Spectral estimates of the p-Laplace Neumann operator in conformal regular domains

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Abstract

In this paper we study spectral estimates of the p-Laplace Neumann operator in conformal regular domains Ω⊂R2. This study is based on (weighted) Poincaré–Sobolev inequalities. The main technical tool is the theory of composition operators in relation with the Brennan's conjecture. We prove that if the Brennan's conjecture holds for any p∈(4/3,2) and r∈(1,p/(2−p)) then the weighted (r,p)-Poincare–Sobolev inequality holds with the constant depending on the conformal geometry of Ω. As a consequence we obtain classical Poincare–Sobolev inequalities and spectral estimates for the first nontrivial eigenvalue of the p-Laplace Neumann operator for conformal regular domains.

Original languageEnglish
Pages (from-to)137-148
Number of pages12
JournalTransactions of A. Razmadze Mathematical Institute
Volume170
Issue number1
DOIs
StatePublished - 1 May 2016

Keywords

  • Conformal mappings
  • Elliptic equations
  • Sobolev spaces

ASJC Scopus subject areas

  • General Mathematics

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