n \mu (t)^{\left( 1-\frac{1}{n}\right) \frac{p}{p-1}} \le \left( \int _0^{\mu (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}} \left( - \mu '(t) + \frac{1}{\beta ^{\frac{1}{p-1}}}\int _{\partial U_t^\text {ext}} \frac{1}{u} \, d\mathcal {H}^{n-1}(x)\right) \end{aligned}$$

(15)

and

$$\begin{aligned} \gamma _n \phi (t)^{\left( 1-\frac{1}{n}\right) \frac{p}{p-1}} = \left( \int _0^{\phi (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}} \left( - \phi '(t) + \frac{1}{\beta ^{\frac{1}{p-1}}}\int _{\partial V_t^\text {ext}} \frac{1}{v} \, d\mathcal {H}^{n-1}(x)\right) . \end{aligned}$$

(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

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

(17)

obtaining

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

(18)

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

$$\begin{aligned} \int _{\partial U_t} g(x) \, d\mathcal {H}^{n-1} = \int _{U_{t}} f\, dx, \end{aligned}$$

where

$$\begin{aligned} g(x)={\left\{ \begin{array}{ll} {\left| \nabla u \right| }^{p-1}&{} \text { if }x \in \partial U_t^{ int },\\ \beta u ^{p-1}&{} \text { if }x \in \partial U_t^{ ext }. \end{array}\right. } \end{aligned}$$

(19)

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

$$\begin{aligned} \hspace{-0.8em} n \omega _n^{\frac{1}{n}} \mu (t)^{\frac{n-1}{n}}&\le P(U_t) = \int _{\partial U_t} \, d\mathcal {H}^{n-1}\end{aligned}$$

(20)

$$\begin{aligned}&\le \left( \int _{\partial U_t}g\, d\mathcal {H}^{n-1}(x)\right) ^{\frac{1}{p}} \left( \int _{\partial U_t}\frac{1}{g^{\frac{1}{p-1}}} \, d\mathcal {H}^{n-1}(x)\right) ^{1-\frac{1}{p}} \end{aligned}$$

(21)

$$\begin{aligned}&= \left( \int _{\partial U_t}g \, d\mathcal {H}^{n-1}(x)\right) ^{\frac{1}{p}} \nonumber \\&\qquad \left( \int _{\partial U_t^{ int }}\frac{1}{{\left| \nabla u\right| }}\, d\mathcal {H}^{n-1}(x) +\frac{1}{\beta ^{\frac{1}{p-1}}} \int _{\partial U_t^{ ext }}\frac{1}{u} \, d\mathcal {H}^{n-1}(x) \right) ^{1-\frac{1}{p}}\end{aligned}$$

(22)

$$\begin{aligned}&\le \left( \int _0^{\mu (t)} f^*(s) \, ds\right) ^{\frac{1}{p}} \left( -\mu '(t) +\frac{1}{\beta ^{\frac{1}{p-1}}} \int _{\partial U_t^{ ext }}\frac{1}{u} \, d\mathcal {H}^{n-1}(x) \right) ^{1-\frac{1}{p}}, \end{aligned}$$

(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 obtain (16). $\square$

Lemma 2.4

For all $\tau \ge v_m$, we have

$$\begin{aligned} \int _0^\tau t^{p-1} \left( \int _{\partial U_t^{ ext } } \frac{1}{ u(x) } \, d \mathcal {H}^{n-1}(x)\right) \, dt \le \frac{1}{p\beta } \int _0^{{\left| \Omega \right| }} f^*(s) \,ds. \end{aligned}$$

(24)

Moreover,

$$\begin{aligned} \int _0^\tau t^{p-1} \left( \int _{\partial V_t \cap \partial \Omega ^\sharp } \frac{1}{ v(x) } \, d \mathcal {H}^{n-1}(x)\right) \, dt = \frac{1}{p\beta } \int _0^{{\left| \Omega \right| }} f^*(s) \,ds. \end{aligned}$$

