MTH 619: Functional Analysis & PDE, Winter, 2013

LECTURE: MWF 1400 - 1450 Kidder 356 CRN 39805
Instructor: R.E. Showalter Kidder 286 show@math.oregonstate.edu
Office Hours: M 1500-1545, W 1000-1045

This two-term course will begin with the elements of Hilbert space, where most basic principles appear in their simplest form. Then we move on to the fundamentals of functional analysis, including the Hahn-Banach Theorem, Open Mapping Theorem, the Closed Range Theorem, and the Uniform Boundedness Principle in Banach space. Topics for Part II include variational principles for minima or saddle points, convex analysis, Lp and Sobolev spaces and corresponding trace spaces of boundary values, semigroup theory, and applications to partial differential equations.

Textbook: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Haim Brezis, Springer, 2011.
Additional materials will be provided for examples, variations and applications.

Schedule:

1. Unbounded operators, Text 2.6. (1/07/2013)
2. Adjoint, examples.
3. No class today. Compare examples with results of Text 2.7.
4. Regular & Maximal Monotone operators, Text 7.1, pp 181-182. (1/14)
Semigroups of Operators
5. The Cauchy Problem, pp. 98-100.
6. Generation of Semigroups, pp. 100-102.
7. MLK Holiday (1/21)
8. "There and Back Again", pp. 102-104 and Text 7.2.
9. Two examples. pp. 95-96, 107-109.
10. Regularity, Text 7.3 and 7.4. (1/28)
11. implicit evolution equations, pp. 127-131.
12. second-order evolution equations, pp. 145-153.
13. Sobolev spaces, Text 9.1. (Read Text 4.1-4.4.) (2/04)
14. Extension and Trace, Text 9.2 and p. 315.
15. Boundary trace for general domains. (Read Text 9.4.)
16. Divergence Theorem (2/11)
17. Advection-Diffusion equations
18. Two models.
Nonlinear elliptic problems
19. Monotone, hemicontinuous operators, pp. 35-37. (2/18)
20. Type M, bounded, coercive is surjective. p. 38.
21. Pseudo-monotone, variational inequality. pp. 39-43.
22. Caratheodory operators, nonlinear Dirichlet BVP. pp. 48, 59-60. (2/25)
23. more BVPs. pp. 61-62.
24. abstract Green-Stokes-Gauss theorem, pp. 63-65.
25. Convex functions, pp. 78-79. (3/04)
26. monotone derivative, the subdifferential, pp. 80-81.
27. Pseudo-monotone + subdifferential. pp. 82-84.
28. Examples. p. 85. (3/11)
29. nonlinear elliptic integrands, pp. 86-88.