Calculus of Variations and Geometric Measure Theory

L. Brasco - G. De Philippis - B. Velichkov

Faber-Krahn inequalities in sharp quantitative form

created by brasco on 03 Jun 2013
modified by dephilipp on 30 Oct 2017


Published Paper

Inserted: 3 jun 2013
Last Updated: 30 oct 2017

Journal: Duke Math. J.
Volume: 164
Number: 9
Pages: 1777-1832
Year: 2015

ArXiv: 1306.0392 PDF


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