Conference on Calculus of Variations in Lille - 3rd edition - July 4-6 2022

Rigidity results for measurable sets

Jul 4, 2022, 9:30 AM

1h

M2 building, Cité Scientifique - Meeting room, 1st floor (Laboratoire Paul Painlevé)

Speaker

Dorin Bucur (Université Savoie Mont Blanc)

Description

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a set with finite Lebesgue measure such that, for a fixed radius $r>0$, the Lebesgue measure of $\mathrm{\Omega }\cap B_r(x)$ is equal to a positive constant when $x$ varies in the essential boundary of $\mathrm{\Omega }$. We prove that $\mathrm{\Omega }$ is a ball (or a finite union of equal balls) provided it satisfies a nondegeneracy condition, which holds in particular for any set of diameter larger than $r$ which is either open and connected, or of finite perimeter and indecomposable. This is a joint work with Ilaria Fragalà.