This course will cover the mean curvature flow from the point of view of the level set method and viscosity solutions. In particular, we will study the anisotropic and crystalline mean curvature flows that serve as models of the evolution of crystals. We will take the point of view of the level set method that allows us to find the solution of the flow as a solution a nonlinear parabolic partial differential equation. Since the most natural notion of generalized solutions are the viscosity solutions, we will spend some time on their introduction and cover some basic properties like the comparison principle and stability. The crystalline mean curvature flow requires us to introduce the notion of facets and the crystalline mean curvature via a connection to the total variation energy. Finally, we will discuss a robust numerical method for the anisotropic mean curvature flow.
Evolution of surfaces and curves have many applications in geometry, material science, image processing, and other fields. Among the most important ones are the evolutions driven by the surface energy, for example the curve shortening flow. The aim of this course is to cover one of the most popular mathematical approaches to this problem, with some discussion of the recent results for surface energies with singular dependence on the normal vector to the surface: the crystalline mean curvature flow.
・Be familiar with the mean curvature flow and its anisotropic variants.
・Understand the level set method for tracking geometric flows.
・Understand fundamentals of the theory of viscosity solutions.
・Learn about numerical methods for mean curvature flows.
・Get acquainted with viscosity solutions for the crystalline mean curvature flow.
anisotropic and crystalline mean curvature flow, viscosity solutions, minimizing movements, level set method, comparison principle
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
This is a standard lecture course. There will be some assignments.
|Course schedule||Required learning|
|Class 1||The lectures will cover the following topics (the order is tentative): ・mean curvature flows ・level set method ・geometric partial differential equations ・viscosity solutions for geometric PDEs ・comparison principle, stability, existence of solutions ・anisotropic and crystalline mean curvature flows ・total variation flow ・facets, notion of crystalline mean curvature, examples of solutions ・viscosity solutions for the crystalline mean curvature flow ・discretization of the anisotropic mean curvature flow: minimizing movements, Chambolle's algorithm, total variation minimization algorithm||Details will be provided during each class session.|
"Giga, Y., Surface evolution equations: A level set approach, Birkhauser Verlag, Basel, 2006 (For those who want to learn more but not required)
Other course material will be announced in the class."