MTH 628: Variational Methods for PDEs, Winter, 2019

LECTURE: MWF 10:00-10:50 STAG 162 CRN 37663
Instructor: R.E. Showalter Kidder 286
Office Hours: Mon & Fri, 14:00-15:30, Fri 11:00-11:30

The existence of solutions to boundary-value problems for elliptic partial differential equations or systems will be posed and solved in Hilbert spaces by variational methods. These include the minimization of convex functions, saddle points and Lagrange multipliers. Examples will be developed for diffusion (porous media), fluid flow (Stokes), and solid mechanics (elasticity). Corresponding evolution equations or systems will be resolved by the semigroup theory for monotone operators. The notions of weak derivatives, Sobolev spaces, boundary trace and topics from linear analysis will be developed as needed.
Grade will be determined by assigned problem sets.

Prerequisite: Basic facts about Hilbert space or consent of instructor. (MTH 627 is NOT prerequisite.)
Contact instructor for questions about prerequisites or content, especially for non-math graduate students.

NOTES will be developed and made available during the term.
Constrained Optimization & Boundary-Value Problems
PDE in 20th Century

1. Lax-Milgram theorem; Poisson equation. (pp. 1-5 in CO-BVP.pdf)(01/07/2019)
2. Abstract Mixed formulation
3. Closed Range Theorem
4. Neumann problem as constrained minimum. (pp. 9-11 in CO-BVP.pdf)(01/14)
5. Dirichlet problem as constrained minimum.
6. Dirichlet-Neumann problem: 3 formulations. (pp. 12-13 in CO-BVP.pdf)
7. MLK Holiday (01/21)
8. Boundary trace, Normal trace, and Stokes' (p. 12 in CO-BVP.pdf)
9. Lagrange multiplier method, (p. 14 in CO-BVP.pdf)
10. Lagrange multipliers, II (01/28)
11. Displacement & Strain (pp. 1-3 in CSM)
12. Force & stress (pp. 4-8 in CSM)
13. Elasticity system (pp. 9-10) (02/04)
14. No class today.
15. Viscoelasticity & Stokes (pp. 11 -12)
16. m-accretive: pp. 17-24 in Ch1.pdf (02/11)
17. Semi-Group to Generator: pp. 24-25.
18. Generator to Semi-Group: 26-27.
19. Parabolic regularity: 29-30. (02/18)
20. Darcy system
21. Mixed form evolution equations, I
22. Snow day (02/25)
23. Snow day
24. Stokes system
25. Mixed form evolution equations, II (03/04)
26. Mixed form evolution equations, III
27. Brinkman equation
30. Implicit Evolution Equaitions.