Spectral estimates of the $p$-Laplace Neumann operator in conformal regular domains

TL;DR

This paper establishes spectral estimates for the $p$-Laplace Neumann operator in conformal regular domains using weighted Poincaré-Sobolev inequalities, contingent on Brennan's conjecture, linking geometry and spectral properties.

Contribution

It introduces new spectral bounds for the $p$-Laplace Neumann operator in conformal domains based on weighted inequalities and composition operator theory, assuming Brennan's conjecture.

Findings

Weighted Poincaré-Sobolev inequalities hold under Brennan's conjecture.

Spectral estimates for the first nontrivial eigenvalue are derived.

Results connect conformal geometry with spectral properties of the $p$-Laplace operator.

Abstract

In this paper we study spectral estimates of the $p$-Laplace Neumann operator in conformal regular domains $\Omega\subset\mathbb R^2$. This study is based on (weighted) Poincar\'e-Sobolev inequalities. The main technical tool is the composition operators theory in relation with the Brennan's conjecture. We prove that if the Brennan's conjecture holds then for any $p\in (4/3,2)$ and $r\in (1,p/(2-p))$ the weighted $(r,p)$-Poincare-Sobolev inequality holds with the constant depending on the conformal geometry of $\Omega$. As a consequence we obtain classical Poincare-Sobolev inequalities and spectral estimates for the first nontrivial eigenvalue of the $p$-Laplace Neumann operator for conformal regular domains.