Nonlocal curvature flows

TL;DR

This paper develops a unified framework for local and nonlocal geometric flows, introducing generalized curvatures and perimeters, and providing existence, uniqueness, and approximation results for these flows.

Contribution

It introduces a comprehensive framework for geometric flows, unifying local and nonlocal cases, and offers new theoretical results and applications.

Findings

Established existence and uniqueness for generalized curvature flows.

Developed a minimizing movements scheme approximating viscosity solutions.

Unified treatment of various geometric motions in the literature.

Abstract

This paper aims at building a unified framework to deal with a wide class of local and nonlocal translation-invariant geometric flows. First, we introduce a class of generalized curvatures, and prove the existence and uniqueness for the level set formulation of the corresponding geometric flows. We then introduce a class of generalized perimeters, whose first variation is an admissible generalized curvature. Within this class, we implement a minimizing movements scheme and we prove that it approximates the viscosity solution of the corresponding level set PDE. We also describe several examples and applications. Besides recovering and presenting in a unified way existence, uniqueness, and approximation results for several geometric motions already studied and scattered in the literature, the theory developed in this paper allows us to establish also new results.