Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems

TL;DR

This paper establishes a sharp lower bound for the first nontrivial Neumann eigenvalue of the p-Laplace operator in Lipschitz domains, using isoperimetric constants without convexity assumptions.

Contribution

It provides a new optimal lower bound for Neumann eigenvalues of the p-Laplace operator applicable to general Lipschitz domains, removing convexity restrictions.

Findings

Derived a sharp lower bound for $_1()$ in Lipschitz domains.

The estimate involves the best isoperimetric constant relative to the domain.

The result applies to both linear and nonlinear Neumann problems.

Abstract

In this paper we prove a sharp lower bound for the first nontrivial Neumann eigenvalue $\mu_1(\Omega)$ for the $p$-Laplace operator in a Lipschitz, bounded domain $\Omega$ in $\R^n$. Our estimate does not require any convexity assumption on $\Omega$ and it involves the best isoperimetric constant relative to $\Omega$.