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We study the asymptotic analysis of a singularly perturbed weakly parabolic system of m equations of anisotropic reaction-diffusion type. Our main result formally shows that solutions to the system approximate a geometric motion of a hypersurface by anisotropic mean curvature. The anisotropy, supposed to be uniformly convex, is explicit and turns out to be the dual of the star-shaped combination of the m original anisotropies.

We prove that the semidiscrete schemes of a Perona-Malik type equation converge, in a long time scale, to a suitable system of ordinary differential equations defined on piecewise constant functions. The proof is based on a formal asymptotic expansion argument, and on a careful construction of discrete sub and supersolutions. Despite the equation has a region where it is backward parabolic, we prove a discrete comparison principle, which is the key tool for the convergence result.

We give a complete classification of the stability properties of the equilibria for the semi-discrete one-dimensional Perona-Malik equation, with Dirichlet boundary conditions. We also give the $\Gamma$-expansion of the corresponding discretized functionals up to the order two, as the discretization parameter goes to zero.

Un software di geometria interattiva non puo' sostituire una dimostrazione rigorosa, come viene esemplificato tramite una particolare costruzione geometrica che mette in evidenza gli errori di arrotondamento del calcolatore.

We provide the complete argument for the proof of step 2 in the demonstration of Lemma 4.2 in the paper "Some results on minimal barriers in the sense of De Giorgi applied to driven motion by mean curvature", 1995 of the same authors.

We show two examples of facet-breaking for three-dimensional polyhedral surfaces evolving by crystalline mean curvature. The analysis shows that creation of new facets during the evolution is a common phenomenon. The first example is completely rigorous, and the evolution after the subdivision of one facet is explicitly computed for short times. Moreover, the constructed evolution is unique among the crystalline flows with the given initial datum. The second example suggests that curved portions of the boundary may appear even starting from a polyhedral set close to the Wulff shape.

In this paper we consider the so called prescribed curvature problem approximated by a singularly perturbed double obstacle variational inequality. We extend [Pao97] with the introduction of the same nonregular potential used for the evolution problem [NPV94c] and we prove an optimal O(e2) error estimate for nondegenerate minimizers (where e stands for the perturbation parameter). Following [Pao97] the result relies on the construction of precise barriers suggested by formal asymptotics combined with the use of the maximum principle. Key ingredients are the construction of a sub(super)solution containing appropriate shape corrections and the use of a modified distance function based on the principal eigenfunction of the second variation of the prescribed curvature functional. This analysis is next extended to a piecewise linear finite element discretization of the elliptic PDE of bistable type to prove the same error extimate for discrete minima using the Rannacher-Scott L\infty-estimates and under appropriated restrictions on the mesh size (h2=O(es) with s > 5/2).

In this paper we present a few numerical simulations of a nonsymmetric anisotropic evolution by mean curvature which leads to the so-called fattening of the interface. The numerical simulations are based on a diffused interface approximation via a bistable reaction-diffusion equation which is then discretized by means of finite elements in space and forward differences in time. An adaptive strategy, together with the use of a dynamic mesh, is used to take advantage of the local behaviour of the equation at hand. However, this particular choice of anisotropy, with a large ratio between the maximal and minimal surface energy, seems to be highly critical for the type of approximation presented here, leading to very expensive computations in terms of CPU time.

In this paper we present and discuss the results of some numerical simulations in order to investigate the mean curvature flow problem in presence of a nonconvex anisotropy. Mathematically, nonconvexity of the anisotropy leads to the ill-posedness of the evolution problem, which becomes forward-backward parabolic. Simulations presented here refer to two different settings: curvature driven vertical motion of graphs (nonparametric setting) and motion in the normal direction by anisotropic mean curvature of surfaces (parametric setting). In the latter case we first relax the problem via an Allen-Cahn type reaction-diffusion equation, in the context of Finsler geometry (diffused interface approximation). Our results suggest three main points. A nonconvex anisotropy and its convexification give rise, for both settings and the discretizations considered, to different evolutions. Wrinkled regions seem to appear only in correspondence to locally concave parts of the anisotropy. Moreover, locally convex regions (interior to the convexification of the so-called Frank diagram) seem to play an important role.