On the principal frequency of non-homogeneous membranes

Abstract

We obtained estimates for first eigenvalues of the Dirichlet boundary value problem for elliptic operators in divergence form (i.e. for the principal frequency of non-homogeneous membranes) in bounded domains $\Omega \subset \mathbb{C}$ satisfying quasihyperbolic boundary conditions. The suggested method is based on the quasiconformal composition operators on Sobolev spaces and their applications to constant estimates in the corresponding Sobolev-Poincaré inequalities. We also prove a variant of the Rayleigh-Faber-Khran inequality for a special case of these elliptic operators.

arXiv comment15 pages. arXiv admin note: text overlap with arXiv:2009.06936, arXiv:1903.11301