Regularity theory of minimal surfaces in Euclidean space
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.
Lecturer
Date
8th March ~ 15th June, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday,Thursday | 13:30 - 15:05 | A3-2a-201 | ZOOM 02 | 518 868 7656 | BIMSA |
Prerequisite
Real analysis, some knowledge of elliptic PDE would be helpful but not required.
Syllabus
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.
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.
Reference
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
2. Simon Leon Lectures on geometric measure theory
3. Fanghua Lin, Xiaoping Yang Geometric Measure Theory - An Introduction
Audience
Undergraduate
, Graduate
Video Public
No
Notes Public
No
Language
Chinese
Lecturer Intro
Dr. Pengyu Le graduated from ETH Zürich in 2018, then became a Van Loo postdoctoral fellow in University of Michigan. He joined BIMSA as an assistant professor in 2021. His research interest lies in differential geometry and general relativity.