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On a weighted anisotropic eigenvalue problem

Nunzia Gavitone, Rossano Sannipoli

TL;DR

This work studies a weighted eigenvalue problem for the anisotropic $(p,q)$-Laplacian with Dirichlet boundary conditions. It develops a full variational framework, proves a Faber–Krahn type inequality, and establishes a Chiti-type reverse Hölder inequality for first eigenfunctions by leveraging convex symmetrization and comparison with symmetric problems on Wulff domains. These results generalize classical Euclidean, unweighted cases to the anisotropic and weighted setting, with equality cases precisely linked to Wulff shapes. The findings provide sharp norm-control of first eigenfunctions and reveal that Wulff shapes optimize both eigenvalues and associated comparison principles, impacting isoperimetric-type questions in anisotropic media.

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 prove a reverse Hölder type inequality for the corresponding eigenfunctions.

On a weighted anisotropic eigenvalue problem

TL;DR

This work studies a weighted eigenvalue problem for the anisotropic -Laplacian with Dirichlet boundary conditions. It develops a full variational framework, proves a Faber–Krahn type inequality, and establishes a Chiti-type reverse Hölder inequality for first eigenfunctions by leveraging convex symmetrization and comparison with symmetric problems on Wulff domains. These results generalize classical Euclidean, unweighted cases to the anisotropic and weighted setting, with equality cases precisely linked to Wulff shapes. The findings provide sharp norm-control of first eigenfunctions and reveal that Wulff shapes optimize both eigenvalues and associated comparison principles, impacting isoperimetric-type questions in anisotropic media.

Abstract

In this paper we deal with a weighted eigenvalue problem for the anisotropic -Laplacian with Dirichlet boundary conditions. We study the main properties of the first eigenvalue and prove a reverse Hölder type inequality for the corresponding eigenfunctions.
Paper Structure (8 sections, 12 theorems, 88 equations)

This paper contains 8 sections, 12 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. The anisotropic norm
  4. Convex symmetrization
  5. The (p,q)-anisotropic Laplacian
  6. The case $\Omega= \mathcal{W}_R$
  7. A Faber-Krahn type inequality
  8. A Chiti type inequality

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb R^n$ be an open, bounded and connected set. Let $1< q\le p$, and let $u$ be an eigenfunction corresponding to the first eigenvalue varcar. Then the following statements hold The equality cases hold if and only if $\Omega$ is a Wulff shape.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • ...and 11 more