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

TL;DR

This paper provides estimates for the first nontrivial eigenvalue of the $p$-Laplace Neumann operator in planar domains using quasiconformal mappings and weighted inequalities, linking spectral theory with Brennan's Conjecture.

Contribution

It introduces a novel approach to estimate eigenvalues via quasiconformal weights and composition operators, connecting spectral analysis with geometric function theory.

Findings

Derived bounds for the $p$-Laplace Neumann eigenvalues in planar domains.

Established a link between eigenvalue estimates and Brennan's Conjecture.

Utilized quasiconformal Jacobians as weights in Poincaré-Sobolev inequalities.

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 Poincar\'e-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.