Colliding Interfaces in Old and New Diffuse-interface Approximations of Willmore-flow

This paper is concerned with diffuse-interface approximations of the Willmore flow. We first present numerical results of standard diffuse-interface models for colliding one dimensional interfaces. In such a scenario evolutions towards interfaces with corners can occur that do not necessarily describe the adequate sharp-interface dynamics. We therefore propose and investigate alternative diffuse-interface approximations that lead to a different and more regular behavior if interfaces collide. These dynamics are derived from approximate energies that converge to the $L^1$-lower-semicontinuous envelope of the Willmore energy, which is in general not true for the more standard Willmore approximation

On Median Filters for Motion by Mean Curvature

The median filter scheme is an elegant, monotone discretization of the level set formulation of motion by mean curvature. It turns out to evolve every level set of the initial condition precisely by another class of methods known as threshold dynamics. Median filters are, in other words, the natural level set versions of threshold dynamics algorithms. Exploiting this connection, we revisit median filters in light of recent progress on the threshold dynamics method. In particular, we give a variational formulation of, and exhibit a Lyapunov function for, median filters, resulting in energy based unconditional stability properties. The connection also yields analogues of median filters in the multiphase setting of mean curvature flow of networks. These new multiphase level set methods do not require frequent redistancing, and can accommodate a wide range of surface tensions.Comment: 41 pages, 8 figure