(25)

Lemma 2.5

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

$$\begin{aligned} \tau \xi ' (\tau ) \le (q-1) \xi (\tau ) + C \quad \forall \tau \ge \tau _0 >0. \end{aligned}$$

Then, we have

$$\begin{aligned} \xi (\tau ) \le \left( \xi (\tau _0) + \frac{C}{q-1}\right) \left( \frac{\tau }{\tau _0}\right) ^{q-1} - \frac{C}{q-1}, \quad \forall \tau \ge \tau _0, \end{aligned}$$

(26)

and

$$\begin{aligned} \xi '(\tau ) \le \left( \frac{(q-1)\xi (\tau _0 )+ C}{\tau _0}\right) \left( \frac{\tau }{\tau _0}\right) ^{q-2}, \quad \forall \tau \ge \tau _0. \end{aligned}$$

(27)

The following Lemma is contained in [14].

Lemma 2.6

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

$$\begin{aligned} \int _{\Omega } fg\;dx=\int _{\Omega ^\sharp } f^\sharp g^\sharp \;dx, \end{aligned}$$

(28)

then, for every $\tau , t\ge 0$ either

$$\begin{aligned} \{ g>\tau \}\subset \{ f>t\}, \quad \text {or }\quad \{ f>t\}\subset \{ g>\tau \}, \end{aligned}$$

(29)

up to a set of measure zero.

We conclude this preliminary paragraph by recalling the classical results contained in [10, Theorem 1.1 and Lemma 2.3].

Theorem 2.7

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

$$\begin{aligned} w_M:={\left\{ \begin{array}{ll} {\left\| w\right\| }_{\infty } &{} \text {if } w\in L^\infty (\Omega )\\ +\infty &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Then, the following are true:

• (i) For almost all $t\in (0,w_M)$,

  $$\begin{aligned} + \infty >-\sigma '(t)\ge \displaystyle \int _{w^{-1}(t)}\dfrac{1}{|\nabla w|}d\mathcal {H}^{n-1} \end{aligned}$$

  (30)

• (ii) $\sigma$ is absolutely continuous if and only if

  $$\begin{aligned} {\left| \left\{ |\nabla w^\sharp |=0 \right\} \cap \left\{ 0<w^\sharp <w_M \right\} \right| }=0. \end{aligned}$$

  (31)

• (iii) If

  $$\begin{aligned} \int _{\mathbb {R}^n} |\nabla w|^p= \int _{\mathbb {R}^n} |\nabla w^\sharp |^p, \end{aligned}$$

  (32)

  and (31) holds, then, up to a translation, $w=w^\sharp$ almost everywhere.

Remark 2.2

We observe that the result (iii) of Theorem 2.7 gives the rigidity of the Pólya-Szegő inequality (see [26]), that is

$$\begin{aligned} \int _{{{\,\mathrm{\mathbb {R}}\,}}^n} |\nabla u^\sharp |^p \, dx\le \int _{{{\,\mathrm{\mathbb {R}}\,}}^n} {\left| \nabla u\right| }^p \, dx, \quad \forall u\in W^{1,p}(\mathbb {R}^n). \end{aligned}$$

(33)

Moreover, in [15], it is proved that the condition

$$\begin{aligned} {\left| \left\{ {\left| \nabla w\right| } =0 \right\} \cap \left\{ 0<w<w_M \right\} \right| }=0 \end{aligned}$$

(34)

implies (31). So, by (iii) in Theorem 2.7, if we have (32) and (34), then, up to a translation, $w^\sharp =w$ almost everywhere.

