On the nonlocal mean curvature

Speaker: Xavier Cabré

Abstract: We will introduce the basic notions and results on the nonlocal mean curvature, a quantity arising from the localized fractional perimeter functional, as introduced in the seminal paper by Caffarelli, Roquejoffre, and Savin. We will then treat some regularity results and a gradient estimate for nonlocal minimal surfaces, that is, surfaces with zero nonlocal mean curvature. This will bring us to some fractional Michael-Simon type geometric Sobolev inequalities on hypersurfaces which involve their nonlocal mean curvature. If time allows, we will present a nonlocal null-Lagrangian or calibration associated with the fractional perimeter functional.