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We extend the theory of viscosity solutions to a class of very singularnonlinear parabolic problems of non-divergence form in a periodic domain of anarbitrary dimension with diffusion given by an anisotropic total variationenergy. We give a proof of a comparison principle, an outline of a proof of thestability under approximation by regularized parabolic problems, and anexistence theorem for general continuous initial data, which extend the resultsrecently obtained by the authors.

Anisotropic total variation flow of non-divergence type on a higher dimensional torus