We observe that the Pólya–Szegő inequality (33) and the relative rigidity result (iii) contained in Theorem 2.7 hold also if we assume $w\in W^{1,p}_0(\Omega )$. Indeed, one can adapt the proof of the Pólya-Szegő inequality (33) to obtain 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 Theorem 1.1, we divide the proof into the following steps. First, we establish 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

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

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

$$\begin{aligned} \int _0^{+\infty } t^{p-1}\mu (t)^{\frac{1}{k}}\, dt=\int _0^{+\infty } t^{p-1}\phi (t)^{\frac{1}{k}}\, dt. \end{aligned}$$

(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 $+\infty$:

$$\begin{aligned} \begin{aligned}&\gamma _n \int _0^{+\infty } t^{p-1}\mu ^{\frac{1}{k}}(t)\, dt \ \le \int _0^{+\infty } \left( \int _0^{\mu (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}} \\&\quad \left( - \mu '(t) + \frac{1}{\beta ^{\frac{1}{p-1}}}\int _{\partial U_t^{ext}} \frac{1}{u} \, d\mathcal {H}^{n-1}\right) t^{p-1} \mu (t)^{\alpha }\, dt\ \\&\quad \le \int _0^{+\infty } t^{p-1} \mu (t)^{\alpha } \left( \int _0^{\mu (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}}(-\mu '(t))\, dt\\ {}&\quad + \frac{{\left| \Omega \right| }^\alpha }{p\beta ^{\frac{p}{p-1}}} \left( \int _0^{{\left| \Omega \right| }}f^*(s ) \, ds\right) ^{\frac{p}{p-1}}, \end{aligned} \end{aligned}$$

(36)

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

$$\begin{aligned} \begin{aligned}&\gamma _n \int _0^{+\infty } t^{p-1}\phi ^{\frac{1}{k}}(t)\, dt\\ {}&= \int _0^{+\infty } \left( \int _0^{\phi (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}} \left( - \phi '(t) + \frac{1}{\beta ^{\frac{1}{p-1}}}\int _{\partial V_t^{ext}} \frac{1}{u} \, d\mathcal {H}^{n-1}\right) t^{p-1} \phi (t)^{\alpha }\, dt\\&= \int _0^{+\infty } t^{p-1} \phi (t)^{\alpha } \left( \int _0^{\phi (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}}(-\phi '(t))\, dt\\&\quad + \frac{{\left| \Omega \right| }^\alpha }{p\beta ^{\frac{p}{p-1}}} \left( \int _0^{{\left| \Omega \right| }}f^*(s ) \, ds\right) ^{\frac{p}{p-1}}. \end{aligned} \end{aligned}$$

(37)

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

$$\begin{aligned} \begin{aligned}&\int _0^{+\infty } \!\!\! \!\!\! t^{p-1} \phi (t)^{\alpha } \left( \int _0^{\phi (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}} \!\!\! \!\!\! (-\phi '(t))\, dt \\&\quad \le \int _0^{+\infty } \!\!\! \!\!\! t^{p-1} \mu (t)^{\alpha } \left( \int _0^{\mu (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}} \!\!\!\!\! (-\mu '(t))\, dt. \end{aligned} \end{aligned}$$

(38)

Setting $\displaystyle {F(l)= \int _0^l \omega ^\alpha \Biggl ( \int _0^\omega f^*(s) \, ds \Biggr )^{\frac{1}{p-1}} \! \!\! d\omega }$, we have

$$\begin{aligned} (F(\mu (t)))'= & {} \mu (t)^\alpha \Biggl ( \int _0^{\mu (t)} f^*(s) \, ds \Biggr )^{\frac{1}{p-1}} \mu '(t), \\ (F(\phi (t)))'= & {} \phi (t)^\alpha \Biggl ( \int _0^{\phi (t)} f^*(s) \, ds \Biggr )^{\frac{1}{p-1}} \phi '(t). \end{aligned}$$

