Spectral Properties of the Neumann-Laplace operator in Quasiconformal Regular Domains

Abstract

In this paper we study spectral properties of the Neumann-Laplace operator in planar quasiconformal regular domains $\Omega \subset {\mathbb{R}}^2$. This study is based on the quasiconformal theory of composition operators on Sobolev spaces. Using the composition operators theory we obtain estimates of constants in Poincaré-Sobolev inequalities and as a consequence lower estimates of the first non-trivial eigenvalue of the Neumann-Laplace operator in planar quasiconformal regular domains.

17 pages, 1 figure. arXiv admin note: text overlap with arXiv:1701.05143

References in corpus (2)

• Brennan's conjecture for composition operators on Sobolev spaces
• Conformal spectral stability for the Dirichlet-Laplace operator