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On a weighted anisotropic eigenvalue problem

Published online by Cambridge University Press:  17 October 2024

Nunzia Gavitone*
Affiliation:
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”. Università degli Studi di Napoli Federico II, 80126, Napoli, Italy
Rossano Sannipoli
Affiliation:
Dipartimento di Matematica “Tullio Levi-Civita”, Universitá degli Studi di Padova, 35131, Padova, Italy
*
Corresponding author: Nunzia Gavitone; Email: [email protected]
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Abstract

In this paper, we deal with a weighted eigenvalue problem for the anisotropic $(p,q)$-Laplacian with Dirichlet boundary conditions. We study the main properties of the first eigenvalue and a reverse Hölder type inequality for the corresponding eigenfunctions.

Type
Research Article
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

1. Introduction

Let $\Omega \subset \mathbb{R}^n$ , $n \,{\geqslant}\, 2$ , be an open, bounded, and connected set and let $p, q$ be such that $1\lt p$ , and $1\lt q\lt p^*$ , where $p^*=np/(n-p)$ if $p\lt n$ and $p^*=\infty$ if $p\,{\geqslant}\, n$ . In this paper, we study the following variational problem:

(1.1) \begin{equation} \lambda ^H_{p,q}(\Omega ) = \inf _{\substack{u\in W^{1,p}_0(\Omega ), \\ u\neq 0}}\frac{\displaystyle \int _\Omega H(\nabla u)^p\,dx}{\bigg (\displaystyle \int _\Omega m|u|^q\,dx\bigg )^{\frac{p}{q}}}, \end{equation}

where $m\in L^{\infty }(\Omega )$ is a positive function and $H\;:\;\mathbb R^n\to [0,+\infty]$ is a $C^1(\mathbb R^n\setminus \{0\})$ convex and positively 1-homogeneous function (see Section 2 for more details).

Obviously, $\lambda ^H_{p,q}(\Omega )$ depends also on $m$ , but to simplify the notation we will omit its dependence.

The Euler–Lagrange equation associated with the minimization problem (1.1) is the following weighted eigenvalue problem for the anisotropic $(p,q)-$ Laplace operator with Dirichlet boundary condition:

(1.2) \begin{equation} \begin{cases} -\mathcal{L}_{p}(u)=\lambda m(x)\| u\|_{q,m}^{p-q}|u|^{q-2}u & \mbox{in}\ \Omega \\[5pt] u=0&\mbox{on}\ \partial \Omega, \end{cases} \end{equation}

where $\| u\|_{q,m}=\| u\|_{L^q(\Omega, m)}$ is the weighted Lebesgue norm of $u$ and $\mathcal{L}_p$ is the so-called anisotropic $p$ -Laplacian operator defined as follows:

(1.3) \begin{equation} \mathcal{L}_{p}(u) = \mathrm{div} (H(\nabla u)^{p-1}H_{\xi }(\nabla u)). \end{equation}

We stress that when $p=q$ and $m(x)\equiv 1$ , (1.1) is the first eigenvalue $\lambda ^H_p(\Omega )$ of the anisotropic $p$ -Laplacian, and it has been studied by many authors (see for instance [Reference Belloni, Ferone and Kawohl9, Reference Della Pietra and Gavitone19] and the references therein). In particular, in [Reference Belloni, Ferone and Kawohl9], it is proved that $\lambda ^{H}_p(\Omega )$ is simple for any $p$ , the corresponding eigenfunctions have a sign, and that a suitable Faber–Krahn inequality holds.

When $H=\mathcal E$ is the usual Euclidean norm, $ \mathcal{L}_{p}(u)$ is the well-known $p$ -Laplace operator and the eigenvalue problem (1.2) reduces to the following:

(1.4) \begin{equation} \begin{cases} -\Delta _p u=\lambda \|u\|_{q,m}^{p-q}|u|^{q-2}u & \mbox{in}\ \Omega \\[5pt] u=0&\mbox{on}\ \partial \Omega . \end{cases} \end{equation}

The spectrum of (1.4) and the first eigenvalue $\lambda ^{\mathcal E}_{p,q}(\Omega )$ , when $p\neq q$ and $m\equiv 1$ , have been studied for instance in the case $p=2$ in [Reference Brasco and Franzina11], for any $p$ in [Reference Franzina and Lamberti25, Reference García Azorero and Peral Alonso26, Reference Ôtani34, Reference Ôtani35] and in [Reference Drábek, Kufner and Nicolosi20] where the authors study also the weighted case. It is known that $\lambda ^{\mathcal E}_{p,q}(\Omega )$ is not simple, in general, for any $1\lt q\lt p^*$ . Indeed in [Reference Idogawa and Ôtani27], the authors prove the simplicity for any $1\lt q\,{\leqslant}\, p$ , while for $p\lt q\lt p^*$ , $\lambda ^{\mathcal E}_{p,q}(\Omega )$ could not be necessary simple. This fact has been observed for instance in [Reference Brasco and Franzina11], in the case $p=2$ , and for any $p$ in [Reference Nazarov32] and [Reference Kawohl, Lucia and Prashanth29], where the authors prove that for every fixed $p\lt q\lt p^*$ the simplicity fails if $\Omega$ is a sufficiently thin spherical shell.

In this paper, we study the main properties of $\lambda ^{H}_{p,q}(\Omega )$ and of the corresponding eigenfunctions. In particular, our aim is to prove a reverse Hölder inequality for them.

In the Euclidean case, if $p=q=2$ and $m\equiv 1$ , in [Reference Chiti15, Reference Chiti16], Chiti proves the following inequality for the first eigenfunctions $v$ corresponding to the first Dirichlet eigenvalue of the Laplace operator $\lambda ^{\mathcal E}_2(\Omega )\equiv \lambda (\Omega )$ :

(1.5) \begin{equation} \|v \|_{L^r(\Omega )} \,{\leqslant}\, C(r,s,n,\lambda (\Omega ))\| v\|_{L^s(\Omega )}, \quad 0\lt s\lt r, \end{equation}

and the equality case is achieved if and only if $\Omega$ is a ball and $v=v^\sharp$ , where the symbol “ $^\sharp$ ” denotes the Schwarz symmetral of a function (see [Reference Kesavan30]). In [Reference Alvino, Ferone and Trombetti3], the authors prove (1.5) for the first eigenfunctions of the $p$ -Laplacian. Moreover, in [Reference Alberico, Ferone and Volpicelli1], the authors extend the result to the weighted case, and the inequality reads as follows:

(1.6) \begin{equation} \|v \|_{L^r(\Omega, m)} \,{\leqslant}\, C(p,r,s,n,\lambda ^{\mathcal E}_{p}(\Omega ))\| v\|_{L^s(\Omega, m)}, \quad 0\lt s\lt r. \end{equation}

The equality sign holds if and only if $\Omega$ is a ball, $v=v^\sharp$ and $m=m^\sharp$ , modulo translation. In the general case $p \neq q$ , in the Euclidean case, a Chiti type inequality is proved in the case $m\equiv 1$ in [Reference Carroll and Ratzkin14] and [Reference Carroll, Fall and Ratzkin13] when $p=2$ and for any $p$ , respectively. More precisely in [Reference Carroll, Fall and Ratzkin13], the authors prove the following inequality:

(1.7) \begin{equation} \|v \|_{L^{r}(\Omega )} \,{\leqslant}\, C(p,q,r,n,\lambda ^{\mathcal E}_{p,q}(\Omega ))\| v\|_{L^q(\Omega )}, \quad q\lt r. \end{equation}

Even in this case, the equality sign holds if and only if $\Omega$ is a ball and $v=v^\sharp$ , modulo translation. The goal of this paper is to prove a Chiti type inequality in the spirit of (1.6) and (1.7) for the first eigenfunctions of the general weighted eigenvalue problem (1.2). We recall that, when $p=q$ and $m\equiv 1$ , the result in the anisotropic setting has been proved in [Reference Belloni, Ferone and Kawohl9]. Our main theorem is the following.

