Space quasiconformal composition operators with applications to Neumann eigenvalues

TL;DR

This paper develops estimates for Neumann eigenvalues of p-Laplace operators in complex space domains using quasiconformal composition operators, enhancing understanding of spectral properties in irregular geometries.

Contribution

It introduces a novel method employing quasiconformal mappings and composition operators to estimate Neumann eigenvalues in domains with quasihyperbolic boundary conditions.

Findings

Derived new bounds for Neumann eigenvalues in quasihyperbolic domains.

Refined estimates for quasi-balls using inverse H"older inequality.

Applied composition operators to improve Sobolev-Poincaré inequalities.

Abstract

In this article we obtain estimates of Neumann eigenvalues of $p$-Laplace operators in a large class of space domains satisfying quasihyperbolic boundary conditions. The suggested method is based on composition operators generated by quasiconformal mappings and their applications to Sobolev-Poincar\'e-inequalities. By using a sharp version of the inverse H\"older inequality we refine our estimates for quasi-balls, that is, images of balls under quasiconformal mappings of the whole space.