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We extend the theory of viscosity solutions to a class of very singular nonlinear parabolic problems of non-divergence form in a periodic domain of an arbitrary dimension with diffusion given by an anisotropic total variation energy. We give a proof of a comparison principle, an outline of a proof of the stability under approximation by regularized parabolic problems, and an existence theorem for general continuous initial data, which extend the results recently obtained by the authors.Furthermore, we assume that F : R n × R → R is a continuous function, nonincreasing in the second variable, i.e.,This makes the operator in (1.1) degenerate parabolic.The symbol ∂W denotes the subdifferential of W . In general, the subdifferential of a convex lower semi-continuous function ϕ on a Hilbert space H endowed with a

We introduce the notions of viscosity super-and subsolutions suitable for singular diffusion equations of non-divergence type with a general spatially inhomogeneous driving term. In particular, the viscosity super-and subsolutions support facets and allow a possible facet bending. We prove a comparison principle by a modified doubling variables technique. Finally, we present examples of viscosity solutions. Our results apply to a general crystalline curvature flow with a spatially inhomogeneous driving term for a graph-like curve.

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