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 $\mathbbR^d$,

$$ (*)\qquad \min\Big{\int_B_1\big(|\nabla u|^2 + \mathds1_{u>0} \big)\, :\,u\in H^1(B_1)\,+ ; Dirichlet : boundary : conditions : on : \partial B_1\Big}.$$

In 1981 Alt and Caffarelli proved that if $u$ is a minimizer of the above problem, then the free boundary $\partial{u>0}\cap B_1$ can be decomposed into a regular part, $Reg\big(\partial{u>0}\big)$, and a singular part, $Sing\big(\partial{u>0}\big)$, where

  • $Reg\big({u>0}\big)$ is locally the graph of a smooth function ;
  • $Sing\big({u>0}\big)$ is a small (possibly empty) set.

Recently, De Silva and Jerison proved that starting from dimension $d= 7$ there are minimal cones with isolated singularities in zero. In particular, the set of singular points $Sing\big({u>0}\big)$ 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 $u$ is a solution of (*), $x_0$ is a point of the free boundary $\partial{u>0}$ and there exists one blow-up limit $u_0=\lim_n\to \infty \fracu(x_0+r_nx)r_n$, which has an isolated singularity in zero, then the free boundary $\partial{u>0}$ is a $C^1$ graph over the cone $\partial{u_0>0}$.
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