Sobolev Extension Operators and Neumann Eigenvalues

TL;DR

This paper links Sobolev extension operator norms to spectral estimates of Neumann eigenvalues, revealing connections between membrane resonances and geometric circle problems.

Contribution

It introduces a novel application of Sobolev extension operator estimates to spectral geometry and eigenvalue bounds in non-convex domains.

Findings

Estimates of Sobolev extension operator norms improve eigenvalue bounds.

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

Extended spectral estimates to non-convex extension domains.

Abstract

In this paper we apply estimates of the norms of Sobolev extension operators to the spectral estimates of of the first nontrivial Neumann eigenvalue of the Laplace operator in non-convex extension domains. As a consequence we obtain a connection between resonant frequencies of free membranes and the smallest-circle problem (initially proposed by J.~J.~Sylvester in 1857).