Abstract

This course aims at presenting some of the ideas behind the (conditional) existence and (weak-strong) uniqueness theory for distributional solutions to mean curvature flow. Focusing on the simple two-phase case, i.e., the evolution of a closed hypersurface, allows for a self-contained and concise presentation, which is accessible for graduate students with some background in PDEs and basic measure theory.

The first lecture provides an overview, basic examples, exercises, and some computational tools. In the second lecture, distributional solutions and a (conditional) closure theorem are presented. If time permits, a relation to the viscosity solution in the two-phase case will be explained. The last lecture is devoted to the weak-strong uniqueness principle in the class of distributional solutions.