OS Partielle Differentialgleichungen: Sharp bounds for eigenfunctions and eigenvalues of non-linear Neumann problems
Time
Thursday, 7. February 2019
15:15 - 16:45
Location
F 426
Organizer
Speaker:
This event is part of an event series „Oberseminar Partielle Differentialgleichungen“.
Referentin: Prof. Dr. Barbara Brandolini (Università degli Studi di Napoli Federico II)
Zusammenfassung: We prove a sharp lower bound for the fi rst nontrivial Neumann eigenvalue μ1(Ω) of the p-Laplace operator (p > 1) in a Lipschitz, bounded domain Ω in Rn. Di fferently 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.