Sharp bounds for Neumann eigenvalues

Abstract

We prove a sharp lower bound for the first nontrivial Neumann eigenvalue ${\mu }_1\left(\Omega \right)$ of the $p$-Laplace operator ($p>1$) in a Lipschitz, bounded domain $\Omega$ in ${ℝ}^n$. Differently from the pioneering estimate by Payne-Weinberger, our lower bound does not require any convexity assumption on $\Omega$, it involves the best isoperimetric constant relative to $\Omega$ and it is sharp, at least when $p=n=2$, as the isoperimetric constant relative to $\Omega$ goes to 0. Moreover, in a suitable class of convex planar domains, our estimate turns out to be better than the one provided by the Payne-Weinberger inequality.

Furthermore, we prove that, when $p=n=2$ and $\Omega$ consists of the points on one side of a smooth curve $\gamma$, within a suitable distance $\delta$ from it, then ${\mu }_1\left(\Omega \right)$ can be sharply estimated from below in terms of the length of $\gamma$, the $L^{\infty }$ norm of its curvature and $\delta$.