Calculus of Variations and Geometric Measure Theory

A. Chambolle - M. Morini - M. Ponsiglione

Nonlocal curvature flows

created by ponsiglio on 03 Sep 2014
modified by morini on 23 Jul 2015


Published Paper

Inserted: 3 sep 2014
Last Updated: 23 jul 2015

Journal: Arch. Ration. Mech. Anal.
Year: 2015
Doi: 10.1007/s00205-015-0880-z


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.

Keywords: Geometric evolution equations, Minimizing movements, Viscosity solutions