On Variations of Neumann Eigenvalues of p-Laplacian Generated by Measure Preserving Quasiconformal Mappings

TL;DR

This paper investigates how measure-preserving quasiconformal mappings affect the first non-trivial eigenvalues of the p-Laplacian in two dimensions, providing lower bounds for certain domains using advanced geometric and functional analysis tools.

Contribution

It introduces new lower estimate techniques for p-Laplacian eigenvalues under quasiconformal mappings, extending spectral theory in geometric analysis.

Findings

Lower bounds for eigenvalues on Ahlfors type domains

Application of reverse H"older inequality in spectral estimates

Connection between quasiconformal mappings and Sobolev space composition operators

Abstract

In this paper we study variations of the first non-trivial eigenvalues of the two-dimensional $p$-Laplace operator, $p>2$, generated by measure preserving quasiconformal mappings $\varphi : \mathbb D\to\Omega$, $\Omega \subset\mathbb R^2$. This study is based on the geometric theory of composition operators on Sobolev spaces with applications to sharp embedding theorems. By using a sharp version of the reverse H\"older inequality we obtain lower estimates of the first non-trivial eigenvalues for Ahlfors type domains.