Based on a series of lectures, this is a collection of self-contained notes devoted to the study of weak solutions of geometric evolution equations, in particular the mean curvature flow.
There are essentially two approaches leading to the definition of weak solutions. One, originating with Brakke, is via geometric measure theory, another one known as the “level-set” approach has been developed independently, first in codimension \(1\), by Chen, Giga & Goto and Evans & Spruck.
The author focuses on the set theoretic approach in codimension \(1\) and higher, while he is presenting in parallel the properties of the distance function. The result is a systematic treatment of the subject which starts with the basic definitions and preliminary results describing the relevant techniques involved. It continues with the consistency of the level set approach with classical solutions, and the level set approach is compared with the barrier approach of De Giorgi. Finally, the level set approach in codimension \(1\) is characterized à la Soner by the properties of the signed distance function from it, and it is related to the asymptotic behaviour of solutions of the reaction-diffusion equation. The rigorous presentation, alternated with the motivation for using these techniques, makes the paper an ideal tool for somebody even remotely interested in this subject.
For the entire collection see [Zbl 0932.00045].