The core-radius approach to supercritical fractional perimeters, curvatures and geometric flows

We consider a core-radius approach to nonlocal perimeters governed by isotropic kernels having critical and supercritical exponents, extending the nowadays classical notion of s-fractional perimeter, defined for 0<1, to the case s>=1. We show that, as the core-radius vanishes, such core-radius regularized s-fractional perimeters, suitably scaled, ?-converge to the standard Euclidean perimeter. Under the same scaling, the first variation of such nonlocal perimeters gives back regularized s-fractional curvatures which, as the core radius vanishes, converge to the standard mean curvature; as a consequence, we show that the level set solutions to the corresponding nonlocal geometric flows, suitably reparametrized in time, converge to the standard mean curvature flow. Furthermore, we show the same asymptotic behavior as the core-radius vanishes and s->s?>=1 simultaneously. Finally, we prove analogous results in the case of anisotropic kernels with applications to dislocation dynamics.