# On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

## TL;DR

This paper studies the second eigenvalue of the anisotropic p-Laplacian, providing bounds and inequalities, and explores the behavior as p approaches infinity, extending classical spectral results to anisotropic operators.

## Contribution

It establishes a Hong-Krahn-Szego type inequality for the second eigenvalue of the anisotropic p-Laplacian and analyzes its limit as p tends to infinity, which is novel for this class of operators.

## Key findings

- Lower bound for the second eigenvalue among sets of fixed measure
- Validity of a Hong-Krahn-Szego type inequality in the anisotropic setting
- As p approaches infinity, the eigenvalue problem converges to a limit problem

## Abstract

Let $\Omega$ be a bounded open set of $\mathbb R^{n}$, $n\ge 2$. In this paper we mainly study some properties of the second Dirichlet eigenvalue $\lambda_{2}(p,\Omega)$ of the anisotropic $p$-Laplacian \[ -\mathcal Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_\xi (\nabla u)\right), \] where $F$ is a suitable smooth norm of $\mathbb R^{n}$ and $p\in]1,+\infty[$. We provide a lower bound of $\lambda_{2}(p,\Omega)$ among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as $p\to+\infty$.