Periodic total variation flow of non-divergence type in Rn

2013/02/04 by Mi-Ho Giga, Yoshikazu Giga, Giga, Mi-Ho +3
Computer Science · Mathematics · #35B51 #35D40 #35K55 #35K67 #35K93 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1302.0618

openalex publication_date 2013/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a new notion of viscosity solutions for a class of very singular nonlinear parabolic problems of non-divergence form in a periodic domain of arbitrary dimension, whose diffusion on flat parts with zero slope is so strong that it becomes a nonlocal quantity. The problems include the classical total variation flow and a motion of a surface by a crystalline mean curvature. We establish a comparison principle, the stability under approximation by regularized parabolic problems, and an existence theorem for general continuous initial data.

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