Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems

Abstract

In this paper we prove a sharp lower bound for the first non-trivial Neumann eigenvalue μ₁(Ω) for the p-Laplace operator (p > 1) in a Lipschitz bounded domain Ω in ℝⁿ. Our estimate does not require any convexity assumption on Ω and it involves the best isoperimetric constant relative to Ω. In a suitable class of convex planar domains, our bound turns out to be better than the one provided by the Payne—Weinberger inequality.