Spectral optimization for the Stekloff--Laplacian: the stability issue

Lorenzo Brasco (1) , Guido de Philippis (2) , Berardo Ruffini (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 SNS - Scuola Normale Superiore di Pisa

We consider the problem of maximizing the first non trivial Stekloff eigenvalue of the Laplacian, among sets with given measure. We prove that the Brock--Weinstock inequality, asserting that optimal shapes for this spectral optimization problem are balls, can be improved by means of a (sharp) quantitative stability estimate. This result is based on the analysis of a certain class of weighted isoperimetric inequalities already proved in Betta et al. (J. of Inequal. \& Appl. 4: 215--240, 1999): we provide some new (sharp) quantitative versions of these, achieved by means of a suitable calibration technique.

Mots clés: Stability for eigenvalues; Weighted isoperimetric inequality

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]; Théorie spectrale [math.SP]

Lorenzo Brasco, Guido de Philippis, Berardo Ruffini. Spectral optimization for the Stekloff--Laplacian: the stability issue. Journal of Functional Analysis, 2012, 262, pp.4675-4710. ⟨10.1016/j.jfa.2012.03.017⟩. ⟨hal-00643643⟩