Flat flows of periodic Lipschitz subgraphs for generalized nonlocal perimeters

TL;DR

This paper establishes the existence, regularity, and properties of flat flows for periodic Lipschitz subgraphs evolving under generalized nonlocal perimeters, including fractional, Riesz-type, and Minkowski perimeters.

Contribution

It introduces a framework for analyzing flat flows of periodic Lipschitz subgraphs driven by generalized nonlocal perimeters, proving existence, regularity, and attractor properties.

Findings

Flat flows exist and are 1/2-Hölder continuous in time.

Flat flows satisfy the semigroup property and decrease the perimeter.

Halfspaces are global minimizers and attractors for the dynamics.

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.