欧式空间极小曲面的正则性理论

The goal of this course is to present the proof of the following remarkable result in the regularity theory of codimension one minimal surfaces in the Euclidean space: the singular set of a locally area minimizing hypersurface in n dimensional Euclidean space has zero (n-1) dimensional Hausdorff measure. The proof presented in the course is due to De Giorgi. We shall cover the theory of the Caccioppoli sets and prove the key De Giorgi lemma for the minimal Caccioppoli set. If the time permits, we shall proceed to show that the dimension of the singular set cannot exceed n-8. The lectures will mainly follow the reference "Minimal Surfaces and Functions of Bounded Variation" by Enrico Giusti.

讲师

乐鹏宇

日期

2023年03月08日至 06月15日

位置

Weekday	Time	Venue	Online	ID	Password
周三,周四	13:30 - 15:05	A3-2a-201	ZOOM 02	518 868 7656	BIMSA

修课要求

Real analysis, some knowledge of elliptic PDE would be helpful but not required.

课程大纲

1. Functions of bounded variation and Caccioppoli sets. Approximation by smooth functions. Compactness in L^1 norm. Coarea formula. Sobolev inequality and isoperimetric inequality.
2. Trace of BV functions. Divergence theorem.
3. The reduced boundary. Blow up. Tangent hyperplane.
4. Regularity of the reduced boundary.
5. De Giorgi lemma. Approximation of minimal sets.
6. Regularity of minimal surfaces.
7. Further topics including minimal cones, first and second variation of the area, etc.

参考资料

1. Enrico Giusti Minimal Surfaces and Functions of Bounded Variation
2. Simon Leon Lectures on geometric measure theory
3. Fanghua Lin, Xiaoping Yang Geometric Measure Theory - An Introduction

听众

Undergraduate , Graduate

视频公开

不公开

笔记公开

不公开

语言

中文

讲师介绍

乐鹏宇博士毕业于苏黎世联邦理工学院，后在美国密西根大学从事博士后研究，2021年9月入职北京雁栖湖应用数学研究院，担任助理研究员，致力于微分几何和与广义相对论相关的基础数学研究。