The spectral estimates for the Neumann–Laplace operator in space domains

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Abstract

In this paper we prove discreteness of the spectrum of the Neumann–Laplacian (the free membrane problem) in a large class of non-convex space domains. The lower estimates of the first non-trivial Neumann eigenvalue are obtained in terms of geometric characteristics of Sobolev mappings. The suggested approach is based on Sobolev–Poincaré inequalities that are obtained with the help of a geometric theory of composition operators on Sobolev spaces. These composition operators are induced by generalizations of conformal mappings that are called as mappings of bounded 2-distortion (weak 2-quasiconformal mappings).

Original languageEnglish
Pages (from-to)166-193
Number of pages28
JournalAdvances in Mathematics
Volume315
DOIs
StatePublished - 31 Jul 2017

Keywords

  • Elliptic equations
  • Quasiconformal mappings
  • Sobolev spaces

ASJC Scopus subject areas

  • Mathematics (all)

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