On the effective sharp stability of the Faber-Krahn inequality

par João Miguel Machado

jeudi 24 sept. 2026, 11:15 → 12:15 Europe/Paris

Amphi Schwartz

The Faber–Krahn inequality states that, among sets of fixed volume, balls minimize the first Dirichlet eigenvalue. Its sharp quantitative form controls the square of the Fraenkel asymmetry by the spectral deficit. Brasco, De Philippis and Velichkov proved this estimate through an argument using compactness, a contradiction step, and the selection principle of Cicalese-Leonardi; consequently, their stability constant is not computable. In this talk, we will discuss a proof with a computable dimensional constant, resolving Open Problem 2 in the survey of Brasco and De Philippis. The argument passes through the Saint–Venant inequality, which states that the torsion functional is maximised by balls. Its two main ingredients are a new level-set characterisation of the torsional deficit and a differential estimate for the distance of the level sets from a moving ball. These ingredients are connected by the strong quantitative isoperimetric inequality of Fusco and Julin. The Kohler–Jobin inequality then transfers the effective sharp stability of the Saint-Venant inequality back to Faber–Krahn.

 

This is joint work with André Guerra (Cambridge) and João P. G. Ramos (IMPA) based on the preprint: https://arxiv.org/html/2609.07772v1