# 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: Tu, Th 11:15 in Research I, 107 Grader: Vladimir Molchanov Email: v.molchanov@iu-bremen.de

Time and Place:
 Lectures: Mo 9:45-11:00, We 11:15-12:30 in East Hall 3

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 March 1.
• 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!)

 05/02/2007: Introduction; linear transport equation 07/02/2007: Laplace equation, fundamental solution 12/02/2007: Poisson equation, mean value formulas 14/02/2007: Harmonic functions: maximum principle, uniqueness for solutions to the Poisson equation, regularity 19/02/2007: Harmonic functions: analyticity 21/02/2007: Harmonic functions: Liouville's theorem, Harnack's inequality 26/02/2007: Dirichlet problem, Green's functions 28/02/2007: Green's function for the half-space 05/03/2007: Energy methods 07/03/2007: Heat equation: fundamental solution, solution formulas 12/03/2007: Mean value formulas for the heat equation, maximum principle 14/03/2007: Maximum principle on unbounded domains 19/03/2007: Regularity 21/03/2007: Energy methods, backward uniqueness 26/03/2007: Wave equation: D'Alembert's formula, energy methods 28/03/2007: Midterm review 11/04/2007: Midterm Exam 16/04/2007: Tools from functional Analysis 18/04/2007: Reaction-Diffusion equation (overview) 23/04/2007: Proof of existence and uniqueness of solutions to the Fisher-Kolmogorov equation I 25/04/2007: Proof of existence and uniqueness of solutions to the Fisher-Kolmogorov equation II 30/04/2007: Special topics 02/05/2007: Special topics 07/05/2007: Presentation 09/05/2007: Presentation 14/05/2007: Presentation 16/05/2007: Review for final exam TBA Final Exam