In this paper we investigate the motion of one-dimensional graphs under anisotropic non-convex mean curvature flow regularized via a Willmore term. Aiming at understanding the evolution problem when we let the regularization parameter tend to zero, we first present rigorous analytical results for the stationary case. Subsequently we discuss the time-dependent problem, focussing mainly on numerical simulations. We discretize by finite elements, and provide a semi-implicit scheme and a number of numerical experiments.
"Willmore-type regularization of mean curvature flow in the presence of a non-convex anisotropy. The graph setting: analysis of the stationary case and numerics for the evolution problem." Adv. Differential Equations 18 (3/4) 265 - 308, March/April 2013.