Spectral Estimates of the $p$-Laplace Neumann operator and Brennan's Conjecture

Abstract

In this paper we obtain estimates for the first nontrivial eigenvalue of the $p$-Laplace Neumann operator in bounded simply connected planar domains $\Omega \subset {\mathbb{R}}^2$. This study is based on a quasiconformal version of the universal weighted Poincaré-Sobolev inequalities obtained in our previous papers for conformal weights. The suggested weights in the present paper are Jacobians of quasiconformal mappings. The main technical tool is the theory of composition operators in relation with the Brennan's Conjecture for (quasi)conformal mappings.

18 pages, 1 figure

References in corpus (2)

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

Cited by in corpus (1)

• On the First Eigenvalue of the Degenerate $p$-Laplace Operator in Non-Convex Domains