# Quasiconformal Mappings and Neumann Eigenvalues of Divergent Elliptic Operators

## TL;DR

This paper investigates how quasiconformal mappings influence the spectral properties of divergence form elliptic operators with Neumann boundary conditions in planar domains, including fractal-like structures.

## Contribution

It introduces a novel approach linking quasiconformal mappings, elliptic operators, and composition operators on Sobolev spaces to analyze spectral properties.

## Key findings

- Spectral properties are characterized in domains satisfying quasihyperbolic boundary conditions.
- Method applies to fractal-type domains, extending classical spectral analysis.
- Connections between quasiconformal mappings and elliptic operator spectra are established.

## Abstract

We study spectral properties of divergence form elliptic operators $-\textrm{div} [A(z) \nabla f(z)]$ with the Neumann boundary condition in planar domains (including some fractal type domains), that satisfy to the quasihyperbolic boundary conditions. Our method is based on an interplay between quasiconformal mappings, elliptic operators and composition operators on Sobolev spaces.