On the principal frequency of non-homogeneous membranes

TL;DR

This paper provides estimates for the fundamental frequency of non-homogeneous membranes modeled by elliptic operators, using quasiconformal mappings and Sobolev inequalities, with a variant of the Rayleigh-Faber-Krahn inequality.

Contribution

It introduces a novel method based on quasiconformal composition operators to estimate eigenvalues for elliptic operators in non-homogeneous membranes.

Findings

Derived bounds for first eigenvalues in non-homogeneous membranes.

Established a variant of the Rayleigh-Faber-Krahn inequality for specific elliptic operators.

Applied quasiconformal mappings to Sobolev space estimates.

Abstract

We obtained estimates for first eigenvalues of the Dirichlet boundary value problem for elliptic operators in divergence form (i.e. for the principal frequency of non-homogeneous membranes) in bounded domains $\Omega \subset \mathbb C$ satisfying quasihyperbolic boundary conditions. The suggested method is based on the quasiconformal composition operators on Sobolev spaces and their applications to constant estimates in the corresponding Sobolev-Poincar\'e inequalities. We also prove a variant of the Rayleigh-Faber-Khran inequality for a special case of these elliptic operators.