A Modica-Mortola type problem involving curvature
Dr. Roger Moser (University of Bath)
Thursday 13th January, 2011 14:00-15:00 325
A possible approach to model faceting of crystal surfaces involves an anisotropic area functional, where the anisotropy is given through a multi-well potential depending on the normal vector of the surface. Due to a lack of convexity, variational problems for such a functional may be challenging and may give rise to undesired phenomena such as infinitesimally fine structures. Therefore, some authors perturb the problem by adding a curvature term to the functional. This gives rise to a singular perturbation problem similar to the Modica-Mortola problems used in the theory of phase transitions. Although the basic ideas from this theory are still applicable, the geometric context and the higher order derivatives give rise to new phenomena and new challenges. I will discuss a framework suitable for the analysis of the problem, using tools from geometric measure theory.