Variational problems with free boundaries for the fractional Laplacian
Luis A. Caffarelli
University of Texas at Austin, USAJean-Michel Roquejoffre
Université Paul Sabatier, Toulouse, FranceYannick Sire
Université Aix-Marseille, Marseille, France
Abstract
We discuss properties (optimal regularity, non-degeneracy, smoothness of the free boundary...) of a variational interface problem involving the fractional Laplacian; Due to the non-locality of the Dirichlet problem, the task is nontrivial. This difficulty is by-passed by an extension formula, discovered by the first author and Silvestre, which reduces the study to that of a co-dimension 2 (degenerate) free boundary.
Cite this article
Luis A. Caffarelli, Jean-Michel Roquejoffre, Yannick Sire, Variational problems with free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12 (2010), no. 5, pp. 1151–1179
DOI 10.4171/JEMS/226