So, equation (38) reads

$$\begin{aligned} \int _0^{+\infty } -t^{p-1}(F(\phi (t)))'\, dt \le \int _0^{+\infty } -t^{p-1}(F(\mu (t)))'\, dt \end{aligned}$$

and, integrating by parts, we get

$$\begin{aligned} \int _0^\infty t^{p-2} F(\phi (t)) \, dt \le \int _0^\infty t^{p-2} F(\mu (t)) \, dt, \end{aligned}$$

as $\mu (t)=\phi (t)=0$ for $t>v_M$. In [Theorem 1.1, [6]], it is proved that

$$\begin{aligned} \int _0^\infty t^{p-2} F(\mu (t)) \, dt \le \int _0^\infty t^{p-2} F(\phi (t)) \, dt, \end{aligned}$$

(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 $\displaystyle {k \le \frac{n(p-1)}{(n-2)p+n}}$, it follows that the function $\displaystyle {h(l)= F(l)l^{-\frac{(n-1)p}{n(p-1)}}}$ is non-decreasing. Hence, we obtain

$$\begin{aligned} \begin{aligned} \int _0^\tau \gamma _n t^{p-1}F(\mu (t)) \, dt&\le \int _0^\tau \Bigl (- \mu '(t) \Bigr ) t^{p-1}\mu (t)^{-\frac{(n-1)p}{n(p-1)} } F(\mu (t)) \\&\quad \left( \int _0^{\mu (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}} \, dt \\&\quad +F({\left| \Omega \right| })\frac{{\left| \Omega \right| }^{-\frac{p(n-1)}{n(p-1)} }}{p\beta ^{\frac{p}{p-1}}} \left( \int _0^{{\left| \Omega \right| }}f^*(s ) \, ds\right) ^{\frac{p}{p-1}}. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \displaystyle {C=F({\left| \Omega \right| })\frac{{\left| \Omega \right| }^{-\frac{p(n-1)}{n(p-1)} }}{p\beta ^{\frac{p}{p-1}}} \left( \int _0^{{\left| \Omega \right| }}f^*(s ) \, ds\right) ^{\frac{p}{p-1}}}, \end{aligned}$$

we obtain

$$\begin{aligned} \tau \int _0^{\tau } \gamma _n t^{p-2} F (\mu (t)) \, dt + \tau H_\mu (\tau )\le & {} \int _{0}^{\tau } \int _0^t r^{p-2} F(\mu (r)) \, dr dt\nonumber \\{} & {} \quad + \int _0^{\tau } H_\mu (t) \, dt +C, \end{aligned}$$

(40)

where

$$\begin{aligned} H_\mu (\tau )=-\int _{\tau }^{+\infty } t^{p-2} \mu (t)^{-\frac{p(n-1)}{n(p-1)}} F(\mu (t)) \biggl ( \int _0^{\mu (t)} f^*(s) \, ds \biggr )^{\frac{1}{p-1}} \, d\mu (t). \end{aligned}$$

Setting now

$$\begin{aligned} \begin{aligned} \xi (\tau )=\int _0^\tau \int _0^t \gamma _n r^{p-2}F(\mu (r)) \, dr +\int _0^\tau H_{\mu }(t) \, dt, \end{aligned} \end{aligned}$$

inequality (40) becomes

$$\begin{aligned} \tau \xi '(\tau ) \le \xi (\tau ) +C. \end{aligned}$$

So, Lemma 2.5, with $\tau _0=v_m$ and $q=2$, gives

$$\begin{aligned} \int _{0}^{\tau } \gamma _n t^{p-2} F(\mu (t)) \, dt + H_\mu (\tau ) \le \left( \frac{\displaystyle {\int _{0}^{v_m} t^{p-2} F(\mu (t) \, dt +H_\mu (v_m) +C}}{v_m}\right) . \end{aligned}$$

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

$$\begin{aligned} \int _{0}^{\tau } \gamma _n t^{p-2} F(\mu (t)) \, dt + H_\mu (\tau )\le \int _{0}^{\tau } \gamma _n F(\phi (t)) \, dt +H_\phi (\tau ), \end{aligned}$$

as $\mu (t)\le \phi (t)= {\left| \Omega \right| }$ for $t \le v_m$. Now, letting $\tau \rightarrow \infty$, one has

$$\begin{aligned} \int _0^\infty t^{p-2} F(\mu (t)) dt \le \int _0^\infty t^{p-2} F(\phi (t)) dt, \end{aligned}$$

since $H_\mu (\tau ), H_\phi (\tau ) \rightarrow 0$.

