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 \mathbb{R}^d,

 (*)\qquad  \min\Big\{\int_{B_1}\big(|\nabla u|^2 + \mathds{1}_{\{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} \frac{u(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
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