Sharp estimates for fundamental frequencies of elliptic operators generated by asymmetric seminorms in low dimensions

TL;DR

This paper derives precise bounds for fundamental frequencies of elliptic operators in low dimensions, utilizing asymmetric Poincare inequalities and anisotropic spectral analysis to improve understanding of spectral optimization.

Contribution

It introduces sharp asymmetric Poincare inequalities with optimal constants in one dimension and applies this to study spectral properties of elliptic operators in the plane with anisotropic norms.

Findings

Established new sharp asymmetric Poincare inequalities with optimal constants.

Developed a comprehensive analysis of fundamental frequencies in the plane.

Advanced spectral optimization techniques for anisotropic elliptic operators.

Abstract

We establish new sharp asymmetric Poincare inequalities in one-dimension with the computation of optimal constants and characterization of extremizers. Using the one-dimensional theory we develop a comprehensive study on fundamental frequencies in the plane and related spectral optimization in the very general setting of positively homogeneous anisotropies.