MTH 619: Functional Analysis & PDE, Winter, 2013
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.
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