Conformal spectral stability estimates for the Dirichlet Laplacian

V. I. Burenkov, V. Gol'dshtein, A. Ukhlov

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains Ω⊂C by reducing it, using conformal transformations, to the weighted eigenvalue problem for the Dirichlet Laplacian in the unit disc 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.

Original languageEnglish
Pages (from-to)1822-1833
Number of pages12
JournalMathematische Nachrichten
Volume288
Issue number16
DOIs
StatePublished - 1 Nov 2015

Keywords

  • 35J40
  • 35P15
  • 47A75
  • 47B25
  • Conformal mappings
  • Eigenvalue problem
  • Elliptic equations
  • Quasidiscs

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