Space Quasiconformal Mappings and Neumann Eigenvalues in Fractal Domains

TL;DR

This paper investigates how quasiconformal domain perturbations affect Neumann eigenvalues of the p-Laplace operator, providing lower bounds in fractal domains through geometric composition operator theory.

Contribution

It introduces a novel approach linking quasiconformal mappings with composition operators to estimate Neumann eigenvalues in fractal-like spaces.

Findings

Lower bounds for Neumann eigenvalues in fractal domains

Connection between quasiconformal mappings and spectral estimates

Application of geometric composition operator theory

Abstract

We study the variation of the Neumann eigenvalues of the $p$-Laplace operator under quasiconformal perturbations of space domains. This study allows to obtain lower estimates of the Neumann eigenvalues in fractal type domains. The suggested approach is based on the geometric theory of composition operators in connections with the quasiconformal mapping theory.