So, we get equality in (36) and, consequently, in (15) for almost every t. Indeed:

$$\begin{aligned}&\gamma _n \int _0^{+\infty } t^{p-1}\mu ^{\frac{1}{k}}(t)\, dt \\&\le \int _0^{+\infty } \left( \int _0^{\mu (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}} \left( - \mu '(t) + \frac{1}{\beta ^{\frac{1}{p-1}}}\int _{\partial U_t^{ext}} \frac{1}{u} \, d\mathcal {H}^{n-1}\right) t^{p-1} \mu (t)^{\alpha }\, dt\\&\le \int _0^{+\infty } t^{p-2} F(\mu (t))\, dt+ \frac{{\left| \Omega \right| }^\alpha }{p\beta ^{\frac{p}{p-1}}} \left( \int _0^{{\left| \Omega \right| }}f^*(s ) \, ds\right) ^{\frac{p}{p-1}}\\&=\int _0^{+\infty } t^{p-2} F(\phi (t))\, dt+\frac{{\left| \Omega \right| }^\alpha }{p\beta ^{\frac{p}{p-1}}} \left( \int _0^{{\left| \Omega \right| }}f^*(s ) \, ds\right) ^{\frac{p}{p-1}}\\&=\int _0^{+\infty } \left( \int _0^{\phi (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}} \left( - \phi '(t) + \frac{1}{\beta ^{\frac{1}{p-1}}}\int _{\partial V_t^{ext}} \frac{1}{v} \, d\mathcal {H}^{n-1}\right) t^{p-1} \phi (t)^\alpha \, dt\\&=\gamma _n \int _0^{+\infty } t^{p-1}\phi ^{\frac{1}{k}}(t)\, dt=\gamma _n \int _0^{+\infty } t^{p-1}\mu ^{\frac{1}{k}}(t)\, dt. \end{aligned}$$

$\square$

In the following Lemma, we prove an auxiliary rigidity result for the solution to the p-Laplace equation with Dirichlet boundary conditions, that we will need to conclude the proof of Theorem 1.1

Lemma 3.2

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

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p w= f &{} \text { in } \Omega \\ w =0 &{} \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(41)

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

$$\begin{aligned} \gamma _n \sigma (t)^{\left( 1-\frac{1}{n}\right) \frac{p}{p-1}} = \left( \int _0^{\sigma (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}} \left( - \sigma '(t) \right) , \quad \text {for a.e. } t\in [0,w_M] \end{aligned}$$

(42)

then, there exists $x_0$ such that

$$\begin{aligned} \Omega =\Omega ^\sharp +x_0, \quad w(\cdot +x_0)=w^\sharp (\cdot ), \quad f(\cdot +x_0)=f^\sharp (\cdot ). \end{aligned}$$

Proof

We recall that w is a weak solution to (41) if and only if

$$\begin{aligned} \int _{\Omega } {\left| \nabla w\right| }^{p-2}\nabla w \nabla \varphi \, dx = \int _{\Omega } f \varphi \, dx, \qquad \forall \varphi \in W_0^{1,p}(\Omega ). \end{aligned}$$

(43)

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

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

one obtains

$$\begin{aligned} \int _{\partial W_t} {\left| \nabla w\right| }^{p-1} \, d\mathcal {H}^{n-1}=\int _{W_t} f(x) \, dx\le \int _0^{\sigma (t)} f*(s)\, ds, \end{aligned}$$

(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,

$$\begin{aligned} n \omega _n^{\frac{1}{n}} \sigma (t)^{\frac{n-1}{n}}&\le P(W_t) = \int _{\partial W_t} \, d\mathcal {H}^{n-1} \end{aligned}$$

