Conformal spectral stability for the Dirichlet-Laplace operator

Abstract

We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains $\Omega \subset \mathbb{C}$ using conformal transformations of the original problem to the weighted eigenvalue problem for the Dirichlet Laplacian in the unit disc $\mathbb{D}$. This allows us to estimate the variation of the eigenvalues of the Dirichlet Laplacian upon domain perturbation via energy type integrals for a large class of "conformal regular" domains which includes all quasidiscs, i.e. images of the unit disc under quasiconformal homeomorphisms of the plane onto itself. Boundaries of such domains can have any Hausdorff dimension between one and two.

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References in corpus (3)

• Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators
• Brennan's conjecture for composition operators on Sobolev spaces
• On the sharpness of a certain spectral stability estimate for the Dirichlet Laplacian

Cited by in corpus (6)

• On the First Eigenvalues of Free Vibrating Membrane in Conformal Regular Domains
• Spectral Estimates of the $p$-Laplace Neumann operator and Brennan's Conjecture
• Spectral Properties of the Neumann-Laplace operator in Quasiconformal Regular Domains
• Space Quasiconformal Mappings and Neumann Eigenvalues in Fractal Domains
• Spectral Stability Estimates of Neumann Divergence Form Elliptic Operators
• On growth monotonicity estimates of the principal Dirichlet-Laplacian eigenvalue