Convergence of supercritical fractional flows to the mean curvature flow

TL;DR

This paper investigates how nonlocal fractional perimeters and their associated geometric flows converge to classical perimeter and mean curvature flow as the core radius diminishes, extending known results to supercritical exponents and anisotropic kernels.

Contribution

It extends the classical fractional perimeter concept to supercritical exponents, demonstrating convergence to Euclidean perimeter and mean curvature flow, including anisotropic cases with applications to dislocation dynamics.

Findings

Gamma-convergence of scaled nonlocal perimeters to Euclidean perimeter

Convergence of nonlocal fractional curvatures to mean curvature

Level set solutions to nonlocal flows converge to classical mean curvature flow

Abstract

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