The spectral estimates for the Neumann-Laplace operator in space domains

TL;DR

This paper proves the discreteness of the Neumann-Laplace spectrum in various non-convex space domains and provides lower eigenvalue bounds using geometric and Sobolev mapping techniques.

Contribution

It introduces a novel approach using composition operators and quasiconformal mappings to estimate eigenvalues for the Neumann-Laplace operator in complex domains.

Findings

Spectrum is discrete in a broad class of non-convex domains.

Lower bounds for the first non-trivial eigenvalue are established.

Method applies to domains with complex geometries using Sobolev and quasiconformal mappings.

Abstract

In this paper we prove discreteness of the spectrum of the Neu\-mann-Lap\-la\-ci\-an (the free membrane problem) in a large class of non-convex space domains. The lower estimates of the first non-trivial eigenvalue are obtained in terms of geometric characteristics of Sobolev mappings. The suggested approach is based on Poincar\'e-Sobolev inequalities that are obtained with the help of the composition operators theory for uniform Sobolev spaces. These composition operators are induced by a generalizations of conformal mappings that are mappings of bounded $2$-dilatation ($2$-quasiconformal mappings).