Theorem 1.1. Let $\Omega \subset \mathbb R^n$ be an open, bounded, and connected set. Let $1\lt q\,{\leqslant}\, p$ , and let $u$ be an eigenfunction corresponding to the first eigenvalue (1.1). Then the following statements hold

  1. i) There exists a constant $C=C(p,q,r,n,\lambda ^{H}_{p,q}(\Omega ))$ such that

    (1.8) \begin{equation} \|u \|_{L^r(\Omega, m)} \,{\leqslant}\, C\,\,\| u\|_{L^q(\Omega, m)}, \quad q\,{\leqslant}\, r; \end{equation}
  2. ii) There exists a constant $C=C(p,q,r,n,\lambda ^{H}_{p,q}(\Omega ))$ such that

    (1.9) \begin{equation} \|u\|_{L^\infty (\Omega )} \,{\leqslant}\, C \|u\|_{L^r(\Omega, m)} \quad 1\,{\leqslant}\, r\lt \infty . \end{equation}

The equality cases hold if and only if $\Omega$ is a Wulff shape and $u$ and $m$ coincide, that is, in $\Omega$ with their convex symmetrization, modulo translation.

We stress that this result gives, in particular, a Chiti type inequality for the eigenfunctions corresponding to the first weighted eigenvalue of the anisotropic $p$ -Laplacian and extend (1.7) to the weighted case. The proof is based on symmetrization techniques and a comparison between the eigenfunctions corresponding to the first eigenvalue (1.1) and the first eigenfunctions of a suitable symmetrical eigenvalue problem.

The structure of the paper is the following. In Section 2, we fix some notation, recall some basic properties of the Finsler norms, and give a brief overview about convex symmetrization. In Section 3, we study the main properties of $\lambda ^H_{p,q}(\Omega )$ and a Faber–Krahn type inequality. In the last section, we prove Theorem 1.1 by using symmetrization arguments.

2. Notations and preliminaries

Throughout this article, $|\cdot |$ denotes the Euclidean norm in $\mathbb{R}^n$ , while $\cdot$ is the standard Euclidean scalar product for $n\geqslant 2$ . Moreover, we denote by $|\Omega |$ the Lebesgue measure of $\Omega \subseteq \mathbb{R}^n$ , by $B_R$ the Euclidean ball centered at the origin with radius $R$ and by $\omega _n$ the measure of the unit ball.

Let $E\subseteq \mathbb{R}^n$ be a bounded, open set and $\Omega \subseteq \mathbb{R}^{n}$ be a measurable set. We recall now the definition of the perimeter of $\Omega$ in $E$ in the sense of De Giorgi, that is,

\begin{equation*} P(\Omega ;\; E)=\sup \left \{ \int _\Omega \textrm {div} \varphi \:dx \;:\;\varphi \in C^{\infty }_c(E;\;\mathbb {R}^n),\;||\varphi ||_{\infty }\leq 1 \right \}. \end{equation*}

The perimeter of $\Omega$ in $\mathbb{R}^n$ will be denoted by $P(\Omega )$ and, if $P(\Omega )\lt \infty$ , we say that $\Omega$ is a set of finite perimeter. Some results relative to the sets of finite perimeter are contained, for instance, in [Reference Maggi31]. Moreover, if $\Omega$ has Lipschitz boundary, we have that

\begin{equation*} P(\Omega )=\mathcal {H}^{n-1}(\partial \Omega ). \end{equation*}

2.1. The anisotropic norm

Let $H\;:\;\mathbb{R}^n \longrightarrow [0,+\infty [$ , $n\,{\geqslant}\, 2$ , be a $C^1(\mathbb{R}^n \setminus \{0\})$ convex function which is positively $1$ -homogeneous, that is,

(2.1) \begin{equation} H(t\xi )= |t| H(\xi ) \,\,\,\,\,\, \forall \xi \in \mathbb{R}^n \,, \,\, \forall t \in \mathbb{R}. \end{equation}

Moreover, let $0\lt \gamma \,{\leqslant}\, \delta$ be positive constants such that

(2.2) \begin{equation} \gamma |\xi | \,{\leqslant}\, H(\xi ) \,{\leqslant}\, \delta |\xi |. \end{equation}

These properties guarantee that $H$ is a norm in $\mathbb{R}^n$ . Indeed by (2.2), we have that $H(\xi )=0$ if and only if $\xi = 0$ . It is homogeneous by (2.1) and the triangular inequality is a consequence of the convexity of the function $H$ : if $\xi, \eta \in \mathbb{R}^n$ , then

\begin{equation*} \frac {H(x+y)}{2}= H\bigg (\frac {x}{2}+\frac {y}{2}\bigg )\,{\leqslant}\, \frac {H(x)}{2}+\frac {H(y)}{2}. \end{equation*}

Because of (2.1), we can assume that the set

\begin{equation*} K = \{\xi \in \mathbb {R}^n \;:\; H(\xi )\,{\leqslant}\, 1 \} \end{equation*}

is such that $|K|=\omega _n$ , where $\omega _n$ is the measure of the unit sphere in $\mathbb{R}^n$ . We can define the support function of $K$ as:

(2.3) \begin{equation} H^{\circ }(x)= \sup _{\xi \in K} \left \lt x, \xi \right \gt, \end{equation}

where $\left \lt \cdot, \cdot \right \gt$ denotes the scalar product in $\mathbb{R}^n$ . $H^{\circ }\;:\;\mathbb{R}^n \longrightarrow [0,+\infty ]$ is a convex, homogeneous function in the sense of (2.1). Moreover, $H$ and $H^{\circ }$ are polar to each other, in the sense that

\begin{equation*} H^{\circ }(x) = \sup _{\xi \neq 0 } \frac {\left \lt x, \xi \right \gt }{H(\xi )} \end{equation*}

and

\begin{equation*} H(x) = \sup _{\xi \neq 0 } \frac {\left \lt x, \xi \right \gt }{H^{\circ }(\xi )}. \end{equation*}

$H$ is the support function of the set:

\begin{equation*} K^{\circ } = \{x \in \mathbb {R}^n \;:\; H^{\circ }(x)\,{\leqslant}\, 1 \}. \end{equation*}

The set $\mathcal{W}=\{x \in \mathbb{R}^n \;:\; H^{\circ }(x)\lt 1 \}$ is the so-called Wulff shape centered at the origin. We set $k_n = |\mathcal{W}|$ . More generally, we will denote by $\mathcal{W}_R(x_0)$ the Wulff shape centered in $x_0 \in \mathbb{R}^n$ the set $R\mathcal{W}+x_0$ , and $\mathcal{W}_R(0)=\mathcal{W}_R$ .

