Periodic total variation flow of non-divergence type inRn

Abstract

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

Existence and Uniqueness for a Crystalline Mean Curvature Flow

“…x. If there exists an admissible field z with tangential divergence div τ z in L 2 (∂E), then the crystalline curvature is given by the tangential divergence of z, where div τ z has minimal L 2 -norm among all admissible fields (see [13,29]). In particular, the crystalline curvature has a nonlocal character.…”

“…3, apart from the new result just mentioned, the only general available notion of a global-in-time solution we are aware of is the minimizing movement motion provided by the ATW scheme; however, no general comparison results have been established so far. In fact, higher-dimensional uniqueness results deal with special classes of initial data (for instance, convex initial data as in [11,15] or polyhedral sets as in [35]) or with very specific anisotropies (see [32], where a comparison principle valid in all dimensions has been established for the anisotropy ./ h j H j g j N j, with N h ¡ e N and j H j the euclidean norm of the orthogonal projection of onto e c N ). In this paper we prove a global-in-time existence and uniqueness (up to possible fattening) result for the crystalline mean curvature flow valid in all dimensions, for arbitrary (possibly unbounded) initial sets and for general crystalline anisotropies , but under the particular choice m h in (1.1).…”

Approximation of General Facets by Regular Facets with Respect to Anisotropic Total Variation Energies and Its Application to Crystalline Mean Curvature Flow

“…We take a different approach using the ideas of the theory of viscosity solutions [24,26,28,29]. The level set formulation of the crystalline mean curvature flow was introduced by the authors in [31].…”

“…Remark 4.9. Definition 4.8 at y p D 0 is a natural extension of the definition that appeared in an earlier paper [28] for anisotropies W smooth outside of the origin. In that case, if appropriate tests are given at y p ¤ 0 where W is smooth, the definition is equivalent to the definition of F-solutions [30].…”

“…The regularization (a) yields a sequence of degenerate parabolic problems that are within the classical viscosity theory [21], while (b) regularizes the crystalline curvature operator by smooth anisotropic curvatures studied in [20] when F comes from a level set formulation, or in [28,29] for general F . Note that both regularizations produce local problems except at ru m D 0 in (b).…”