A geometric evolution problem

A traditional approach to compression moulding of polymers involves the study of a generalized Hele-Shaw flow of a power-law fluid, and leads to the p-Poisson equation for the instantaneous pressure in the fluid. By studying the convex dual of an equivalent extremal problem, one may let the power-law index of the fluid tend to zero. The solution of the resulting extremal problem, referred to as the asymptotically dual problem, is known to have the property that the flow is always directed towards the closest point on the boundary. In this paper we use this property to introduce the concept of boundary velocity in the case of piece-wise C-2 domains with only convex corners, and we also give an explicit solution to the asymptotically dual problem in this case. This involves the study of certain topological properties of the ridge of planar domains. With use of the boundary velocity, we define a geometric evolution problem and the concept of classical solutions of it. We prove a uniqueness theorem and use a comparison principle to study the persistence of corners. We actually estimate "waiting times" for corners, in terms of geometric quantities of the initial domain.