Lucia De Luca

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 $0<s<1$, to the case $s\ge 1$\,. We show that, as the core-radius vanishes, such core-radius regularized $s$-fractional perimeters, suitably scaled, $\Gamma$-converge to the standard Euclidean perimeter. 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. Finally, we prove analogous results in the case of anisotropic kernels with applications to dislocation dynamics. Keywords: Fractional perimeters; $\Gamma$-convergence; Local and nonlocal geometric evolutions; Viscosity solutions; Level set formulation; Fractional mean curvature flow; Dislocation dynamics

We introduce a notion of uniform convergence for local and nonlocal curvatures. Then, we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general theory to characterize the limits of $s$-fractional mean curvature flows as $s\to 0^{+}$ and $s\to 1^{-}$. In analogy with the $s$-fractional mean curvature flows, we introduce the notion of $s$-Riesz curvature flows and characterize its limit as $s\to 0^{-}$. Eventually, we discuss the limit behavior as $r\to 0^{+}$ of the flow generated by a regularization of the $r$-Minkowski content.