Math 266C Spring 12: MW 9:30-10:45AM, MS 5138

Instructor: Inwon Kim.  www.math.ucla.edu/~ikim

Textbook: L. C. Evans, Partial Differential Equations, Chapter 5, 8, 9 and 10. Some handouts in class.

Prerequisite: Math 266A-B or equivalent. We will quickly go through review materials as needed in the class.

Grading: The grades will be based on weekely assigned homework problems.


Homework 1: Evans Chapter 5: Problems 3,5,6,8,10. Due on Wed. 4/11.
Homework 2 is Here . Due on Wed. 4/18.
Correction to Homework 2 the original homework problems 3.-6. refer to the first edition of Evans. The corresponding problems in the second edition are: 12, 14, 15,18 (b)(c) in p307.
Homework 3 is Here . Due on Wed. 4/25.
Homework 4 is Here . Due on Wed. 5/2.
Homework 5: Evans Chapter 8: 18,20. Due on Wed. 5/9.
Homework 6: Evans Chapter 5: prove the details of step 1 of the proof of Theorem 2 in section 5.9. (p302). Evans Chapter 7: Problems 4,5. Chapter 9: Problems 3,4. Due on Wed. 5/16.
Homework 7 is Here . Due on Wed. 5/23.
Homework 8 (part I) is Here . Due on 6/8.

Here is a tentative schedule, which probably covers too much.

Week 1-2: Sobolev space: smooth approximations by mollifiers. Extensions, Traces, Imbedding Theorems, compactness, Poincare inequalities. Applications to the Dirichlet problem.

Week 3-4: Calculus of variations: First and second variation, weak solutions and regularity, constraints, mountain pass theorem, examples.

Week 5-6: Gradient flows, Fixed point methods, geometric arguments.

Week 7-8: Elliptic and parabolic equations and maximum principle.

Week 9-10: Hamilton-Jacobi equations and optimal control