Sharp isoanisotropic estimates for fundamental frequencies of membranes and connections with shapes

TL;DR

This paper establishes sharp bounds for the fundamental frequencies of anisotropic membranes in the plane, characterizes extremizers, and connects shape optimization with spectral properties, proving a key conjecture for p-Laplace operators.

Contribution

It provides a complete solution for anisotropic operators in the plane, including optimal constants, extremizers, and a link between shape and spectral properties, extending existing spectral optimization results.

Findings

Supremum of anisotropic fundamental frequencies over fixed-area membranes is infinite for nonzero anisotropy.

Maximization conjecture for p-Laplace fundamental frequencies is proved for all p ≠ 2.

Characterization of extremizers and optimal constants in anisotropic spectral bounds.

Abstract

The underlying motivation of the present work lies on a cornerstone question in spectral optimization that consists of determining sharp lower and upper uniform estimates for fundamental frequencies of a set of uniformly elliptic operators on a fixed membrane. We solve completely the problem in the plane for the general class of anisotropic operators in divergence form generated by arbitrary norms, which also includes the computation of optimal constants and the characterization of corresponding anisotropic extremizers (if they exist). Our approach is based on an isoanisotropic optimization formulation which, in turn, demands to be addressed within the broader environment of nonnegative, convex and 1-homogeneous anisotropies. A fine and detailed analysis of least energy levels associated to anisotropies with maximum degeneracy leads to a central connection between shapes and fundamental frequencies of rather degenerate elliptic operators. Such a linking also permits to establish that the supremum of anisotropic fundamental frequencies over all fixed-area membranes is infinite for any nonzero anisotropy. As a by-product, the well-known maximization conjecture for fundamental frequencies of the p-Laplace operator is proved for any p other than 2.