Faber-Krahn inequalities in sharp quantitative form

Lorenzo Brasco (1) , Guido de Philippis (2) , Bozhidar Velichkov

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 HCM - Hausdorff Center for Mathematics

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 Poincaré-Sobolev constant for the embeddings $W^{1,2}_0(\Omega)\hookrightarrow L^q(\Omega)$.

Mots clés: Stability for eigenvalues; regularity for free boundaries; torsional rigidity

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]; Théorie spectrale [math.SP]

Lorenzo Brasco, Guido de Philippis, Bozhidar Velichkov. Faber-Krahn inequalities in sharp quantitative form. Duke Mathematical Journal, 2015, 164 (9), pp.1777-1831. ⟨10.1215/00127094-3120167⟩. ⟨hal-00831975⟩