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

TL;DR

This paper investigates the asymptotic behavior of the first eigenvalues and eigenfunctions of the p-Laplacian with Robin boundary conditions as p approaches infinity, revealing their convergence to solutions of the infinity-Laplacian eigenproblem.

Contribution

It establishes the convergence of eigenfunctions to viscosity solutions of the infinity-Laplacian eigenproblem under Robin boundary conditions as p tends to infinity.

Findings

Eigenfunctions converge to viscosity solutions of the infinity-Laplacian eigenproblem.

The limit eigenfunctions satisfy a specific eigenvalue problem for the infinity-Laplacian.

The study extends understanding of p-Laplacian behavior under Robin boundary conditions for large p.

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.