Approximation of mean curvature flow by diffusion/redistancing schemes

We study time-discrete approximation schemes for anisotropic mean curvature flow of the diffusion/redistancing type: one step consists in applying an operator to the anisotropic signed distance function of the current set, and in taking as the new set the region where the result is nonpositive. The notion of solution is the distributional one of Chambolle, Morini and Ponsiglione, which is well-posed for an arbitrary convex anisotropy, and in particular in the crystalline case. The main result is an abstract scheme, together with the identification of the properties of the operator that make it converge. Besides a list of structural axioms, the operator is required to satisfy three properties: a comparison with the Wulff shape, which produces the estimates on the discrete evolutions and hence the compactness, and a stability and a consistency property, which are the two halves of the single statement that the incremental quotient of one step is the divergence of a vector field lying in the subdifferential of the anisotropy at the gradient of the limit distance. Every family satisfying these properties generates evolutions which converge, up to a subsequence, to a superflow with the prescribed initial condition, and generically to a flow. We then verify the properties for three families of schemes. Two of them are built on the anisotropic p-Laplacian, in an elliptic and in a parabolic form, and are defined for an arbitrary norm; the third is built on the convolution with a kernel, and is completely explicit but restricted to the anisotropies induced by a scalar product. A first chapter, of exploratory character, derives a variational formulation of the flow from the principle of virtual powers, in a framework designed for mechanical systems whose space of configurations carries no linear structure.