Fractional Perimeter and Nonlocal Minimal Surfaces

This work aims to present a study of the principal results about the fractional perimeter and the regularity theory of nonlocal minimal surfaces. The fractional perimeter is a variation of the classical notion of Caccioppoli perimeter, recently introduced by L. Caffarelli, J.-M. Roquejoffre and O. Savin, that takes into account also long-range point-wise interactions between sets, modulated by a kernel with polynomial decay. That type of nonlocal minimal surfaces arises naturally, for instance, in the study of fractals and phase transitions models. First of all, we de�fine the fractional perimeter, and applying classical direct methods of Calculus of Variations we prove existence and compactness results for the corresponding minimizers. Then we deal with the regularity properties of such objects. Finally, we study a calibration result concerning nonlocal perimeters within Carnot groups.