Generalized motion by nonlocal curvature in the plane. (English) Zbl 1004.35075

The authors consider the level-set solutions for a general planar anisotropic curvature flow when the anisotropic curvature effect is nonlocal due to a singular interfacial energy. They introduce a new notion of solutions leading to a comparison principle and stability results. A new technique called slicing, and which is not limited to nonlocal curvature flow equations, converts the level-sets of solutions into graph-like functions. The approach establishes the convergence of the crystalline anisotropic flow to the anisotropic curvature flow with smooth interfacial energy.


35K65 Degenerate parabolic equations
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35B65 Smoothness and regularity of solutions to PDEs
35A25 Other special methods applied to PDEs
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