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

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

Research output: Contribution to journalArticle

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

Fingerprint

Conformal mapping
Spectral Properties
Fractals
Mathematical operators
Derivatives
Derivative
Geometry
Estimate
Conformal Geometry
Rayleigh quotient
Non-Lipschitz
Quasiconformal
Conformal Mapping
Operator
Unit Disk
Bounded Domain
Fractal
Eigenvalue

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Integral estimates of conformal derivatives and spectral properties of the Neumann-Laplacian. / Gol'dshtein, V.; Pchelintsev, V.; Ukhlov, A.

In: Journal of Mathematical Analysis and Applications, Vol. 463, No. 1, 01.07.2018, p. 19-39.

Research output: Contribution to journalArticle

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