The following properties hold for $H$ and $H^{\circ }$ :

(2.4) \begin{equation} H_{\xi }(\xi )\cdot \xi = H(\xi ), \;\;\;\; H^{\circ }_{\xi }(\xi )\cdot \xi = H^{\circ }(\xi ), \end{equation}
(2.5) \begin{equation} H(H^{\circ }_{\xi }(\xi ))= H^{\circ }(H_{\xi }(\xi ))=1 \,\,\,\,\, \forall \xi \in \mathbb{R}^n \setminus \{0\}, \end{equation}
(2.6) \begin{equation} H^{\circ }(\xi )H_{\xi }(H^{\circ }_{\xi }(\xi ))=H(\xi )H^{\circ }_{\xi }(H_{\xi }(\xi )) = \xi \,\,\, \forall \xi \in \mathbb{R}^n \setminus \{0\}. \end{equation}

An example of an anisotropic norm that satisfies the above-mentioned properties is the following. Let $r\in (1,+\infty )$ and let us consider

\begin{equation*} H(\xi ) = \bigg (\sum _{i=1}^n \left \vert \xi _i\right \vert ^r\bigg )^{\frac {1}{r}}, \end{equation*}

known in literature also as $r$ -norm. With this choice, the highly nonlinear operator $\mathcal{L}_p(u)$ , defined in (1.3), becomes

\begin{equation*} \mathcal {L}_p(u)= \sum _{k=1}^n\frac {\partial }{\partial x_k} \bigg (\bigg ( \sum _{i=1}^n \left \vert \frac {\partial u}{\partial x_i}\right \vert ^r\bigg )^{\frac {p-r}{r}}\left \vert \frac {\partial u}{\partial x_k}\right \vert ^{r-2}\frac {\partial u}{\partial x_k}\bigg ). \end{equation*}

We stress that for $r=p$ , $\mathcal{L}_p(u)$ is the so called pseudo- $p$ -Laplace operator. Examples of non-smooth anisotropic norm can be found in [Reference Bellettini, Caselles, Chambolle and Novaga8] and references therein, where the authors consider a crystalline anisotropy and the associated Wulff shape is a polyhedron.

If $E \subset \mathbb{R}^n$ is an open, bounded set with Lipschitz boundary and $\Omega$ is an open subset of $\mathbb{R}^n$ , we can give a generalized definition of perimeter of $\Omega$ with respect to the anisotropic norm as follows (see for instance [Reference Amar and Bellettini6]):

\begin{equation*} P_H(\Omega, E)= \int _{\partial ^* \Omega \cap E} H(\nu )\,d\mathcal {H}^{n-1}, \end{equation*}

where $\partial ^*\Omega$ is the reduced boundary of $\Omega$ (for the definition see [Reference Evans and Gariepy21]), $\nu$ is its Euclidean outer normal, and $\mathcal{H}^{n-1}$ is the $(n-1)-$ dimensional Hausdorff measure in $\mathbb{R}^n$ . Clearly, if $E$ is open, bounded and Lipschitz, then the outer unit normal exists almost everywhere and

(2.7) \begin{equation} P_H(E,\mathbb{R}^n)\;:\!=\;P_H (E) = \int _{\partial E} H(\nu )\,d\mathcal{H}^{n-1}. \end{equation}

By (2.2), we have that

\begin{equation*} \gamma P(E) \,{\leqslant}\, P_H(E)\,{\leqslant}\, \delta P(E). \end{equation*}

In [Reference Alvino, Trombetti and Lions5], it is shown that if $u\in W^{1,1}(\Omega )$ , then for, that is, $t\gt 0$

(2.8) \begin{equation} -\frac{d}{dt}\int _{\{u\gt t\}} H(\nabla u)\,dx = P_H(\{u\gt t\},\Omega ) = \int _{\partial ^*\{u\gt t\}\cap \Omega } \frac{H(\nabla u)}{|\nabla u|}\,d\mathcal{H}^{n-1}. \end{equation}

Moreover, an isoperimetric inequality for the anisotropic perimeter holds (for instance see [Reference Alvino, Ferone, Lions and Trombetti2, Reference Busemann12, Reference Dacorogna and Pfister18, Reference Figalli, Maggi and Pratelli23, Reference Fonseca and Müller24])

(2.9) \begin{equation} P_H( E ) \,{\geqslant}\, n k^{\frac{1}{n}}_n |E|^{1-\frac{1}{n}}. \end{equation}

2.2. Convex symmetrization

Let $\Omega \subset \mathbb{R}^n$ be an open, bounded, and connected set. Let $f: \Omega \longrightarrow [0,+\infty ]$ be a measurable function. The decreasing rearrangement $f^*$ of $f$ is defined as follows:

\begin{equation*} f^*(s)= \inf \{t\,{\geqslant}\, 0 \;:\; \mu (t)\lt s\} \,\,\,\,\,\, s\in [0,|\Omega |], \end{equation*}

where

\begin{equation*} \mu (t) = |\{x\in \Omega \;:\; |f(x)|\gt t\}|, \end{equation*}

is the distribution function of $f$ . We recall that the Schwarz symmetrand of $f$ is a radially spherically function defined as follows:

\begin{equation*} f^{\sharp }(x) = f^*(\omega _n |x|^n) \,\,\,\,\,\,\,\, x\in \Omega ^{\sharp }. \end{equation*}

where $\Omega ^{\sharp }$ is the ball centered at the origin such that $|\Omega ^{\sharp }|=|\Omega |$ . The convex symmetrization $f^{\star }$ of $f$ , instead, is a function symmetric with respect to $H^{\circ }$ defined as follows:

\begin{equation*} f^{\star }(x) = f^*(k_n (H^{\circ }(x))^n) \,\,\,\,\,\,\,\, x\in \Omega ^{\star }, \end{equation*}

where $\Omega ^\star$ is a Wulff shape centered at the origin and such that $|\Omega ^{\star }|=|\Omega |$ (see [Reference Alvino, Ferone, Lions and Trombetti2]). We stress that both $f^{\star }$ and $f^{\sharp }$ are defined by means the decreasing rearrangement $f^*$ , but they have different symmetry. In particular, it is well known that the functions $f$ , $f^*$ , $f^{\sharp }$ , and $f^{\star }$ are equimeasurable, that is,

\begin{equation*} |\{f\gt t\}| =|\{f^{\sharp } \gt t\}| = |\{f^* \gt t\}| = |\{f^{\star }\gt t\}| \,\,\,\,\, t\,{\geqslant}\, 0. \end{equation*}

As a consequence, if $f\in L^p(\Omega )$ , $p\,{\geqslant}\, 1$ , then

(2.10) \begin{equation} \| f\|_{L^p(\Omega )}\| =\| f^{\sharp } \|_{L^p(\Omega ^{\sharp })} = \| f^* \|_{L^p([0,|\Omega |])} = \|f^{\star }\|_{L^p(\Omega ^{\star })}. \end{equation}

Regarding the norm of the gradient, a generalized version of the well-known Pólya–Szegö inequality holds and it states (see for instance [Reference Alvino, Ferone, Lions and Trombetti2])

Theorem 2.1. (Pólya–Szegö principle). If $w \in W^{1,p}_0(\Omega )$ for $p\,{\geqslant}\, 1$ , then we have that

\begin{equation*} \int _{\Omega }H(\nabla u)^p\,dx \,{\geqslant}\, \int _{\Omega ^{\star }} H(\nabla u^{\star })^p\,dx. \end{equation*}

where $\Omega ^\star$ is the Wulff Shape such that $|\Omega ^\star |=|\Omega |$ .

