Jean-Michel Roquejoffre
(Mathématiques pour l'Industrie et la Physique)
"Free Boundary Problems for the Fractional Laplacian"
We discuss here local properties
- optimal regularity, nondegeneracy, smoothness -
of a free boundary problem involving the fractional Laplacian, generalising the
classical phase transition problem for the standard Laplacian with gradient
jump.
Our equations are relevant models for boundary reactions, but also to
reaction-diffusion processes involving non-Gaussian diffusion.
The nonlocality of the fractional laplacian renders the problem nontrivial,
and the key tool is the Caffarelli-Silvestre extension formula,
which transforms the model
into a codimension 2 free boundary problem.
Joint work with L. Caffarelli and Y. Sire.