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.Comment: 36 pages, 2 figure
