Published Paper
Inserted: 3 jun 2013
Last Updated: 30 oct 2017
Journal: Duke Math. J.
Volume: 164
Number: 9
Pages: 1777-1832
Year: 2015
Abstract:
The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. In this paper we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman. More generally, the result applies to every optimal Poincar\'e-Sobolev constant for the embeddings $W^{1,2}_0(\Omega)\hookrightarrow L^q(\Omega)$.
Keywords: Stability for eigenvalues, Torsional rigidity, regularity for free boundaries
Download: