LECTURE: MWF 1100 -- 1150 StAg 132 CRN 16060 Instructor: R.E. Showalter Kidder 298B show@math.oregonstate.edu
Prerequisite: MTH 512 or MTH 622 or consent of instructor.
Final Exam: TBA
Textbook: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. It is in the series Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, 1997, and is available on-line. Some examples will be taken from Hilbert Space Methods for Partial Differential Equations, and additional notes will be available for later material.The goal is to introduce a selection of topics of analysis of partial differential equations. We begin with an overview of Hilbert space methods for linear problems in one spatial dimension. Most of the major notions appear in this setting, and they are motivated by the classical Dirichlet and Neumann boundary-value problems. Then we extend these notions to monotone operators and corresponding nonlinear problems. We include a discussion of partial differential equations for fluid flow and for transport and flow through porous media, mixed formulations and Lagrange multipliers for problems with constraints, and techniques of homogenization and upscaling for problems with multiple scales.Schedule:
Chapter I. Linear Problems ... an Introduction
1. I.1 BVP in 1D, pp. 1-2.(9/28)
2. pp. 3-5.
3. pp. 5-6.
4. I.2 Variational Method pp. 6-8. (10/05)
5. pp. 8-9.
6. pp. 9-10.
7. pp. 10-12. (10/12)
8. I.3 Applications to BVP pp. 12-16.
9. A.3 Flow in porous media pp. 253-254.
10. II.4 Sobolev Spaces pp. 51-54. (10/19)
11. Boundary Trace on half-space pp. 54-56.
12. Boundary Trace on Manifold pp. 56-57.
13. Porous Media Equation (10/26)
14. Remarks on Cauchy problem
15. Advection-Diffusion BVPs
16. a unilateral boundary-value problem (11/02)
17. m-accretiuve operators, pp. 18-21
18. I.5 The Cauchy problem, pp. 22-24
19. Hille-Yosida Theorem, pp. 25-26 (11/09)
20. parabolic regularity, pp. 27-29
21. Stokes System, pp. 3 - 5.
22. Homogenization (11/16)
23. --
24. --
25. Two-scale Convergence (11/23)
26. --
27. Thanksgiving
28. Highly-heterogeneous case (11/30)
29. --
30. --