| 05/02/2013: |
Introduction; linear transport equation
|
| 07/02/2013: |
Laplace equation, fundamental solution
|
| 12/02/2013: |
Poisson equation, mean value formulas
|
| 14/02/2013: |
Harmonic functions: maximum principle,
uniqueness for solutions to the Poisson equation, regularity
|
| 19/02/2013: |
Harmonic functions: analyticity
|
| 21/02/2013: |
Harmonic functions: Liouville's
theorem, Harnack's inequality
|
| 26/02/2013: |
Dirichlet problem, Green's functions
|
| 28/02/2013: |
Green's function for the half-space; energy methods
|
| 05/03/2013: |
Heat equation: fundamental solution, solution
formulas
|
| 07/03/2013: |
Mean value formulas for the heat equation,
maximum principle
|
| 12/03/2013: |
Maximum principle on unbounded domains
|
| 14/03/2013: |
Regularity
|
| 19/03/2013: |
Midterm review
|
| 21/03/2013: |
Midterm Exam
|
| 02/04/2013: |
Energy methods, backward uniqueness
|
| 04/04/2013: |
Wave equation: D'Alembert's formula, energy
methods
|
| 09/04/2013: |
First order PDEs, method of characteristics
|
| 11/04/2013: |
Boundary conditions, local existence theory, examples
|
| 16/04/2013: |
Scalar conservation laws, Hamilton Jacobi
equations, connections to Hamilton's equation of motion in
classical mechanics and the Hamilton variational principle;
Legendre transform
|
| 18/04/2013: |
Legendre transform, Hopf-Lax formula
|
| 23/04/2013: |
Weak solutions for Hamilton Jacobi equations,
uniqueness, semiconcavity; uniqueness theorem
without proof
|
| 25/04/2013: |
Integral solutions for conservation laws
|
| 30/04/2013: |
Shocks, rarefaction waves, entropy condition
|
| 02/05/2013: |
No class
|
| 07/05/2013: |
Lax-Oleinik solutions. Afternoon
Presentations: 15:45-17:00 in Research I, 107
|
| 14/05/2013: |
Presentations
|
| 16/05/2013: |
Review for final exam
|
| 30/05/2013: |
Final Exam, 9:00-11:00 in the Research
II Lecture Hall
|