For the sake of completeness, we will state the result concerning the equality case of the PólyaSzegö inequality, whose proof is contained in [Reference Ferone and Volpicelli22] for the generic anisotropic case and in [Reference Ziemer and Brothers38] for the Euclidean case.

Theorem 2.2. Let $u$ be a non-negative function in $W^{1,p}(\mathbb{R}^n)$ , for $1\lt p\lt +\infty$ , such that

\begin{equation*} |\{|\nabla u^\star |=0\}\cap \{0\lt u^\star \lt \operatorname {ess\;sup} u\}|=0. \end{equation*}

Then

\begin{equation*} \int _{\mathbb {R}^n}H(\nabla u)^p\,dx = \int _{\mathbb {R}^n} H(\nabla u^{\star })^p\,dx \end{equation*}

if and only if $u=u^\star$ a.e. in $\mathbb{R}^n$ , up to translations.

Obviously, Theorem 2.2 can holds true in the case of a $W^{1,p}_0(\Omega )$ function.

We conclude this section by recalling some known properties about rearrangements that we will use in the proof of the main theorem. The following result is the well-known Hardy–Littlewood inequality (see [Reference Kesavan30]):

(2.11) \begin{equation} \int _{\Omega } |f(x)g(x)| \,dx \,{\leqslant}\, \int _0^{|\Omega |} f^*(s)g^*(s)\,ds. \end{equation}

So, if we consider $g$ as the characteristic function of the set $\{x\in \Omega \;:\; u(x)\gt t\}$ , for some measurable function $u\;:\;\Omega \rightarrow \mathbb{R}$ and $t\,{\geqslant}\, 0$ , then we get

(2.12) \begin{equation} \int _{\{u\gt t\}} f(x)\,dx \,{\leqslant}\, \int _0^{\mu (t)} f^*(s)\,ds, \end{equation}

where, again, $\mu (t)$ is the distribution function of $u$ . Finally, we recall the definition of dominated rearrangements (see for instance [Reference Alvino, Lions and Trombetti4] and [Reference Chong and Rice17]).

Definition 2.3. Let $f,g \in L^1(\Omega )$ be nonnegative functions. We say that $g$ is dominated by $f$ and write $g\prec f$ if the following two statements hold

  1. (i) $\displaystyle \int _{0}^s g^* (t)\,dt \,{\leqslant}\, \displaystyle \int _{0}^s f^* (t)\,dt$ ;

  2. (ii) $\displaystyle \int _{0}^{|\Omega |} g^* (t)\,dt =\displaystyle \int _{0}^{|\Omega |} f^* (t)\,dt$ .

In [Reference Alvino, Lions and Trombetti4], the authors prove the following result:

Proposition 2.4. Let $f,g,h$ be positive and such that $hf, hg\in L^1(\Omega )$ . Let $F$ be a convex, nonnegative function such that $F(0)=0$ . If $hg \prec hf$ Then

\begin{equation*} \int _0^{|\Omega |}h^* F(g^*) \, dt \,{\leqslant}\, \int _0^{|\Omega |}h^* F(f^*) \, dt. \end{equation*}

Moreover, if $F$ is strictly convex, the equality holds if and only if $f^*\equiv g^*$ , that is, in $[0,|\Omega |]$ .

3. The (p,q)-anisotropic Laplacian

In this section, we study the main properties of (1.1) and the corresponding minimizers. Let $\Omega \subset \mathbb{R}^n$ , $n\,{\geqslant}\, 2$ be an open, bounded, and connected set. Let $m\in L^{\infty }(\Omega )$ be a positive function and $p, q$ be such that $1\lt p\lt \infty$ and $1\lt q\lt p^*$ , where $p^*=np/(n-p)$ , if $p\lt n$ , and $p^*=\infty$ , if $p\,{\geqslant}\, n$ . A function $v\in W_0^{1,p}(\Omega )$ is a weak solution to the problem (1.2) corresponding to $\lambda$ if

(3.1) \begin{equation} \int _{\Omega }(H(\nabla v))^{p-1}H_{\xi }(\nabla v) \cdot \nabla \varphi \, dx=\lambda \|v\|_{q,m}^{p-q}\int _{\Omega }m(x)\,|v|^{q-2}\,v \,\varphi \,dx, \end{equation}

for every $\varphi \in W_0^{1,p}(\Omega )$ . By standard argument of calculus of variations, it is not difficult to prove the following result:

Theorem 3.1. Let $n\,{\geqslant}\, 2$ and $\Omega \subset \mathbb{R}^n$ , be an open, bounded set and let $p,q$ and $m$ be as above. Then $\lambda ^H_{p,q}(\Omega )$ , defined in (1.1), is strictly positive and actually a minimum. Moreover, any minimizer is a weak solution to the problem (1.2), with $\lambda =\lambda ^H_{p,q}(\Omega )$ , and has constant sign on every connected component.

As regard the simplicity, we have

Theorem 3.2. Let $n\,{\geqslant}\, 2$ and $\Omega \subset \mathbb{R}^n$ , be an open, bounded, and connected set and let $p$ and $m$ be as above and let $1\lt q\,{\leqslant}\, p$ . Then $\lambda ^H_{p,q}(\Omega )$ is simple, that is, there exists a unique corresponding eigenfunction up to multiplicative constants.

The proof of the previous result is contained in [Reference Kawohl, Lucia and Prashanth29], where the authors consider a more general class of quasilinear operators. We stress that this result was already proved when $H=\mathcal{E}$ and $m\equiv 1$ in the paper [Reference Kawohl28] and in the case of a positive and essentially bounded weight in [Reference Nazarov33]. Finally, we have the following:

Theorem 3.3. Let $n\,{\geqslant}\, 2$ and let $\Omega \subset \mathbb{R}^n$ be an open, bounded, and connected set. Let $p$ and $m$ be as above and let $1\lt q\,{\leqslant}\, p$ . Any nonnegative function $v \in W_0^{1,p}(\Omega )$ , which is a weak solution to the problem (1.2), for some $\lambda \gt 0$ , is a first eigenfunction, that is $\lambda =\lambda ^H_{p,q}(\Omega )$ .

