SMS scnews item created by Daniel Daners at Fri 23 Mar 2018 1614
Type: Seminar
Distribution: World
Expiry: 27 Mar 2018
Calendar1: 27 Mar 2018
CalLoc1: AGR Carslaw 829
CalTitle1: PDE Seminar: Sharp bounds for Neumann eigenvalues (Brandolini)
Auth: daners@dora.maths.usyd.edu.au

PDE Seminar

Sharp bounds for Neumann eigenvalues

Brandolini

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.

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