(45)

$$\begin{aligned}&\le \left( \int _{\partial W_t} {\left| \nabla w\right| }^{p-1}\, d\mathcal {H}^{n-1}(x)\right) ^{\frac{1}{p}} \left( \int _{\partial W_t}\frac{1}{{\left| \nabla w\right| }} \, d\mathcal {H}^{n-1}(x)\right) ^{1-\frac{1}{p}} \end{aligned}$$

(46)

$$\begin{aligned}&\le \left( \int _0^{\sigma (t)} f^*(s) \, ds\right) ^{\frac{1}{p}} \left( -\sigma '(t) \right) ^{1-\frac{1}{p}}. \end{aligned}$$

(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 $\left\{ w>t \right\}$ 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 $\left\{ t_k \right\}$ such that

• 1. $t_k\rightarrow t$;

• 2. $t_k>t_{k+1}$;

• 3. $\{w>t_k\}$ is a ball for all k.

Since $\displaystyle {\left\{ w>t \right\} =\cup _k\left\{ w>t_k \right\} }$ can be written as an increasing union of balls, $\{w>t\}$ is a ball for all t and, in particular, ${{\,\mathrm{\Omega }\,}}=\{w>0\}$ is a ball too and we obtain that ${{\,\mathrm{\Omega }\,}}=x_0+{{\,\mathrm{\Omega }\,}}^\sharp$.

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) also implies equality in Hölder inequality, i.e.

$$\begin{aligned} \int _{\partial W_t} \, d\mathcal {H}^{n-1} =\left( \int _{\partial W_t} {\left| \nabla w\right| }^{p-1}\, d\mathcal {H}^{n-1}(x)\right) ^{\frac{1}{p}} \left( \int _{\partial W_t}\frac{1}{{\left| \nabla w\right| }} \, d\mathcal {H}^{n-1}(x)\right) ^{1-\frac{1}{p}}. \end{aligned}$$

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 $C_t\ne 0$ for almost every t. Indeed, (44) and the positivity of f ensure us that

$$\begin{aligned} P(W_t) C_t^{p-1}=\int _{\partial W_t} {\left| \nabla w\right| }^{p-1} \, d\mathcal {H}^{n-1}=\int _{W_t} f(x) \, dx>0. \end{aligned}$$

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

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p z= f^\sharp &{} \text { in } \Omega ^\sharp \\ z =0 &{} \text { on } \partial \Omega ^\sharp , \end{array}\right. } \end{aligned}$$

(48)

and it has the following explicit form:

$$\begin{aligned} z(x)=\int _{\omega _n {\left| x\right| }^n}^{{\left| \Omega \right| }} \frac{1}{\gamma _n} \left( \int _0^s f^\star (r)\, dr\right) ^{1/(p-1)}\frac{1}{s^{(1-1/n)(p/(p-1))}} \, ds, \end{aligned}$$

so it easily follows that

$$\begin{aligned} {\left| \left\{ |\nabla w^\sharp |=0 \right\} \cap \left\{ 0<w^\sharp <w_M \right\} \right| }=0. \end{aligned}$$

(49)

Using (ii) in Theorem 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

$$\begin{aligned} -\sigma '(t)= \int _{\partial W^\sharp _t} \frac{1}{{\left| \nabla w^\sharp \right| }}=\frac{P(\partial W^\sharp _t)}{C^\sharp _t}, \end{aligned}$$

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

$$\begin{aligned} -\sigma '(t) = \int _{\partial W_t} \frac{1}{{\left| \nabla w\right| }}=\frac{P(\partial W_t)}{C_t}. \end{aligned}$$

