Teaching / Course archive
Math 581: Winter 2013
PDE 2
Course Record
Overview
PDE II course on constant coefficient operators, semilinear evolution equations, strongly elliptic systems, distributions, Fourier transform, Sobolev spaces, and functional analytic methods.
Schedule and Topics
| Date | Topic |
|---|---|
| W 1/16 | Cauchy-Kovalevskaya theorem. |
| F 1/25 | Subspaces of distributions and compactly supported distributions. |
| F 2/8 | Fundamental solutions and Schwartz's theorem on hypoellipticity. |
| W 2/20 | Hormander's characterization of hypoellipticity. |
| W 2/27 | Wave front set and microlocal regularity theorem. |
| W 3/20 | Inhomogeneous equations and Duhamel's principle. |
| W 3/27 | Multiplication in Sobolev spaces and Navier-Stokes equations. |
| F 4/5 | Boundary value problems in half space and Lopatinsky-Shapiro conditions. |
| F 4/12 | Garding inequality and interior regularity. |
| T 4/16 | Regularity up to the boundary. |
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
- Overview of elliptic theory, Lopatinsky-Shapiro condition, strong ellipticity, and Garding inequality
- Variational minimization problems and semilinear elliptic problems, if time permits
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.
Assignments and Projects
Assignments and Exams
- Assignments 1-6.
- Course project.
Projects
- T. Salvador: Mean field games
- B. Landon: Introductory scattering theory
- A. Kumar: Kato's theorem on the Coulomb Hamiltonian
- M.-A. Mandich: Probabilistic approach to the Dirichlet problem
Local Materials
Notes
- Scattering pdf