Analysis Seminar

The Brakke flow is a measure-theoretic generalization of the mean curvature flow which describes the evolution by mean curvature of surfaces with singularities. In the first part of the talk, I am going to discuss global existence and large time asymptotics of solutions to the Brakke flow with fixed boundary when the initial datum is given by any arbitrary rectifiable closed subset of a convex domain which disconnects the domain into finitely many "grains". Such flow represents the motion of material interfaces constrained at the boundary of the domain, and evolving towards a configuration of mechanical equilibrium according to the gradient of their potential energy due to surface tension. In the second part, I will focus on the case when the initial datum is already in equilibrium (a generalized minimal surface): I will prove that, in presence of certain singularity types in the initial datum, there always exists a non-constant solution to the Brakke flow. This suggests that the class of dynamically stable minimal surfaces, that is minimal surfaces which do not move by Brakke flow, may be worthy of further study within the investigation on the regularity properties of minimal surfaces. Based on joint works with Yoshihiro Tonegawa (Tokyo Institute of Technology).