Optimal anisotropies for p-Laplace type operators in the plane

TL;DR

This paper derives sharp bounds and characterizes extremizers for fundamental frequencies of p-Laplace operators in the plane, revealing anisotropic effects and stability properties with applications to isoperimetric inequalities.

Contribution

It provides the first sharp anisotropic estimates for p-Laplace operators, including extremizer characterizations and stability results, extending classical isoperimetric inequalities.

Findings

Optimal constants for anisotropic p-Laplace operators are established.

Rigidity and attainability of extremizers are characterized.

Lower bounds remain positive without uniform ellipticity.

Abstract

Sharp lower and upper uniform estimates are obtained for fundamental frequencies of $p$-Laplace type operators generated by quadratic forms. Optimal constants are exhibited, rigidity of the upper estimate is proved, anisotropic attainability of the lower estimate is derived as well as characterization of anisotropic extremizers for circular and rectangular membranes. Sharp quantitative anisotropic inequalities associated with lower constants are also established, providing as a by-product information on anisotropic stability. When the uniform ellipticity condition is relaxed, we show that the optimal lower constant remains positive, while anisotropic extremizers no longer exist. Our sharp lower estimate can be viewed as an isoanisotropic counterpart of the Faber-Krahn isoperimetric inequality in the plane.