| Date |
Sections From the Books |
Topics |
What Students Should Do |
| Week 1 |
| 04-04 |
Evans, Ch8.1 |
Introduction to the Calculus Variations |
Start HW1, read Evans Ch 8.1, particularly 8.1.4.c |
| 04-06 |
Evans, Ch8.1 |
Examples, Problems |
Read Lax's Amer. Math. Monthly, 1999, pp497--501 |
| Week 2 |
| 04-11 |
Evans, Ch8.2 |
The Minimizers |
Start HW2, read the proof of Dirichlet Principle on Jost's PDE book |
| 04-13 |
Evans, Ch8.2 |
The Existence of Minimizers |
Read Evans 8.2.4.a, Theorem 10 and 6.3 |
| Week 3 |
| 04-18 |
Evans, Ch8.3 |
Weak Solutions |
Start HW 3, Read Examples on page 5-6 of Morrey |
| 04-20 |
Evans, Ch8.3 |
The Regularity of the Weak Solutions |
Read Evans Ch 8.3 and De Giorgi-Nash Regularity from GT Ch8, HL Ch4 or Morrey Ch 5.3-5.4 |
| Week 4 |
| 04-25 |
Evans, Ch8.4 |
Variational Problems with Constrains |
Start HW4 |
| 04-27 |
Evans, Ch8.4 |
Variational Problems with Constrains II |
Read Evans Ch8.4, particularly 8.4.4. |
| Week 5 |
| 05-02 |
Evans, Ch8.5 |
Saddle Points and the Mountain Pass-I |
Start HW5. Check into Rabinowitz's CBMS book Minimax Method in Critical Point Theory |
| 05-04 |
Evans, Ch8.5 |
Saddle Points and the Mountain Pass-II |
Read Evans Ch8.5 |
| Week 6 |
| 05-09 |
Evans, Ch 8.6 |
Conservation Law-Noether's Theorem-I |
Start HW6. GF, Sec.38 for Conservation Laws in Physics and Gauge Groups |
| 05-11 |
Evans, Ch 8.6 |
Conservation Law-Monotonicity formula and Stress-Energy Tensor |
Read Ch1.3 of Hélein on Conservation Law and Symmetry; |
| Week 7 |
| 05-16 |
Gutièrrez: Ch 1.1 |
Monge-Ampère Measures |
Start HW7. Pick a Presentation (Four in HW7-10). Review Linear Algebra from Lax, Measure and Radon Measure from Folland (Thm 7.8). |
| 05-18 |
Gutièrrez: Ch 1.2-1.3 |
Monge-Ampère Equation-Generalized Solutions |
Read EG, Theorem 2 of Ch 3.1.2, Chapter 1 of Busemann's Convex Surfaces and EG, Ch 6.3-6.4. Review EG, Ch 1.8-1.9 on Radon Measure and Riesz-Markov Theorem. |
| Week 8 |
| 05-23 |
Gutièrrez: Ch 1.4 |
Monge-Ampère Equation - Maximum Principle |
Start HW8. Review the Perron method, pp23-26 of GT or Ch6.1 of HL. |
| 05-25 |
Gutièrrez: Ch 1.5 |
Dirichlet Problem |
Read Gutièrrez: Ch 1.3, 1.7 on Viscosity Solutions. Also Ch 1.6 and Ch1.8. |
| Week 9 |
| 05-30 |
Gutièrrez: Ch 4.1-4.2 |
Pogorelov Interior Estimate |
Compare with p73-76 of Pogorelov 1978 book. Check my CPAM 2021 paper
|
| 06-01 |
Gutièrrez: Ch 4.3 |
Jorgens-Calabi-Pogorelov Theorem |
Read page 454-456 of GT on Evans' Theorem, and Pogorelov's proof (1972 paper and 1978 book). |
| Week 10 |
| 06-06 |
Gutièrrez: Ch 2.1 |
Evans Estimate module Krylov-Safanov Harnack Estimate |
Present a HW or a Project. Check Calabi 1958 paper (Also Caffarelli-Nirenberg-Spruck third order estimate) and Nomizu-Sasaki's Affine Differential Geometry |
| 06-08 |
Gutièrrez: Ch 4.1 |
Krylov-Safanov Harnack Estimate |
Read Gutièrrez: Ch 2.2-2.3. Compare with Ch9.8 of GT |
