## Variational approach to the regularity of the singular free boundaries

### Mardi 23 janvier 2018 14:00-15:00 - Bozhidar Velichkov - Université Grenoble Alpes

Résumé : In this talk we will present some recent results on the structure of the free boundaries of the (local) minimizers of the Bernoulli problem in ,

In 1981 Alt and Caffarelli proved that if $u$ is a minimizer of the above problem, then the free boundary can be decomposed into a regular part, , and a singular part, , where

• is locally the graph of a smooth function ;
• is a small (possibly empty) set.

Recently, De Silva and Jerison proved that starting from dimension there are minimal cones with isolated singularities in zero. In particular, the set of singular points might not be empty.
The aim of this talk is to describe the structure of the free boundary around a singular point. In particular, we will show that if is a solution of (*), is a point of the free boundary and there exists one blow-up limit , which has an isolated singularity in zero, then the free boundary is a graph over the cone .
Our approach is based on the so called logarithmic epiperimetric inequality, which is a purely variational tool for the study of free boundaries and was introduced in the framework of the obstacle problem in a series of works in collaboration with Maria Colombo and Luca Spolaor.

Lieu : IMO ; salle 3L8.

Variational approach to the regularity of the singular free boundaries  Version PDF
août 2019 :

Rien pour ce mois

 Département de Mathématiques Bâtiment 307 Faculté des Sciences d'Orsay Université Paris-Sud F-91405 Orsay Cedex Tél. : +33 (0) 1-69-15-79-56 Département Laboratoire Formation