MTH 627: Partial Differential Equations (Fall 2015):
Nonlinear Cauchy Problems and SemiGroups

LECTURE: MWF 1500 - 1550 Weniger 287 CRN 20036
Instructor: R.E. Showalter Kidder 286 show@math.oregonstate.edu
Office Hours: Mondays 1400-1430, Wednesdays 1330-1400 and by appointment.
We begin with the development of some fundamental results for maximal monotone (multi-valued) operators which provide a powerful tool to develop a large class of nonlinear partial differential equations and related variational inequalities. Applications include problems with free-boundary or phase change, hysteresis, unilateral constraints, .... Then we turn to the nonlinear semigroup generation theory and its application to corresponding nonlinear parabolic or hyperbolic PDEs or systems. The notions of weak derivatives, Sobolev spaces and boundary trace will be developed as needed.
Prerequisite: MTH 622 or MTH 512 or consent of instructor.
Final Exam: TBA
Textbook : Monotone Operators in Banach Space and Nonlinear Partial Differential Equations
The relevant sections will be made available in pdf format.
Chapter II
Chapter IV

Schedule:
1. Lp spaces Read pp. 44-49 of Section II.3 in Chapter II.
2. Read pp. 35-37 of Section II.3 in Chapter II. (9/28)
3. Type M, bounded, coercive is surjective Read p. 38 of Section II.3.
4. Read general examples of p. 40.
5. Sobolev spaces Read pp. 51-53 of Section II.4. (10/5)
6. Trace Read pp. 53-55 of Section II.4.
7. Read pp. 56-58 of Section II.4.
8. Elliptic BVPs Read pp. 59-60 of Section II.5. (10/12)
9. Exercise #1
10.Read pp. 61-62 of Section II.4.
11. Convex functions Read pp. 78-81 of Section II.7. (10/19)
12. Calculus pp. 82-83
13. No class on Friday. Attend the Colloquium on Thursday, 10/22.
14. Examples pp. 85-88. (10/26)
15. Subgradient in Hilbert space pp. 155-157 of Section IV.1.
16. M-accretive operators pp. 157-159.
17. Resolvent pp. 159-160. Exercise #2 due 11/9. (11/02)
18. Yosida approximation pp. 161-162.
19. Approximation of a subgradient pp. 162-163.
20. Examples pp. 163-165. (11/09)
No class Wednesday 11/11.
21. Sums of m-accretive pp. 166-167.
22. Doubly Nonlinear Elliptic example (11/16)
23. Cauchy problem for subgradients pp. 168-169.
24. Cauchy problem for m-acretive pp. 171-173.
25. Nonlinear Semigroup Theorem pp. 173-175. (11/23)
26. No class Wednesday 11/25. Exercise #3 due Wednesday 12/9 at noon.
27. Subgradient flows pp. 175-178. (11/30)
28. u' + A(u) = f, u(0) = u_0
29. Examples