Integral estimates of conformal derivatives and spectral properties of the Neumann-Laplacian

V. Gol'dshtein, V. Pchelintsev, A. Ukhlov

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Abstract

In this paper we study integral estimates of derivatives of conformal mappings φ:D→Ω of the unit disc D⊂C onto bounded domains Ω that satisfy the Ahlfors condition. These integral estimates lead to estimates of constants in Sobolev–Poincaré inequalities, and by the Rayleigh quotient we obtain spectral estimates of the Neumann–Laplace operator in non-Lipschitz domains (quasidiscs) in terms of the (quasi)conformal geometry of the domains. Specifically, the lower estimates of the first non-trivial eigenvalues of the Neumann–Laplace operator in some fractal type domains (snowflakes) were obtained.

Original languageEnglish
Pages (from-to)19-39
Number of pages21
JournalJournal of Mathematical Analysis and Applications
Volume463
Issue number1
DOIs
Publication statusPublished - 1 Jul 2018

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Keywords

  • Conformal mappings
  • Elliptic equations
  • Quasiconformal mappings
  • Sobolev spaces

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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