# Partial Differential Equations

### Syllabus

Summary:
This class is an introduction to the theory of partial differential equations. The main topics are: classification of PDEs, linear prototypes (transport equation, Poisson equation, heat equation, wave equation); functional setting, function spaces, variational methods, weak and strong solutions; examples of nonlinear parabolic PDEs, introduction to conservation laws; exact solution techniques, transform methods, power series solutions, asymptotics.

Contact Information:
 Instructor: Marcel Oliver Email: m.oliver@iu-bremen.de Phone: 200-3212 Office hours: M 15:00, W 11:00 in Research I, 107 Grader: Shaowu Tang Email: s.tang@iu-bremen.de

Time and Place:
 Lectures: Tu 9:45-11:00, Th 8:15--9:30 in East Hall 8

Reommended Textbook:
L.C. Evans: Partial Differential Equations

• The final grade will be computed as a grade point average with the following weights:

 Homework: 20% Presentation: 20% Midterm Exam: 20% Final Exam: 40%

• Each participant will give an approximately half-hour presentation toward the end of the semester. A topic should be chosen until September 21.
• Each of the individual scores will be converted to IUB grade points before the overall weighted grade point average is computed.

### Class Schedule (subject to change!)

 31/08/2004: No class 02/09/2004: Introduction; linear transport equation 07/09/2004: Laplace equation, fundamental solution 09/09/2004: Poisson equation, mean value formulas 14/09/2004: Harmonic functions: maximum principle, uniqueness for solutions to the Poisson equation, regularity 16/09/2004: Harmonic functions: analyticity 21/09/2004: Harmonic functions: Liouville's theorem, Harnack's inequality 23/09/2004: Dirichlet problem, Green's functions 28/09/2004: Green's function for the half-space; energy methods 30/09/2004: Heat equation: fundamental solution, solution formulas 05/10/2004: Mean value formulas for the heat equation, maximum principle 07/10/2004: Regularity 12/10/2004: Energy methods, backward uniqueness 14/10/2004: Wave equation: D'Alembert's formula, energy methods 19/10/2004: Tools from functional Analysis I 21/10/2004: Tools from functional Analysis II 26/10/2004: Midterm review 28/10/2004: Midterm Exam 02/11/2004: Reaction-Diffusion equation (overview) 04/11/2004: Proof of existence and uniqueness of solutions to the Fisher-Kolmogorov equation I 09/11/2004: Proof of existence and uniqueness of solutions to the Fisher-Kolmogorov equation II 11/11/2004: Special topics 16/11/2004: Special topics 18/11/2004: Special topics 23/11/2004: Presentation 25/11/2004: Presentation 30/11/2004: Presentation 02/12/2004: Review for final exam 09/12/2004: Final Exam, 11:00-13:00 in the Research II Lecture Hall