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Bozhidar Velichkov
Lundi 5 février 2018
Bozhidar Velichkov (LJK Grenoble)
Approche variationnelle à la régularité des frontières libres singulières.
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 1981 Alt and Caffarelli proved that if u is a minimizer, then the
free boundary ∂u > 0 can be decomposed into a regular part, Reg (∂u > 0), and a singular part, Sing (∂u > 0), where
• Reg (∂u > 0) is locally the graph of a smooth function ;
• Sing (∂u > 0) 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(∂u >0) 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 and x is a point of the free boundary ∂u > 0 and there exists one blow-up limit v which has an isolated singularity in zero, then the free boundary ∂u > 0 is a C^1 graph over the cone
∂v > 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.