Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle

TL;DR

This paper derives sharp bounds for the first Dirichlet eigenvalue of anisotropic p-Laplacian operators using the maximum principle and the -function method, linking eigenvalues to geometric domain properties.

Contribution

It introduces a novel application of the -function method to obtain optimal eigenvalue bounds for anisotropic p-Laplacian operators.

Findings

Established sharp lower and upper bounds for _{F}(p,\u2206)

Connected eigenvalue estimates to geometric quantities of the domain

Enhanced understanding of eigenvalue behavior via maximum principle

Abstract

In this paper we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue $\lambda_{F}(p,\Omega)$ of the anisotropic $p$-Laplacian, $1<p<+\infty$. Our aim is to enhance how, by means of the $\mathcal P$-function method, it is possible to get several sharp estimates for $\lambda_{F}(p,\Omega)$ in terms of several geometric quantities associated to the domain. The $\mathcal P$-function method is based on a maximum principle for a suitable function involving the eigenfunction and its gradient.