Proof. The proof is similar to the one contained in [Reference Brasco and Franzina10, Theorem 5.1], and it follows standard arguments and a general Picone inequality. For the reader convenience and sake of completeness, we write the main steps. Let $v$ be a non-negative weak solution to the problem (1.2) corresponding to $\lambda$ . By the strong maximum principle, we have that $v\gt 0$ in $\Omega .$ Let $u$ be the first positive eigenfunction corresponding to $\lambda ^H_{p,q}(\Omega )$ such that

(3.2) \begin{equation} \|u\|_{L^q(\Omega, m)}=\|v\|_{L^q(\Omega, m)}. \end{equation}

Then,

(3.3) \begin{equation} \int _{\Omega }(H(\nabla u))^p\,dx=\lambda ^H_{p,q}(\Omega )\left (\int _{\Omega }m(x) \,u^q \,dx \right )^{\frac pq}. \end{equation}

Being $v$ a weak positive solution to (1.2) corresponding to $\lambda$ , we can chose $\varphi =\displaystyle \frac{u^q}{v^{q-1}}$ as test function in (3.1) obtaining

(3.4) \begin{align} \int _{\Omega }\left ((H(\nabla v)\right )^{p-1}H_{\xi }(\nabla v ) \cdot \nabla \left ( \frac{u^q}{v^{q-1}}\right )\,dx &=\lambda \|m^{\frac{1}{q}} v\|_q^{p-q}\int _{\Omega }m(x)\,u^q\,dx \nonumber \\[5pt] &= \lambda \left (\int _{\Omega }m(x) \,u^q \,dx \right )^{\frac pq}, \end{align}

where last equality follows by (3.2). In the left-hand side, we can apply the general Picone inequality (see Proposition 2.9 in [Reference Brasco and Franzina10]) and we have

\begin{equation*} \int _{\Omega }\left (H(\nabla v)\right )^q \left (H(\nabla u)\right )^{p-q} \,dx \,{\geqslant}\, \lambda \left (\int _{\Omega }m(x) \,u^q \,dx \right )^{\frac pq}. \end{equation*}

By the Hölder inequality, the normalization (3.2) and (3.3) we get that $\lambda ^{H}_{p,q}(\Omega )\,{\geqslant}\, \lambda$ , that implies $u=v$ .

3.1. The case $\Omega = \mathcal{W}_R$

In this subsection, we study the problem (1.2) when $\Omega$ is a Wulff shape. In this case, the eigenfunctions inherit some symmetry properties. Let be $\Omega = \mathcal{W}_R$ and let $m\in L^{\infty }(\mathcal{W}_R)$ be a positive function such that $m(x)=m^{\star }(x)$ . Then problem (1.2) becomes

(3.5) \begin{equation} \begin{cases} -\mathcal{L}_{p}(v)=\lambda m^{\star }(x) \| v\|_{q, m^\star }^{p-q}|v|^{q-2}v & \mbox{in}\ \mathcal{W}_R\\[5pt] v=0&\mbox{on}\ \partial \mathcal{W}_R. \end{cases} \end{equation}

The following result holds

Proposition 3.4. Let $1\lt p\lt \infty$ and $1\lt q\,{\leqslant}\, p$ . Let $v \in C^1(\overline{\Omega })\cap C^{1,\alpha }(\Omega )$ be a first positive eigenfunction to the problem (3.5). Then there exists a decreasing function $\rho (r)$ , $r \in [0,R]$ , such that $\rho \in C^{\infty }((0,R))\cap C^1([0,R])$ , $\rho ^{\prime }(0)=0$ , and $v(x)=\rho (H^o(x))$ .

Proof. By the simplicity, we can assume that $\|v\|_{L^q(\mathcal W_R,m^{\star })}=1$ . Let $B_R$ be the ball centered at the origin with radius $R\gt 0$ , and let us consider the weighted p-Laplace eigenvalue problem in $B_R$ :

(3.6) \begin{equation} \begin{cases} -\Delta _p z=\lambda \tilde m(|x|)\|z\|_{q,m^{\star }}^{p-q}|z|^{q-2}z & \mbox{in}\ B_R\\[5pt] z=0&\mbox{on}\ \partial B_R, \end{cases} \end{equation}

where $\tilde m(r)=m^*(k_n r^n)$ , $0 \,{\leqslant}\, r\,{\leqslant}\, R$ . Let $z$ be the positive eigenfunction corresponding to the first eigenvalue $\lambda ^{\mathcal E}_{p,q}(B_R)$ to the problem (3.6), such that $\|z\|_{L^q(B_R,\tilde{m})}=\|v\|_{L^q(\mathcal W_R,m^{\star })}=1$ . Then uniqueness guarantees that $z$ is radially symmetric, which means that there exists a positive one-dimensional function $\rho _p \;:\; r\in [0,R]\to \mathbb{R}^+$ such that $z(x)=\rho _p(|x|)$ , and $\rho _p$ solves the following problem:

(3.7) \begin{equation} \begin{cases} -(p-1)|\rho ^{\prime }_p|^{p-2}\rho ^{\prime \prime }_p+\frac{n-1}{r}|\rho ^{\prime }_p|^{p-1}= \lambda ^{\mathcal E}_{p,q}(B_R)\tilde m|\rho _p|^{q-2}\rho _p,& r \in (0,R) \\[5pt] \rho ^{\prime }_p(0)=\rho _p(R)=0. \end{cases} \end{equation}

In particular, integrating equation (3.7), it is possible to see that $\rho ^{\prime }_p$ is zero only when $r=0$ and consequently that $\rho _p$ is strictly decreasing in $[0,R]$ . Now we can come back to the anisotropy. Indeed if we consider $w=\rho _p(H^\circ (x))$ , then using properties (2.4)-(2.6) and the regularity of $H$ , by construction, we obtain that $w(x)$ is a solution to problem (3.5), which is positive and radial with respect to the anisotropic norm. The simplicity and Theorem 3.3 imply that $v=w$ , and this concludes the proof.

Remark 3.5. We stress that the proof of the previous result shows that the first eigenvalue $\lambda ^H_{p,q}(\mathcal W_R)$ coincides with the first eigenvalue of problem (3.6).

3.2. A Faber–Krahn type inequality

Theorem 3.6. Let $\Omega \in \mathbb{R}^n$ , $n\,{\geqslant}\, 2$ , be an open, bounded, and connected set and let $1\lt q\,{\leqslant}\, p$ . Then

(3.8) \begin{equation} \lambda ^H_{p,q}(\Omega )\,{\geqslant}\, \lambda ^H_{p,q}(\Omega ^{\star }), \end{equation}

where $\Omega ^\star$ is the Wulff shape such that $|\Omega ^\star |=|\Omega |$ . The equality case holds if and only if $\Omega =\Omega ^{\star }$ and $m=m^{\star }$ , that is, in $\Omega$ , up to translations, where $m^{\star }$ is the convex symmetrization of $m$ .

Proof. We argue as in [Reference Belloni, Ferone and Kawohl9]. We observe that $\lambda ^H_{p,q}(\Omega ^{\star })$ has the following variational characterization:

(3.9) \begin{equation} \lambda ^H_{p,q}(\Omega ^{\star }) = \inf _{\substack{w\in W^{1,p}_0(\Omega ^{\star }), \\ w \neq 0}}\frac{\displaystyle \int _{\Omega ^{\star }} H(\nabla w)^p\,dx}{\bigg (\displaystyle \int _{\Omega ^{\star }} m^{\star }|w|^q\, dx\bigg )^{\frac{p}{q}}}. \end{equation}

The Faber–Krahn inequality is a straightforward application of the Pólya–Szegö principle and the Hardy–Littlewood inequality. Indeed if $u$ is a positive eigenfunction corresponding to $\lambda ^H_{p,q}(\Omega )$ , then

(3.10) \begin{equation} \lambda ^H_{p,q}(\Omega ) = \frac{\displaystyle \int _\Omega H(\nabla u)^p\,dx}{\bigg (\displaystyle \int _\Omega m{u}^q\,dx\bigg )^{\frac{p}{q}}} \,{\geqslant}\, \frac{\displaystyle \int _{\Omega ^{\star }} H(\nabla u^{\star })^p\,dx}{\bigg (\displaystyle \int _{\Omega ^{\star }}m^{\star }(u^{\star })^q\,dx\bigg )^{\frac{p}{q}}}\,{\geqslant}\, \lambda ^H_{p,q}(\Omega ^{\star }). \end{equation}

