数学特別講義3

科目ナンバリング U-SCI00 17103 LJ55 開講年度・開講期 2021 ・ 通年集中
単位数 1 単位 授業形態
配当学年 1回生以上 対象学生
使用言語 英語 曜時限 集中
教員 Pozar Norbert (非常勤講師)
Svadlenka Karel (理学研究科 准教授)
授業の概要・目的 ☆応用数学Ⅱ(Applied MathematicsⅡ)☆
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 appear in material science as models of the evolution of crystals, in image processing and other fields. 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 of 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.
到達目標 You will become familiar with the mean curvature flow and its anisotropic variants, understand the basic level set method for tracking geometric flows and understand the fundamentals of the theory of viscosity solutions. You will be able to perform basic computations with viscosity solutions of geometric flows.
授業計画と内容 1. Introduction, examples of curvature flows and applications, description of surface evolution (1回)
2. Level set method, viscosity solutions for the mean curvature flow (1回)
3. Anisotropic mean curvature flow, examples of solutions (2回)
4. Total variation flow and the crystalline mean curvature flow (1回)
履修要件 Basic knowledge of linear algebra and calculus of functions of multiple variables is expected. Some knowledge of basic partial differential equations is recommended.
授業外学習(予習・復習)等 Review of basic notions in the calculus of functions of multiple variables is recommended: partial derivatives, gradient, chain rule, implicit function theorem.
参考書等 Reference texts will be discussed during the lectures.
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