Teaching / Course archive
Math 580: Fall 2018
PDE 1
Course Record
Overview
PDE I course focused on harmonic functions, the Dirichlet problem, Sobolev spaces, weak solutions, elliptic regularity, spectral theory, and heat equation examples.
Schedule and Topics
| Date | Topic |
|---|---|
| W 9/5 | Examples of harmonic functions, Green's identities, and fundamental solutions. |
| W 9/12 | Harnack inequality, Liouville's theorem, maximum principles, and Green's function. |
| W 10/3 | Minimization of the Dirichlet energy. |
| F 10/5 | Sobolev spaces, strong derivatives, and weak solutions. |
| W 10/24 | Poisson problems and moduli of continuity. |
| W 10/31 | Higher regularity and Sobolev's lemma. |
| W 11/14 | Resolvent of the Laplacian. |
| W 11/21 | Hilbert-Schmidt theory. |
| F 11/23 | Laplace eigenproblems, Rayleigh quotient, and explicit examples. |
| F 11/30 | Weyl's law. |
Topic List
- Green's identities, harmonic functions, Harnack inequality, and maximum principles
- Fundamental solutions, Green's function, Poisson's formula, and the Dirichlet problem
- Sobolev spaces, weak derivatives, variational formulation, and elliptic regularity
- Spectral theory of compact self-adjoint operators and Laplace eigenvalues
- Spectral resolution of heat and wave equations
Books and Related Pages
Books
- Lawrence C. Evans, Partial differential equations. AMS, 1998.
- Fritz John, Partial differential equations. Springer, 1982.
- Jurgen Jost, Partial differential equations. Springer, 2007.
- Qing Han, A basic course in partial differential equations. AMS, 2011.
Assignments and Projects
Assignments and Exams
- Assignments 1-5.
- Final exam.
Archive Note
The referenced assignment files are not present locally.