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On the solutions to $p$-Poisson equation with Robin boundary conditions when $p$ goes to $+\infty$

Vincenzo Amato, Alba Lia Masiello, Carlo Nitsch, Cristina Trombetti

Abstract

We study the behaviour, when $p \to +\infty$, of the first $p$-Laplacian eigenvalues with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that the limit of the eigenfunctions is a viscosity solution to an eigenvalue problem for the so-called $\infty$-Laplacian.

On the solutions to $p$-Poisson equation with Robin boundary conditions when $p$ goes to $+\infty$

Abstract

We study the behaviour, when , of the first -Laplacian eigenvalues with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that the limit of the eigenfunctions is a viscosity solution to an eigenvalue problem for the so-called -Laplacian.
Paper Structure (7 sections, 16 theorems, 124 equations)

This paper contains 7 sections, 16 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Viscosity solutions
  4. The $\infty$-eigenvalue problem
  5. The first Robin $\infty$-eigenvalue
  6. The $p$-Poisson equation
  7. The limit PDE

Key Result

Lemma 2.1

Given $f, g \in W^{1,\infty}(\Omega)$, then

Theorems & Definitions (40)

  • Lemma 2.1
  • proof
  • Definition 2.1
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 30 more