Flat flows of periodic Lipschitz subgraphs for generalized nonlocal perimeters

Abstract

We prove the existence and the 1/2-H\"older continuity in time of flat flows for periodic Lipschitz subgraphs, whose evolution is governed by the gradient flow of generalized nonlocal perimeters. Moreover, we show that the flat flow satisfies the semigroup property and, as a consequence, the generalized perimeter decreases along the evolution. Finally, we prove that halfspaces are global minimizers of the generalized nonlocal perimeters and act as attractors for the dynamics. Our theory covers several generalized perimeters, including fractional and Riesz-type perimeters (defined on entire periodic subgraphs through suitable renormalization procedures) and the Minkowski pre-content.