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

$$\begin{aligned} \frac{P(\partial W_t)}{C_t}=\frac{P(\partial W^\sharp _t)}{C^\sharp _t}. \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} \int _\Omega {\left| \nabla w\right| }^p\, dx&=\int _0^{+\infty } \int _{\partial W_t} {\left| \nabla w\right| }^{p-1} \, d\mathcal {H}^{n-1}=\int _0^{+\infty } C_t^{p-1}P(W_t)\, dt \, d\mathcal {H}^{n-1}\\&=\int _0^{+\infty } \left( C_t^\sharp \right) ^{p-1}P(W_t^\sharp )\, dt \, d\mathcal {H}^{n-1}= \int _0^{+\infty } \int _{\partial W^\sharp _t} |\nabla w^\sharp |^{p-1} \, d\mathcal {H}^{n-1}\\&=\int _{\Omega ^\sharp } |\nabla w^\sharp |^p\, dx. \end{aligned} \end{aligned}$$

By (iii) in Theorem 2.7, we conclude that $w=w^\sharp$.

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

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

$$\begin{aligned} \int _{W_t} f(x) \, dx =\int _0^{\sigma (t)} f^*(s) \, ds. \end{aligned}$$

Moreover, for all $\tau \in [0, w_M)$, there exists a sequence $\left\{ \tau _k \right\}$ such that

• 1. $\tau _k\rightarrow \tau$;

• 2. $\tau _k>\tau _{k+1}$;

• 3. $\displaystyle \int _{W_{\tau _k}} f(x) \, dx =\displaystyle \int _0^{\sigma (\tau _k)} f^*(s) \, ds$,

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

$$\begin{aligned} \int _0^{\sigma (\tau )} f^*(s) \, ds= \lim _{k} \int _0^{\sigma (\tau _k)} f^*(s)=\lim _k \int _{W_{\tau _k}} f(x) \, dx =\int _{W_\tau } f(x)\;dx. \end{aligned}$$

By Lemma 2.6, we have that for all $\tau$, and $\alpha _\tau$, it holds

$$\begin{aligned} \text {either }\{w>\tau \}\subset \{f>\alpha _\tau \}\quad \text {or }\quad \{f>\alpha _\tau \}\subset \{w>\tau \}. \end{aligned}$$

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

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

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p w= f &{} \text { in } \Omega \\ w =0 &{} \text { on } \partial \Omega , \end{array}\right. } \quad {\left\{ \begin{array}{ll} -\Delta _p z= f^{\sharp } &{} \text { in } \Omega ^\sharp \\ z =0 &{} \text { on } \partial \Omega ^\sharp . \end{array}\right. } \end{aligned}$$

(50)

If $w^\sharp (x)=z(x)$, for all $x\in \Omega ^\sharp$, then there exists $x_0\in {{\,\mathrm{\mathbb {R}}\,}}^n$ such that

$$\begin{aligned} \Omega =\Omega ^\sharp +x_0, \qquad w(\cdot +x_0)=w^\sharp (\cdot ), \qquad f(\cdot + x_0)=f^\sharp (\cdot ). \end{aligned}$$

Proof

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

$$\begin{aligned} n \omega _n^{\frac{1}{n}} \sigma (t)^{\frac{n-1}{n}}\le \left( \int _0^{\sigma (t)} f^*(s) \, ds\right) ^{\frac{1}{p}} \left( -\sigma '(t) \right) ^{1-\frac{1}{p}}. \end{aligned}$$

(51)

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

$$\begin{aligned} z^*(t)=w^*(t) \le \int _{\sigma (t)}^{{\left| \Omega \right| }} \frac{1}{\gamma _n} \left( \int _0^s f^*(r)\, dr\right) ^{1/(p-1)}\frac{1}{s^{(1-1/n)(p/(p-1))}} \, ds= z^*(t). \end{aligned}$$

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

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 centered at the origin 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 $x_0\in {{\,\mathrm{\mathbb {R}}\,}}^n$ such that

$$\begin{aligned} \Omega =\Omega ^\sharp +x_0, \qquad u(\cdot +x_0)=u^\sharp (\cdot ), \qquad f(\cdot + x_0)=f^\sharp (\cdot ). \end{aligned}$$

