Two cases of squares evolving by anisotropic diffusion

Abstract

We are interested in an anisotropic singular diffusion equation in the plane and in its regularization. We establish existence, uniqueness and the basic regularity of solutions to both equations. We construct explicit solutions showing the creation of facets, i.e., flat parts of graphs of solutions. Inspired by the formula for solutions, we rigorously prove that both equations create ruled surfaces out of convex initial data. We also notice that at each positive time, the solutions do not have strict (local) extrema either. We present results of numerical experiments suggesting that the two flows do not seem to differ much. Possible applications to the image reconstruction is pointed out, too.