Conformal Spectral Stability Estimates for the Neumann Laplacian

TL;DR

This paper investigates how the eigenvalues of the Neumann Laplacian change when the domain's shape varies, using conformal mappings and energy estimates, even for domains with fractal boundaries.

Contribution

It introduces conformal spectral stability estimates for the Neumann Laplacian in planar domains with irregular, fractal-like boundaries, extending previous stability results.

Findings

Derived bounds for eigenvalue variations under domain perturbations

Applicable to domains with boundaries of Hausdorff dimension between one and two

Established energy integral estimates for spectral stability

Abstract

We study the eigenvalue problem for the Neumann-Laplace operator in conformal regular planar domains $\Omega\subset\mathbb{C}$. Conformal regular domains support the Poincar\'e inequality and this allows us to estimate the variation of the eigenvalues of the Neumann Laplacian upon domain perturbation via energy type integrals. Boundaries of such domains can have any Hausdorff dimension between one and two.