Estimates of Dirichlet eigenvalues of divergent elliptic operators in non-Lipschitz domains

TL;DR

This paper provides spectral estimates for divergence form elliptic operators with Dirichlet boundary conditions in non-Lipschitz domains, using quasiconformal mappings and Sobolev space techniques.

Contribution

It introduces a novel approach combining quasiconformal composition operators with spectral analysis of elliptic operators in irregular domains.

Findings

Derived new spectral bounds for elliptic operators in non-Lipschitz domains

Established connections between quasiconformal mappings and weighted Sobolev inequalities

Extended classical spectral estimates to irregular geometric settings

Abstract

We study spectral estimates of the divergence form uniform elliptic operators $-\textrm{div}[A(z) \nabla f(z)]$ with the Dirichlet boundary condition in bounded non-Lipschitz simply connected domains $\Omega \subset \mathbb C$. The suggested method is based on the quasiconformal composition operators on Sobolev spaces with applications to the weighted Poincar\'e-Sobolev inequalities.