MATHEON-ICM WORKSHOP on Free Boundaries and Materials Modeling - Abstract

Rybka, Piotr

A singular weighted mean curvature flow in the plane: special curves, generic driving

We want to gain insight into the details of evolution of 3-D ice crystals which are hexagonal prisms (or are approximated by circular cylinders). The evolutions in question is governed by a Stefan-type problem the Gibbs-Thomson law and a kinetic undercooling. Any axial section of either the prism of the cylinder is a rectangle. Thus we study a driven singular weighted mean curvature flow in the plane with rectangles as initial curves. We expect that the rectangles will bend due the presence of the forcing term. We will justify this picture, we define what we mean by bent rectangles. We will prove an existence and uniqueness result for the singular WMC with bent rectangles as initial data. We will assume a generic form of the driving conforming to a physical constraint and under a simplifying assumption for the kinetic coefficient.