Conformal composition operators with applications to Dirichlet eigenvalues

TL;DR

This paper develops spectral estimates for the first Dirichlet eigenvalue of the degenerate p-Laplace operator in complex domains using conformal analysis, extending results to non-rectifiable boundaries.

Contribution

It introduces a conformal analysis approach to estimate Dirichlet eigenvalues in complex domains with irregular boundaries, advancing spectral theory methods.

Findings

Spectral estimates valid for non-rectifiable boundary domains

Application of conformal analysis to degenerate p-Laplace operators

Extension of eigenvalue bounds to complex geometries

Abstract

This paper is concerned with spectral estimates for the first Dirichlet eigenvalue of the degenerate $p$-Laplace operator in bounded simply connected domains $\Omega \subset \mathbb C$. The proposed approach relies on the conformal analysis of the elliptic operators, which allows us to obtain spectral estimates in domains with non-rectifiable boundaries.