MTH 627: Advanced Partial Differential Equations (Fall 2017)


LECTURE: MWF 1500 - 1550 Kidder 280 CRN 26821
Instructor: R.E. Showalter Kidder 286 show@math.oregonstate.edu
Office Hours: Wed 9-11, after class and by appointment.
We develop the notions of weak derivatives, Sobolev spaces, boundary trace and weak formulations of elliptic boundary-value problems and of initial-boundary-value problems for parabolic or hyperbolic PDEs. These are resolved by variants of the closed-range theorem, variational methods, and the generation theory of semigroups of contractions. Extensions to nonlinear problems and systems will be presented (as Supplementry material) according to the interests of the class.
Prerequisite: MTH 622 or MTH 513 or consent of instructor.
Questions to the instructor about prerequisites and expectations are welcome.
Textbook : Hilbert Space Methods for PDEs. It is also available from Dover.
Supplementary notes.pdf will be available.
Grades are based on problem sets (Homework) and progress relative to previous experience.

Schedule:
1. HSM Chapter 1: pp. 1 - 18.
2. HSM Chapter 2: pp. 31 - 35, generalized functions, derivatives.
3. antiderivatives, pp. 36-39. (9/24)
4. HSM pp. 40 - 42, 59-60. Sobolev space, H^1(G).
5. HSM pp. 43 - 48, localization, Boundary Trace.
6. Divergence Theorem and Chapter 3: pp. 59-61. (10/02)
7. Lax-Milgram theorem I.5.4 & III.2. pp. 21-22, 61-64.
8. Boundary-value problems, Green's operator, pp. 64-67.
9. Examples pp. 68-70. (10/09)
10. pp. 71 - 72
11. Coercive forms, elliptic equations. Problems 2.4 and 3.2 on p. 90.
12. Accretive operators. pp. 83-87. (10/16)
13. HSM Chapter 4: pp. 95 - 99, Cauchy problems to Semigroups.
14. HSM pp. 100 - 102, Semigroups to Generators.
15. HSM pp. 102 - 104, Generators to Semigroups. (10/23)
16. HSM pp. 105 - 109, Monotone operators.
17. No class today.
18. HSM Chapter 5: pp. 127-132, Implicit Evolution Equations. (10/30)
19. HSM Chapter 6: pp. 145-153, Wave Equations.
20. HSM Chapter 7: pp. 169-172, Minimization.
21. pp. 172-174. (11/06)
22. pp. 174-176.
23. No class today: Problems due Monday 11/13: 2.4, 3.2, 4.4 and 5.3 on pp. 90-91.
24. pp. 176-178. Variational Inequalities: a boundary constraint. (11/13)
25. Examples: contact problems.
26. Variational Inequalities: non-symmetric case.
27. Interface problem, pp. 71-73. (11/20)
28. Parabolic regularity, I (11/27)
29. Parabolic regularity, II
30. Systems ...