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

2016/07/02 by Gol'dshtein, Vladimir, Ukhlov, Alexander · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1607.00487

Abstract

In this paper we prove discreteness of the spectrum of the Neu\-mann-Lap\-la\-ci\-an (the free membrane problem) in a large class of non-convex space domains. The lower estimates of the first non-trivial eigenvalue are obtained in terms of geometric characteristics of Sobolev mappings. The suggested approach is based on Poincaré-Sobolev inequalities that are obtained with the help of the composition operators theory for uniform Sobolev spaces. These composition operators are induced by a generalizations of conformal mappings that are mappings of bounded 2-dilatation (2-quasiconformal mappings).

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• Estimates of variational eigenvalues on metric measure spaces

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