PDE Seminar Abstracts

Sharp bounds for Neumann eigenvalues

Barbara Brandolini
Università degli Studi di Napoli “Federico II”, Italy
Tue 27th Mar 2018, 2-3pm, Carslaw Room 829 (AGR)

Abstract

We prove a sharp lower bound for the first nontrivial Neumann eigenvalue μ1(Ω) of the p-Laplace operator (p > 1) in a Lipschitz, bounded domain Ω in n. Differently from the pioneering estimate by Payne-Weinberger, our lower bound does not require any convexity assumption on Ω, it involves the best isoperimetric constant relative to Ω and it is sharp, at least when p = n = 2, as the isoperimetric constant relative to Ω 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 Ω consists of the points on one side of a smooth curve γ, within a suitable distance δ from it, then μ1(Ω) can be sharply estimated from below in terms of the length of γ, the L norm of its curvature and δ.