講義名 数学特別講義F（Special lectures on advanced topics in Mathematics F） 科目コード：MTH.E436
開講学期 ３Q 単位数 2--0--0
担当 Pozar Norbert 非常勤講師（金沢大学理工研究域 准教授）
【講義の概要とねらい】
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.
【到達目標】
・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.
【キーワード】
anisotropic and crystalline mean curvature flow, viscosity solutions, minimizing
movements, level set method, comparison principle
【学生が身につける力】
専門力
【授業の進め方】
通常の講義形式で行う．また、適宜レポートを課す．
【授業計画・課題】
| 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 |
課題は講義中に指示する
【教科書】
使用しない
【参考書、講義資料等】
"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."
【成績評価の基準及び方法】
レポート課題(100%)による．
【関連する科目】
MTH.C351： 函数解析
【履修の条件（知識・技能・履修科目等）】
なし