# Spectral Properties of the Neumann-Laplace operator in Quasiconformal Regular Domains

## TL;DR

This paper investigates the spectral characteristics of the Neumann-Laplace operator in planar quasiconformal domains, providing estimates for eigenvalues using composition operator theory on Sobolev spaces.

## Contribution

It introduces new estimates for Poincaré-Sobolev constants and eigenvalues in quasiconformal domains via composition operator analysis.

## Key findings

- Derived bounds for Poincaré-Sobolev constants.
- Established lower estimates for the first non-trivial eigenvalue.
- Linked spectral properties to quasiconformal domain regularity.

## Abstract

In this paper we study spectral properties of the Neumann-Laplace operator in planar quasiconformal regular domains $\Omega\subset\mathbb R^2$. This study is based on the quasiconformal theory of composition operators on Sobolev spaces. Using the composition operators theory we obtain estimates of constants in Poincar\'e-Sobolev inequalities and as a consequence lower estimates of the first non-trivial eigenvalue of the Neumann-Laplace operator in planar quasiconformal regular domains.