Let us now consider the equality case. From (3.10), Pólya–Szegö principle and Hardy–Littlewood inequality, we get

\begin{equation*} 1\,{\leqslant}\, \frac {\displaystyle \int _\Omega H(\nabla u)^p\,dx}{\displaystyle \int _{\Omega ^{\star }} H(\nabla u^{\star })^p\,dx}= \frac {\bigg (\displaystyle \int _{\Omega ^{\star }} m^{\star }(u^{\star })^q\,dx\bigg )^{\frac {p}{q}}}{\bigg (\displaystyle \int _\Omega m u^q\,dx\bigg )^{\frac {p}{q}}}\,{\leqslant}\, 1. \end{equation*}

It follows that

(3.11) \begin{equation} \int _\Omega H(\nabla u)^p\,dx\,dx=\int _{\Omega ^{\star }} H(\nabla u^{\star })^p\,dx, \end{equation}

and

(3.12) \begin{equation} \bigg (\displaystyle \int _\Omega m u^q\,dx\bigg )^{\frac{p}{q}}=\bigg (\displaystyle \int _{\Omega ^{\star }} m^{\star }(u^{\star })^q\,dx\bigg )^{\frac{p}{q}}. \end{equation}

The thesis follows from (3.11), Theorem 2.2 and (3.12).

4. A Chiti type inequality

In this section, we prove a reverse Hölder inequality for the eigenfunctions corresponding to $\lambda ^H_{p,q}(\Omega )$ . We first prove the following proposition as in the spirit of the Talenti result contained in [Reference Talenti37] (see also [Reference Alberico, Ferone and Volpicelli1Reference Alvino, Ferone and Trombetti3, Reference Amato, Barbato, Masiello and Paoli7, Reference Sannipoli36]).

Proposition 4.1. Let $\Omega \subset \mathbb{R}^n$ , $n\,{\geqslant}\, 2$ , be an open, bounded, and connected set, $1\lt q\,{\leqslant}\, p$ , and let $m\in L^{\infty }(\Omega )$ be a positive function. Let $u$ be a positive eigenfunction corresponding to $\lambda ^H_{p,q}(\Omega )$ . Then we have that

(4.1) \begin{equation} (-{u^*}^{\prime }(s))^{p-1} \,{\leqslant}\, n^{-p} k_n^{-\frac{p}{n}}\lambda ^H_{p,q}(\Omega ) \| u\|_{q,m}^{p-q}\int _0^{s} m^*u^*(r)^{q-1}\,dr, \qquad s\in [0,|\Omega |]. \end{equation}

In particular, the equality case holds if and only if $\Omega = \Omega ^{\star }$ and $m=m^{\star }$ , that is, in $\Omega$ , up to translations, where $m^{\star }$ is the convex symmetrization of $m$ .

Proof. We argue exactly as in the proof of [Reference Belloni, Ferone and Kawohl9, Lemma 3.6]. Let $u$ be a weak solution to the problem (1.2) corresponding to the first eigenvalue $\lambda ^H_{p,q}(\Omega )$ , that is,

\begin{equation*} \begin {cases} -\mathcal {L}_{p}(u)=\lambda ^H_{p,q}(\Omega ) m\| u\|_{q,m}^{p-q}u^{q-1} & \mbox {in}\ \Omega \\[5pt] u=0&\mbox {on}\ \partial \Omega . \end {cases} \end{equation*}

Let $t,h\gt 0$ and let us choose as a test function in (3.1) the following function in $W^{1,p}_0(\Omega )$ :

\begin{equation*} \varphi _h = \begin {cases} 0 & u\,{\leqslant}\, t\\[5pt] u-t & t\lt u\,{\leqslant}\, t+h\\[5pt] h & u\gt t+h. \end {cases} \end{equation*}

By standard arguments, we have

(4.2) \begin{equation} -\frac{d}{dt}\int _{\{u\gt t\}} H(\nabla u)^p\,d\mathcal{H}^{n-1}=\lambda _{p,q}^H(\Omega ) \|u\|_{q,m}^{p-q}\int _{\{u\gt t\}}mu^{q-1}\,dx. \end{equation}

Recalling that the anisotropic perimeter can be written as follows:

\begin{equation*} -\frac {d}{dt}\int _{\{u\gt t\}} H(\nabla u)\,d\mathcal {H}^{n-1} = P_H(\{u\gt t\}), \end{equation*}

by Hölder inequality, we get

\begin{equation*} P_H(\{u\gt t\})\,{\leqslant}\, \bigg (-\frac {d}{dt}\int _{\{u\gt t\}} H(\nabla u)^p\,d\mathcal {H}^{n-1}\bigg )^{\frac {1}{p}}(-\mu ^{\prime }(t))^{1-\frac {1}{p}}. \end{equation*}

Therefore, the isoperimetric inequality (2.9) gives

\begin{equation*} (-\mu ^{\prime }(t))^{1-p}\bigg (-\frac {d}{dt}\int _{\{u\gt t\}} H(\nabla u)^p\,d\mathcal {H}^{n-1}\bigg ) \,{\geqslant}\, n^pk_n^{\frac {p}{n}}\mu (t)^{p-\frac {p}{n}} \end{equation*}

Since $\mu ^{\prime }(t)=\frac{1}{{u^*}^{\prime }(\mu (t))}$ and (4.2) holds true, we have

\begin{equation*} (-{u^*}^{\prime }(\mu (t)))^{p-1} \,{\leqslant}\, n^{-p}k_n^{-\frac {p}{n}} \lambda ^H_{p,q}(\Omega )\| u\|_{q,m}^{p-q}\mu (t)^{\frac {p}{n}-p}\int _{\{u\gt t\}} m(x)u^{q-1} \,d\mathcal {H}^{n-1}. \end{equation*}

Using (2.12) and calling $s=\mu (t)$ , we have

\begin{equation*} (-{u^*}^{\prime }(s))^{p-1} \,{\leqslant}\, n^{-p}k_n^{-\frac {p}{n}}\lambda ^H_{p,q}(\Omega )\| u\|_{q,m}^{p-q}\,s^{\frac {p}{n}-p}\int _0^{s} m^* (u^*)^{q-1} \,dr. \end{equation*}

An application of the Hardy–Littlewood inequality gives the desired result.

The main tool we use in order to prove Theorem 1.1 is a suitable comparison result between $u$ and an eigenfunction $z$ of a suitable eigenvalue problem. More precisely, let $\Omega ^{\star }_\lambda$ be the Wulff shape centered at the origin such that $\lambda ^H_{p,q}(\Omega )$ is the first eigenvalue to the following symmetric problem:

(4.3) \begin{equation} \begin{cases} -\mathcal{L}_{p}(z)=\mu m^\star \| z\|_{q,m^\star }^{p-q}z^{q-1} & \mbox{in}\ \Omega ^{\star }_\lambda \\[5pt] z=0&\mbox{on}\ \partial \Omega ^{\star }_\lambda, \end{cases} \end{equation}

We stress that the Faber–Krahn inequality (3.8) implies that

(4.4) \begin{equation} |\Omega |\,{\geqslant}\, |\Omega ^{\star }_\lambda |, \end{equation}

and hence $m^\star$ is well defined in $\Omega ^\star _\lambda$ .

