Principal frequencies of free non-homogeneous membranes and Sobolev extension operators

TL;DR

This paper establishes estimates for the fundamental frequencies of free non-homogeneous membranes using quasiconformal mappings and Sobolev extension operators, linking membrane vibrations to geometric and elliptic operator theories.

Contribution

It introduces a novel approach connecting divergence form elliptic operators with quasiconformal mappings to analyze membrane frequencies.

Findings

Derived estimates for principal frequencies of non-homogeneous membranes.

Established a connection between membrane frequencies and the smallest-circle problem.

Applied Sobolev extension operators within quasiconformal geometry framework.

Abstract

Using the quasiconformal mappings theory and Sobolev extension operators, we obtain estimates of principal frequencies of free non-homogeneous membranes. The suggested approach is based on connections between divergence form elliptic operators and quasiconformal mappings. Free non-homogeneous membranes we consider as circular membranes in the quasiconformal geometry associated with non-homogeneity of membranes. As a consequence we get a connection between principal frequencies of free membranes and the smallest-circle problem (initially suggested by J.~J. Sylvester in 1857).