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MTH 628: Variational Methods for PDEs, Winter, 2019

**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**

**CSM**

**Ch1.pdf**

**PDE in 20th Century**

**Schedule:**

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

28.

29.

30. Implicit Evolution Equaitions.