Flat flows of periodic Lipschitz subgraphs for generalized nonlocal perimeters
Lucia De Luca, Antonia Diana, Marcello Ponsiglione
TL;DR
The paper analyzes flat flows for periodic Lipschitz subgraphs evolving under the gradient flow of generalized nonlocal perimeters. It constructs a minimizing movements scheme in the periodic setting, proves discrete Hölder-in-time estimates, and passes to a Hölder-1/2 flat flow with a semigroup property, ensuring the generalized perimeter decreases along the evolution. It then demonstrates that periodic halfspaces minimize all considered generalized perimeters and can act as attractors for the dynamics, with convergence to a halfspace established under suitable conditions. A broad class of perimeters is covered, including fractional, Riesz-type, and Minkowski pre-content functionals, illustrating the framework’s versatility in both local and nonlocal contexts. The results have implications for understanding asymptotic shapes in nonlocal geometric evolutions on periodic structures.
Abstract
We prove the existence and the 1/2-Hölder 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.