Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators

TL;DR

This paper establishes sharp bounds for the first eigenvalue and torsional rigidity of certain anisotropic nonlinear elliptic operators, advancing understanding of their spectral and geometric properties.

Contribution

It provides the first sharp bounds for the eigenvalues and torsional rigidity of a broad class of anisotropic nonlinear operators, including p-Laplace types.

Findings

Sharp upper bounds for the first Dirichlet eigenvalue.

A stability result via isoperimetric deficit.

Sharp lower bounds for anisotropic p-torsional rigidity.

Abstract

We prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the p-Laplace and the pseudo-p-Laplace operators. Moreover, we prove a stability result by means of a suitable isoperimetric deficity. Finally, we give a sharp lower bound for the anisotropic p-torsional rigidity.