Teaching / Course archive
Math 581: Winter 2012
PDE 2
Course Record
Overview
PDE II course on topological vector spaces, distributions, constant coefficient operators, hypoellipticity, Fourier transform, boundary value problems, semilinear evolution equations, elliptic theory, and spectral theory.
Schedule and Topics
| Date | Topic |
|---|---|
| Wed 1/11 | Idea of distributions and topological vector spaces. |
| Wed 1/18 | Locally convex spaces, seminorms, and Frechet spaces. |
| Wed 1/25 | Distributions and Radon measures. |
| Fri 2/10 | Constant coefficient operators, fundamental solutions, and hypoellipticity. |
| Fri 2/17 | Fourier transform and Liouville's theorem. |
| Fri 3/2 | Problems in half-space, Cauchy problem, and Petrowsky well-posedness. |
| Wed 3/14 | Strong hyperbolicity and inhomogeneous Cauchy problem. |
| Wed 3/21 | Semilinear evolution equations. |
| Wed 3/28 | Elliptic boundary value problems and Garding inequality. |
| Fri 4/13 | Spectral theory and semigroups. |
Topic List
- Cauchy-Kovalevskaya theorem, Holmgren's theorem, and characteristic surfaces
- Distributions, convolution, Fourier transform, and fundamental solutions
- Constant coefficient operators, hypoellipticity, and the Malgrange-Ehrenpreis theorem
- Ellipticity, parabolicity, hyperbolicity, and well-posedness
- Sobolev spaces, semilinear evolution equations, and applications to Navier-Stokes type problems
- Wave front sets and microlocal regularity
- Lopatinsky-Shapiro condition, Garding inequality, and spectral theory
Books and Related Pages
Books
- Gerald B. Folland, Introduction to partial differential equations. Princeton, 1995.
- Gregory I. Eskin, Lectures on linear partial differential equations. AMS, 2011.
- Joseph Wloka, Partial differential equations. Cambridge, 1987.
- Francois Treves, Basic linear partial differential equations. Dover, 2006.
- Walter Rudin, Functional Analysis. McGraw-Hill.
- Walter A. Strauss, Nonlinear wave equations. AMS, 1990.
- Terence Tao, Nonlinear dispersive equations. AMS, 2006.
Assignments and Projects
Assignments and Exams
- Assignments 1-7.
- Final project.
Projects
- Morgane Henry: Wave maps
- Ibrahim Al Balushi: Nonlinear diffusion
- Olga Yakovlenko: Introduction to the Ricci flow
- Yang Guo: Einstein equations
- Sebastien Picard: Introduction to the Yang-Mills equations
- Spencer Frei: DeGiorgi-Nash-Moser regularity theory
- Olivier Mercier: Mean curvature flow
- Joshua Lackman: Pseudodifferential operators
- Andrew MacDougall: Some topics in semiclassical analysis
Local Materials
Notes
- OlivierMercier pdf
- Ricci pdf
- Semiclassical pdf