Let $z$ be a positive eigenfunction for the problem (4.3) corresponding to the first eigenvalue $\lambda ^H_{p,q}(\Omega )$ , and we observe that repeating the same argument as before, by Proposition 3.4, for any $1\lt q\,{\leqslant}\, p$ we have

(4.5) \begin{equation} (-{z^*}^{\prime }(s))^{p-1} = n^{-p} k_n^{-\frac{p}{n}}\lambda ^H_{p,q}(\Omega ) \| z^*\|_{q,m^*}^{p-q}\int _0^{s} m^*z^*(r)^{q-1}\,dr. \end{equation}

The following proposition gives a comparison result between the eigenfunctions $u$ and $z$ when they are normalized with respect to the $L^{\infty }$ norm.

Proposition 4.2. Let $\Omega \subset \mathbb{R}^n$ , $n\,{\geqslant}\, 2$ , be an open, bounded, and connected set, $1\lt q\,{\leqslant}\, p$ and let $m\in L^{\infty }(\Omega )$ be a positive function. Let $u$ be a positive solution to the problem (1.2) corresponding to $\lambda ^H_{p,q}(\Omega )$ and let $z$ be a positive eigenfunction to the problem (4.3) corresponding to $\lambda ^H_{p,q}(\Omega )$ such that

\begin{equation*}\|u\|_{L^{\infty }(\Omega )}=\|z\|_{L^{\infty }(\Omega ^\star _\lambda )}\end{equation*}

Then

\begin{equation*} u^*(s)\,{\geqslant}\, z^*(s), \qquad \forall s \in [0,|\Omega ^\star _\lambda |], \end{equation*}

where $u^*$ and $z^*$ are, respectively, the decreasing rearrangements of $u$ and $z$ . The equality case holds if and only if $\Omega =\Omega ^{\star }_\lambda$ and $m=m^{\star }$ , that is, in $\Omega$ , up to translations, where $m^\star$ is the convex symmetrization of $m$ .

Proof. First of all we stress that, if $|\Omega |=|\Omega ^\star _\lambda |$ , then there is nothing to prove, since Faber–Krahn inequality implies that $u^*(s)=z^*(s)$ .

Moreover, we have $u^*(|\Omega ^\star _\lambda |)\gt z^*(|\Omega ^\star _\lambda |)=0$ . Then, the following definition is well posed:

\begin{equation*} s_0=\inf \{s\in [0,|\Omega ^\star _\lambda |]\,:\; u^*(t)\,{\geqslant}\, z^*(t), \, \forall t \in [s,|\Omega ^\star _\lambda |]\}. \end{equation*}

By definition, $u^*(s_0)=z^*(s_0)$ , and we want to prove that $s_0=0$ . We proceed by contradiction supposing that $s_0\gt 0$ . Then under this assumption, $u^*$ and $z^*$ coincide in $0$ and $s_0$ and we have

(4.6) \begin{equation} \begin{cases} u^*(s)\lt z^*(s) & s\in (0,s_0)\\[5pt] u^*(s)\,{\geqslant}\, z^*(s) & s\in (s_0,|\Omega ^\star _\lambda |). \end{cases} \end{equation}

By (4.1), (4.5), and (4.6), we have that

\begin{equation*} -{u^*}^{\prime }(t)\,{\leqslant}\, -{z^*}^{\prime }(t), \quad \text { for every } t\in (0,s_0). \end{equation*}

Integrating between $(0,s)$ , with $s\in (0,s_0)$ , being $u^*(0)=z^*(0)$ , we get

\begin{equation*} u^*(s)\,{\geqslant}\, z^*(s), \qquad \forall s \in (0,s_0), \end{equation*}

which is in contradiction with the definition of $s_0$ . Hence, $s_0=0$ , and the proof is completed.

As an immediately consequence of the previous result, we get the following scale-invariant inequality for any $r\gt 0$ :

(4.7) \begin{equation} \frac{\|u\|_{L^r(\Omega, m)}}{\|u\|_{L^\infty (\Omega )}}\,{\geqslant}\, \frac{\|z\|_{L^r(\Omega ^{\star }_{\lambda },m^\star )}}{\|z\|_{L^\infty (\Omega _{\lambda }^{\star })}}. \end{equation}

When the functions $u$ and $z$ are normalized with respect to the weighted $L^q$ -norm, we get the following comparison result.

Theorem 4.3. Let $\Omega \subset \mathbb{R}^n$ be an open, bounded, and connected set, $1\lt q\,{\leqslant}\, p$ and let $m\in L^{\infty }(\Omega )$ be a positive function. Let $u$ be a positive solution to the problem (1.2) corresponding to $\lambda ^H_{p,q}(\Omega )$ and let $z$ be a positive eigenfunction to the problem (4.3) corresponding to $\lambda ^H_{p,q}(\Omega )$ such that

(4.8) \begin{equation} \int _{\Omega } m\,u^{q}\,dx = \int _{\Omega ^{\star }_\lambda } m^\star z^{q}\,dx. \end{equation}

Then we have

(4.9) \begin{equation} \int _0^{s} m^* \,(u^*)^{r}\,dt\,{\leqslant}\, \int _0^{s} m^*\,(z^*)^{r}\,dt, \quad s\in [0,|\Omega ^\star _\lambda |], \quad q\,{\leqslant}\, r \end{equation}

where $u^*$ , $m^*$ , and $z^*$ are, respectively, the decreasing rearrangements of $u$ , $m$ , and $z$ , and $m^\star$ is the convex symmetrization of $m$ . The equality case holds if and only if $\Omega =\Omega ^{\star }$ , $z=u=u^*$ , and $m=m^{\star }$ , that is, $\Omega$ , up to translations.

Proof. If $|\Omega |=|\Omega ^{\star }_\lambda |$ , the conclusion is trivial. Let be $|\Omega |\gt |\Omega ^{\star }_\lambda |$ , since $u$ and $z$ verify (4.8), by (4.7) it holds that

\begin{equation*} u^*(0)=\|u\|_{L^{\infty }(\Omega )} \,{\leqslant}\, \|z\|_{L^{\infty }(\Omega ^{\star }_{\lambda })}= z^*(0), \end{equation*}

If $u^*(0)=z^*(0)$ , then Proposition 4.2 and the normalization (4.8) imply that $u^*(s)=z^*(s)$ for every $s\in [0,|\Omega ^{\star }_\lambda |]$ and than the claim follows trivially.