Proof

We claim that the superlevel sets $\left\{ u>t \right\}$ are balls for every $t\in [0,u_M)$. Equality in (15) implies the equality in (20), i.e.

$$\begin{aligned} n \omega _n^{\frac{1}{n}} \mu (t)^{\frac{n-1}{n}} = P(U_t), \quad \text {for a. e. } t\in [0,u_M], \end{aligned}$$

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, ${{\,\mathrm{\Omega }\,}}=\{u>u_m\}$ is a ball and we obtain that ${{\,\mathrm{\Omega }\,}}=x_0+{{\,\mathrm{\Omega }\,}}^\sharp$.

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 $\sigma (t)={\left| \left\{ w>t \right\} \right| }$ satisfies (42). Since $\left\{ w>t \right\} =\left\{ u>t+u_m \right\}$, we have $\sigma (t)=\mu (t+u_m)$ for all $t\in [0, u_M-u_m]$. Moreover, we have

$$\begin{aligned} \int _{\partial U_t} \frac{1}{u}\, d\mathcal {H}^{n-1}= 0, \quad \forall t>u_m. \end{aligned}$$

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

$$\begin{aligned} \gamma _n \sigma (t)^{\left( 1-\frac{1}{n}\right) \frac{p}{p-1}}&= \gamma _n \mu (t+u_m)^{\left( 1-\frac{1}{n}\right) \frac{p}{p-1}} \\&= \left( \int _0^{\mu (t+u_m)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}} \\&\qquad \left( - \mu '(t+u_m) + \frac{1}{\beta ^{\frac{1}{p-1}}}\int _{\partial U_{t+u_m}^\text {ext}} \frac{1}{u} \, d\mathcal {H}^{n-1}(x)\right) \\&=\left( \int _0^{\sigma (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}} \left( - \sigma '(t) \right) , \end{aligned}$$

for all $t\in (0, u_M-u_m)$. So, we can conclude by Lemma 3.2. $\square$

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, i. e.

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

implies the following equality for almost every $t\in (0,u_M)$:

$$\begin{aligned} \gamma _n \mu (t)^{\left( 1-\frac{1}{n}\right) \frac{p}{p-1}} =\left( \int _0^{\mu (t)}f^*(s ) \, ds\right) ^{\frac{1}{p-1}} \left( - \mu '(t) + \frac{1}{\beta ^{\frac{1}{p-1}}}\int _{\partial U_t^\text {ext}} \frac{1}{u} \, d\mathcal {H}^{n-1}(x)\right) , \end{aligned}$$

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

Now, we are in position to apply Proposition 3.4, and, so, there exists $x_0\in {{\,\mathrm{\mathbb {R}}\,}}^n$ such that

$$\begin{aligned} \Omega =\Omega ^\sharp +x_0, \qquad u(\cdot +x_0)=u^\sharp (\cdot ), \qquad f(\cdot + x_0)=f^\sharp (\cdot ). \end{aligned}$$

$\square$

Remark 3.1

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

$$\begin{aligned} {\left\| u\right\| }_{L^{pk,p}(\Omega )} \le {\left\| v\right\| }_{L^{pk,p}(\Omega ^{\sharp })}, \quad \quad \textrm{if} \,\; \displaystyle {0 <k \le \frac{n(p-1)}{n(p-1)-p}}, \end{aligned}$$

(52)

that is an improvement of (7) in the case $p\le 2n$. We stress that the proof of Theorem 1.1 can be adapted to the case $f\equiv 1$ and $p\le 2n$, although now the admissible k varies in a wider range.

4 Conclusions and Open Problems

In conclusion, the main novelty of the paper consists in proving a rigidity result for the p-Laplacian Poisson problem with Dirichlet boundary conditions that can be used to deduce the rigidity property in the case of Robin boundary conditions.

Open problem 4.1 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 [27].

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