MTH 622: Differential and Integral Equations of Mathematical Physics


LECTURE: MWF 1300 - 1350 Weniger 149A CRN 32478
Instructor: R.E. Showalter Kidder 286 show@math.oregonstate.edu
The topic of the first half of this second term is a classical treatment of the elliptic Laplace - Poisson equation. We obtain useful representations and properties of solutions and then characterize the corresponding boundary-value problems by a minimization principle. This provides the transition to Hilbert space methods for boundary-value problems.
Prerequisite: 6 credits of senior-level analysis.
Final Exam: Monday, March 14, 1400.
Textbook : R. Guenther and J. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations.
Class NOTES : The Potential Equation.pdf, Variational Method in Hilbert Space.pdf

1. Sturm-Liouville BVPs: an elementary example. Text: pp. 219-231. (1/3/11)
2. The Divergence Theorem ; Conservation laws. Text: pp. 5-10.
3. Poisson equation, BVPs and Uniqueness. Text: 295-299, 306-308. Notes: p. 1-2.
4. Poisson's formula for circle. Text: 300-305. (1/10)
5. Singular solutions in R^N.
6. Text: 309-310. Notes: pp. 3-4. Fundamental Integral Representation. Exercise 4, p. 4 of Notes.
7. Text: 318. Notes: pp. 5-6. Subharmonic Functions. Mean Value Theorems.
8. Text: 314-320. Notes: p. 7. Maximum principle.
PROBLEMS: Notes: pp. 4-5, Exercise 4; p. 12, Exercise 2. Due 2/2.
9. Text: 311-313. Notes: pp. 8-9. Green's function. (1/24)
10. Notes: pp. 9 - 10. The Half-Space, Quadrant, The Sphere.
11. Notes: pp. 11 - 12. Poisson's formula.
12. Mean-value property, Reflection. Notes: p. 13. (1/31)
13. Weierstrass' Theorem, Harnack inequality, Monotone convergence theorem. Notes: pp. 14-15.
14. The Dirichlet Problem, Perron's method. Notes: pp. 15-16.
15. Barriers, regular boundary, and Green's function. pp. 16-17. (2/7)
Variational Method in Hilbert Space.pdf
16. Section 1: Function spaces; the weak formulation. pp. 1 - 4.
17. derivatives. pp. 5 - 7.
18. anti-derivatives, scalar-product spaces. pp. 7 - 8; Text: pp. 254-255; 448-449. (2/14)
19. Section 2: Hilbert space, Sobolev spaces. pp. 9 - 10.
20. Continuity, dual space. pp. 11 - 12.
21. Minimization Principle. pp. 12 - 13. (2/21)
Exercise 1 and Exercise 2 (`if' part only) on p. 8. Due 2/28.
22. Projection, Riesz operator. pp. 14 - 15.
23. Examples. pp. 15 - 18.
24. Variational inequalities. (2/28)
25. Section 3: Approximation of solutions pp. 19 - 20.
26. approximate solution pp. 21 - 22.
27. interpolation error pp. 22 - 23. (3/7)
28. solution error p. 24.
29.
Exercise 9, page 18. Due 3/15.