Let $u^*(0)\lt z^*(0)$ . Since $u^*(|\Omega ^{\star }_\lambda |)\gt z^*(|\Omega ^{\star }_\lambda |)$ , we can consider

\begin{equation*} s_0 = \sup \{s\in (0,|\Omega ^{\star }_\lambda |) \;:\; u^*(t)\,{\leqslant}\, z^*(t) \text { for } t \in [0,s]\}. \end{equation*}

Obviously, $0\lt s_0\lt |\Omega ^{\star }_\lambda |$ , $u^*(s_0)=z^*(s_0)$ and $u^*\,{\leqslant}\, z^*$ in $[0,s_0]$ . We want to show that $u^*\gt z^*$ in $[s_0,|\Omega ^{\star }_\lambda |]$ . Indeed, if we suppose by contradiction that there exists $s_1\gt s_0$ such that $u^*(s_1)=z^*(s_1)$ and $u^*(s)\gt z^*(s)$ for $s\in (s_0,s_1)$ , we can construct the following function:

\begin{equation*} w^*(s)= \begin {cases} z^*(s) & s \in [0,s_0]\cup [s_1,|\Omega ^{\star }_\lambda |]\\[5pt] u^*(s) & s \in [s_0,s_1]. \end {cases} \end{equation*}

It is straightforward to check that

\begin{equation*} \int _{\Omega }H(\nabla w)^p\,dx = n^pk^p_n\int _{\Omega } (-{w^*}^{\prime }(k_n H^{\circ }(x)^n))^pH^{\circ }(x)^{p(n-1)}\,dx. \end{equation*}

Applying Coarea Formula and considering the change of variables $s=k_n t^n$ , we get

\begin{equation*} \int _{\Omega } H(\nabla w)^p\,dx = n^p k_n^{\frac {p}{n}}\int _0^{|\Omega ^{\star }_\lambda |} s^{p-\frac {p}{n}}(-{w^*}^{\prime }(t))^p\,dt. \end{equation*}

Thanks to the normalization (4.8) and the definition of $w$ , we have that

\begin{equation*} \| u\|_{L^q(\Omega, m)} = \| z\|_{L^q(\Omega ^\star _\lambda, m^\star )} \,{\leqslant}\, \| w\|_{L^q(\Omega ^\star _\lambda, m^\star )}, \end{equation*}

then by (4.1) and (4.5), we have that

(4.10) \begin{equation} (-{w^*}^{\prime }(s))^{p-1} \,{\leqslant}\, n^{-p} k_n^{-\frac{p}{n}}\lambda ^H_{p,q}(\Omega ) \| w^*\|_{q,m^*}^{p-q}s^{\frac{p}{n}-p}\int _0^{s} m^*(r)({w^*})^{q-1}(r)\,dr. \end{equation}

Multiplying (4.10) by $-w^{\prime }$ , rearranging the terms and integrating between $0$ and $|\Omega ^{\star }_\lambda |$ , we get

\begin{equation*} \begin {aligned} n^pk_n^{\frac {p}{n}}&\int _0^{|\Omega ^{\star }_\lambda |} s^{p-\frac {p}{n}}(-{w^*}^{\prime }(s))^p\,ds\,{\leqslant}\, \\[5pt] &\,{\leqslant}\, \lambda ^H_{p,q}(\Omega ) \| w^*\|_{q,m^*}^{p-q} \int _0^{|\Omega ^{\star }_\lambda |}(-{w^*}^{\prime }(s))\int _0^sm^*(r)({w^*})^{q-1}(r)\,dr\,ds. \end {aligned} \end{equation*}

An integration by parts allows us to conclude that

\begin{equation*} \frac {\displaystyle \int _{\Omega ^\star _\lambda } H(\nabla w)^p\,dx}{\bigg (\displaystyle \int _{\Omega ^\star _\lambda }m^\star w^q\,dx\bigg )^{\frac {p}{q}}}=\frac {\displaystyle n^pk_n^{\frac {p}{n}}\int _0^{|\Omega ^{\star }_\lambda |} s^{p-\frac {p}{n}}(-w^{\prime }(s))^p\,ds}{\bigg (\displaystyle \int _0^{|\Omega ^{\star }_\lambda |}m^*(s)({w^*})^q(s)\,ds\bigg )^{\frac {p}{q}}}\,{\leqslant}\, \lambda ^H_{p,q}(\Omega )=\lambda ^H_{p,q}(\Omega ^{\star }_\lambda ). \end{equation*}

By the minimality and the simplicity of $\lambda ^H_{p,q}$ , and the definition of $w^*$ , it must be $w^*(s)=z^*(s)$ for every $s\in [0,|\Omega ^{\star }_\lambda |]$ , but this is a contradiction since in $(s_0,s_1)$ we have that $u^*(s)\gt z^*(s)$ .

In this way, we have proved that there exists a unique point $s_0$ where $u^*$ and $z^*$ can cross each other, and such that

(4.11) \begin{equation} \begin{cases} u^*(s)\,{\leqslant}\, z^*(s) & s\in [0,s_0]\\[5pt] u^*(s)\,{\geqslant}\, z^*(s) & s\in [s_0,|\Omega ^{\star }_\lambda |]. \end{cases} \end{equation}

If we extend $z^*$ to be zero in $[|\Omega ^{\star }_\lambda |, |\Omega |]$ , by (4.8) and (4.11) then we have that for every $s\in [0,|\Omega |]$

(4.12) \begin{equation} \int _0^{s} m^*(t)({u^*(t)})^{q}\,dt\,{\leqslant}\, \int _0^{s} m^*(t)({z^*(t)})^{q}\,dt. \end{equation}

Indeed (4.11) implies that the function:

\begin{equation*}G(s)=\int _0^{s} {m^*}(t)((z^*))^{q}-(u^*)^q\,dt, \qquad s \in [0,|\Omega |] \end{equation*}

has a maximum in $s_0$ and cannot be negative in any point. This proves (4.12). Finally, inequality (4.9) follows easily by (4.12) by using Proposition 2.4 being $m^{\star }(u^{\star })^q \prec m^{\star }z^q$ .

Proof of Theorem 1.1. The proof of statement i) follows directly from (4.8) and (4.9), indeed we have

\begin{equation*} \bigg (\int _{\Omega } m u^{r}\,dx\bigg )^{\frac {1}{r}}\,{\leqslant}\, \bigg (\int _{\Omega ^{\star }_\lambda } m^\star z^{r}\,dx\bigg )^{\frac {1}{r}} = \frac {\displaystyle \bigg (\int _{\Omega ^{\star }_\lambda } m^\star z^{r}\,dx\bigg )^{\frac {1}{r}} }{\displaystyle \bigg (\int _{\Omega ^{\star }_\lambda } m^\star z^{q}\,dx\bigg )^{\frac {1}{q}} } \bigg (\int _{\Omega } m u^{q}\,dx\bigg )^{\frac {1}{q}}. \end{equation*}

Therefore, we have

\begin{equation*} \|u \|_{L^r(\Omega, m)} \,{\leqslant}\, C\,\,\| u\|_{L^q(\Omega, m)}, \quad q \,{\leqslant}\, r\lt +\infty ; \end{equation*}

with

\begin{equation*} C = \displaystyle \frac {\|z \|_{L^r(\Omega ^{\star }_{\lambda },m^{\star })}}{\|z \|_{L^q(\Omega ^{\star }_{\lambda },m^{\star })}}. \end{equation*}

The proof of statement ii) follows immediately by Proposition 4.2 with $C= \frac{\|z\|_{\infty }}{\|z\|_{L^r(\Omega ^\star, m^\star )}}$ and the theorem is completely proved.

Acknowledgments

This work has been partially supported by MiUR PRIN 2017 grant “Qualitative and quantitative aspects of nonlinear PDEs,” the MiUR PRIN-PNRR 2022 grant “Linear and Nonlinear PDE’s: New directions and Applications,” and GNAMPA of INdAM. The second author acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Pisa, CUP I57G22000700001.

Conflicts of interest

The authors declare that there is no conflict of interest.

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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