In this talk I present the problems that I have studied in my Ph.D. project.
We have considered a core-radius approach to non-local perimeters governed by isotropic kernels having critical and supercritical exponents, extending the nowadays classical notion of s-fractional perimeter, defined for 0<s<1, to the case s greater or equal to 1.
We show that, as the core-radius vanishes, such core-radius regularized s-fractional perimeters, suitably scaled, Gamma-converge to the standard Euclidean perimeter. We also show that the geometric flow of the core radius regularized s-fractional perimeters, suitably scaled, converge to the classical mean curvature flow.
We consider functionals of Riesz type and geometrical type, as for instance non-local perimeters, showing that it is possible to obtain them as Gamma limits of interaction functionals among rigid spheres