### 2020年度　数学最先端特別講義L   Special lectures on current topics in Mathematics L

Pozar Norbert  利根川 吉廣

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MTH.E642

2

2020年度

3Q
シラバス更新日
2020年9月18日

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### 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 aquinted 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(ディグリー・ポリシー)

 ✔ 専門力 教養力 コミュニケーション力 展開力(探究力又は設定力) 展開力(実践力又は解決力)

授業計画 課題

### 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."

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### 関連する科目

• MTH.C351 ： 函数解析

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