Conformal spectral stability for the Dirichlet-Laplace operator

2014/06/17 by В. И. Буренков, Victor Burenkov, Vladimir Gol’dshtein +5 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analytic and geometric function theory #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #math.SP

paper · pdf · doi:10.48550/arxiv.1406.4340

14

openalex publication_date 2014/06/17 · arxiv created 2015/04/06 · arxiv updated 2015/04/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains Ω⊂ℂ using conformal transformations of the original problem to the weighted eigenvalue problem for the Dirichlet Laplacian in the unit disc \mathbbD. 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.

Cited by

• On growth monotonicity estimates of the principal Dirichlet-Laplacian\n eigenvalue
• Conformal Spectral Stability Estimates for the Neumann Laplacian

Related

• Conformal Spectral Stability Estimates for the Neumann Laplacian
• Conformal composition operators with applications to Dirichlet eigenvalues
• On Conformal Spectral Gap Estimates of the Dirichlet-Laplacian
• Spectral estimates of the Dirichlet-Laplace operator in conformal regular domains
• Spectral stability estimates for the Dirichlet and Neumann Laplacian in rough domains