MTH 623: Differential and Integral Equations of Mathematical Physics (Spring 2015)


LECTURE: MWF 1300 - 1350 Weniger 275 CRN 51948
Instructor: R.E. Showalter Kidder 286 show@math.oregonstate.edu
Office Hours: Mondays 1400-1540 and by appointment.
We first describe the notions of weak derivatives, Sobolev spaces and boundary trace. These are used to develop the variational theory of elliptic boundary-value problems. Then the generation theory of linear semigroups of contractions is developed and applied to initial-boundary-value problems for parabolic or hyperbolic PDEs and systems.
Prerequisite: MTH 622 or MTH 513 or consent of instructor.
Final Exam: Wednesday, June 10, 1800.
Textbooks : Hilbert Space Methods for PDEs. It is also available from Dover.

Schedule:
Distributions, Sobolev spaces, and Trace.
1. HSM Chapter 2: pp. 31 - 40, distributions. (03/30)
2. HSM pp. 40 - 41, Sobolev space, H^m(G).
3. HSM pp. 43 - 48, localization, Boundary Trace.
Elliptic Boundary-Value Problems.
4. HSM Chapter 3: pp. 59 - 62, Text: Section 11-4,5, pp. 458-464. (04/06)
5. HSM pp. 62 - 65, general Green's operator.
6. HSM pp. 66 - 68, 2nd order Elliptic BVPs.
7. HSM: pp. 68 - 70, non-symmetric parts. (04/13)
8. HSM: pp. 71 - 72, Interface and Periodic BVPs.
9. HSM: pp. 53-54 (Chapt 2), compactness of V -> H.
10. HSM: pp. 86 - 89, eigenfunction expansions. (04/20)
11. No class today.
Semi-groups of Operators.
12. HSM Chapter 4: pp. 95 - 99, Cauchy problems to Semigroups.
13. HSM pp. 100 - 103, Semigroups to Generators & Cauchy problems. (04/27)
14. Problems Discussion: Chapter IV: 2.2 and 3.1. (Due 05/04.)
15. Problems Discussion: Assume a(u,v) + \lambda (u,v)_H is V-elliptic for every \lambda > 0. Define A by Corollary 3.2 in Chapter III.
Show \lambda (\lambda + A)^{-1} is a contraction on H for every \lambda > 0.
16. HSM p. 103, Review, Generators to Cauchy problems. (05/04)
17. pp. 103-106.
18. pp. 107-108.
19. Parabolic regularity. (05/11)
20. HSM pp. 74 - 77, strongly elliptic , Garding's inequality.
21. HSM pp. 119 - 124.
22. The porous medium equation.pdf (05/18)
23. advection-diffusion equations
24. Problems Discussion: Discuss the parabolic equation obtained from (0.1) with the boundary conditions (0.2.b) and initial conditions. Repeat for (0.1) with the boundary conditions (0.4.b) and initial conditions.
25. Holiday (05/25)
26. Stokes System, pp. 1 - 3.
27. SS pp. 3 - 5, weak & strong solutions.
28. SS pp. 6 - 7, normal trace.
29. SS pp. 7 - 8, mixed problem for Stokes system.
30. homogenization

Final Exercise Due Wednesday 6/10, 1800 = 6pm. Office hours Tuesday 6/9/2015: 1130 - 1400.