Università degli Studi di Napoli “Federico II”, Italy
Tue 27th Mar 2018, 2-3pm, Carslaw Room 829 (AGR)
We prove a sharp lower bound for the first nontrivial Neumann eigenvalue of the -Laplace operator () in a Lipschitz, bounded domain in . Differently from the pioneering estimate by Payne-Weinberger, our lower bound does not require any convexity assumption on , it involves the best isoperimetric constant relative to and it is sharp, at least when , as the isoperimetric constant relative to 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 and consists of the points on one side of a smooth curve , within a suitable distance from it, then can be sharply estimated from below in terms of the length of , the norm of its curvature and .
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