Optimal lower bounds for eigenvalues of 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 ${\mathbb{R}}^n$. Our estimate does not require any convexity assumption on $\Omega$ and it involves the best isoperimetric constant relative to $\Omega$.