On the First Eigenvalue of the Degenerate \(\varvec{p}\)-Laplace Operator in Non-convex Domains

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Abstract

In this paper we obtain lower estimates of the first non-trivial eigenvalues of the degenerate p-Laplace operator, \(p>2\), in a large class of non-convex domains. This study is based on applications of the geometric theory of composition operators on Sobolev spaces that permits us to estimate constants of the Poincaré–Sobolev inequalities. On this base we obtain lower estimates of the first non-trivial eigenvalues for Ahlfors-type domains (i.e. quasidiscs). This class of domains includes some snowflake-type domains with fractal boundaries.

Keywords

Elliptic equations Sobolev spaces Composition operators 

Mathematics Subject Classification

35P15 46E35 30C65 

Notes

Acknowledgements

The first author was supported by the United States-Israel Binational Science Foundation (BSF Grant No. 2014055). The second author was supported by RFBR Grant No. 18-31-00011.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBen-Gurion University of the NegevBeer ShevaIsrael
  2. 2.Division for Mathematics and Computer SciencesTomsk Polytechnic UniversityTomskRussia
  3. 3.Department of General MathematicsTomsk State UniversityTomskRussia

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