On the second eigenvalue of the infinity Laplacian with Robin boundary conditions

TL;DR

This paper investigates the asymptotic behavior of the second eigenvalues of the p-Laplacian with Robin boundary conditions as p approaches infinity, establishing their convergence to the second eigenvalue of the infinity Laplacian.

Contribution

It provides a rigorous analysis of the second eigenvalues' limit for the p-Laplacian with Robin conditions, linking it to the infinity Laplacian's eigenvalues.

Findings

Second eigenvalues converge to the infinity Laplacian's second eigenvalue as p→∞.

The limit eigenfunctions are characterized in the context of the infinity Laplacian.

The results depend on certain regularity conditions of the domain.

Abstract

We study the behaviour, as $p \to +\infty$, of the second eigenvalues of the $p$-Laplacian with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that, up to some regularity of the set, the limit of the second eigenvalues is actually the second eigenvalue of the so-called $\infty$-Laplacian.