Generalized Motion¶by Nonlocal Curvature in the Plane

Abstract

This paper considers the level-set equation for a general planar anisotropic curvature flow equation when the interfacial energy is very singular so that the anisotropic curvature effect is nonlocal. A new notion of solutions is introduced to establish an analytic foundation of the level-set method including a comparison principle and stability results. The main idea behind the proofs is to convert the level-sets of solutions into graph-like functions. This new procedure is called slicing and it is not limited to nonlocal curvature flow equations. Our theory is useful for establishing the convergence of a crystalline algorithm as well as for justifying the crystalline flow as a limit of anisotropic curvature flow with smooth interfacial energy.