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The anisotropic $\infty$-Laplacian eigenvalue problem with Neumann boundary conditions

Gianpaolo Piscitelli

Abstract

We analize the limit problem of the anisotropic $p$-Laplacian as $p\rightarrow\infty$ with the mean of the viscosity solution. We also prove some geometric properties of eigenvalues and eigenfunctions. In particular, we show the validity of a Szegö-Weinberger type inequality.

The anisotropic $\infty$-Laplacian eigenvalue problem with Neumann boundary conditions

Abstract

We analize the limit problem of the anisotropic -Laplacian as with the mean of the viscosity solution. We also prove some geometric properties of eigenvalues and eigenfunctions. In particular, we show the validity of a Szegö-Weinberger type inequality.

Paper Structure

This paper contains 5 sections, 15 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The limiting problem
  4. Proof of the Main Result
  5. Geometric properties of the first $\infty$-eigenvalue

Key Result

Proposition 2.1

Let $\Omega$ be a convex set in $\mathbb{R}^n$. Then The equality sign holds if and only if $\Omega$ is equivalent to a Wulff shape.

Theorems & Definitions (33)

  • Proposition 2.1
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Definition 3.6
  • ...and 23 more