Two inequalities for the first Robin eigenvalue of the Finsler Laplacian

TL;DR

This paper establishes two inequalities for the first Robin eigenvalue of the Finsler Laplacian in bounded domains, analyzing its asymptotic behavior as the boundary parameter approaches zero.

Contribution

It introduces novel inequalities for the Finsler Laplacian's first eigenvalue with Robin boundary conditions, extending spectral analysis in Finsler geometry.

Findings

Derived two inequalities for the first Robin eigenvalue

Analyzed asymptotic behavior as boundary parameter tends to zero

Extended spectral results to Finsler Laplacian context

Abstract

Let \Omega be a bounded connected, open set of \R^n with Lipschitz boundary. Let F be a suitable norm in \R^n and let \Delta_F u be the so-colled Finsler Laplacian. In this paper we prove two inequalities for the first eigenvalue of \Delta_F with Robin boundary conditions involving a positive function \beta. As a consequence of our result we obtain the asymptotic behavior of this eigenvalue when \beta is a positive constant which goes to zero.