MTH 628, Winter, 2005

MTH 628: Advanced Topics in Partial Differential Equations, Winter, 2005

LECTURE:	MWF 1500 - 1550	Gilkey Hall 104	CRN 28379
Instructor:	R.E. Showalter	Kidder 368	show@math.oregonstate.edu

After a review of Lp spaces, we construct and describe the corresponding Sobolev spaces together with the trace map onto boundary values. Existence theorems are established for monotone operators from a Banach space to its dual, and these are applied to resolve quasi-linear elliptic boundary-value-problems. Then we develop the variational theory for rather general convex functions and the corresponding problems associated with the generalized derivative or subgradient .
The prerequisite material for the course consists of some familiarity with Lp spaces and related analysis and either some experience or motivation from differential equations or boundary-value problems. In particular, the preceding MTH 627 course is not a prerequisite.

Prerequisite: MTH 627 or consent of instructor.
Final Exam: Tuesday, March 15, 1200.
Textbook: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. It is available on-line!

Schedule:
Chapter II. Nonlinear Stationary Problems
1. II.3 Lp spaces, p. 44. (1/03)
2. Interpolation & approximation, pp. 45 - 47.
3. Substitution operators, pp. 46, 48 - 49.
4. II.2 Existence theorems, pp.35, 37. (1/10)
5. monotone, type M, pp. 38, 39.
6. ----------------------------
7. II.4 Sobolev spaces, p.51 - 52.
8. p.52 - 54.
9. Trace, pp. 54 - 55. (1/24)
10. pp. 55 - 56.
11. pp. 56 - 57.
12. -------------------(1/31)
13. II.5 Elliptic boundary-value problems, pp. 59 - 60.
14. Exercises due 2/21.
15. Discussion day. (2/7)
16. Discussion day.
17. Dirichlet, Neumann, and Robin Problems, pp. 39, 61.
18. Periodic, Interface Problems, p. 62-63. (2/14)
19. Oblique derivative Problems, Coercivity conditions, p. 63-64.
20. Discussion day.
21. abstract Green's theorem, pp. 64-65. (2/21)
22. Examples.
23. Examples.
24. Stokes System. (2/28)
25. The Strong Solution.
26. The Normal Trace.
27. The Mixed Problem. (3/7)
28. Navier-Stokes System.
29. Further examples. Final Exercise due 3/15.

The notes above on the Stokes System were updated and cosmetic corrections were made on 3/10/05.