Alex Freire, University of Tennessee

Networks of curves or surfaces moving by curvature

Alex Freire, University of Tennessee

December 3, 2009

Abstract: In the solidification of certain metal alloys, one observes the following pattern: a fast phase separation stage is followed by a much slower coarsening stage, in which the interfaces separating the phases move so as to minimize the total perimeter- that is, move by a curvature law.  This problem can be modeled by a parabolic system with a small parameter. Furthermore, one observes that, at the junctions where three interfaces of the network meet, the angles between them are constant throughout the motion.

In the limit when the parameter goes to zero we obtain a differential-geometric evolution problem with non-standard boundary conditions. Even for networks of curves a general global existence theorem is still missing. I will discuss the p.d.e. model, what is known for curve networks and very recent results on local and global existence for triple-junction networks of surfaces moving by mean curvature. There will be movies.