Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems

In this paper we prove a sharp lower bound for the first non-trivial Neumann eigenvalue $mu_1(Omega)$ for the $p$-Laplace operator ($p > 1$) 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 $Omega$. In a suitable class of convex planar domains, our bound turns out to be better than the one provided by the Payne–Weinberger inequality.