|ALINA STANCU, Courant Institute of Mathematical Sciences, New York, New York 10012, USA|
|Asymptotic behavior of a crystalline evolution|
Motion by crystalline curvature is viewed as a typical example of geometric evolution by a nonsmooth boundary energy. Assume that a planar curve is endowed with an energy density defined on a finite set of normal directions. It is natural then to consider the restricted class of piecewise linear curves with just this ordered set of normals. These curves do not have a motion by curvature in the conventional geometric sense, but, following M. Gurtin and J. Taylor, one can still define the so-called crystalline curvature flow which is analogous to the motion by weighted curvature for smooth planar curves.
We consider Gurtin's defintion, with no driving term, for closed convex curves, so that the inward normal velocity of each segment of an admissible polygonal curve as above is inversely proportional to the length of the segment, where the proportionality factor is only required to be positive. Our results show that, if the curve has more than four sides, it will shrink to a point while approaching the shape of a homothetic solution to the flow. This implies the existence of at least one self similar solution for any flow associated to an energy density defined on more than four unitary directions. The number of homothetic solutions will be discussed based on the properties of the energy density.