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

By V. Gol’dshtein, A. Ukhlov. 2016

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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 Poincare-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).

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/409b16f8f4e4c166a83f4984055a17e464beaa94