# 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@jacobs-university.de Phone: 200-3212 Office hours: Mo, Fr 11:15 in Research I, 107 Grader: Mahmut Çalik Email: m.calik@jacobs-university.de

Time and Place:
 Lectures: We, Fr 9:45 in East Hall 3

Recommended 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: 10% Midterm Exam: 30% Final Exam: 40%

• Each participant will give an approximately half-hour presentation toward the end of the semester. A topic should be chosen until October 1.
• Each of the individual scores will be converted to Jacobs grade points before the overall weighted grade point average is computed.

### Class Schedule (subject to change!)

 03/09/2008: Introduction; linear transport equation 05/09/2008: Laplace equation, fundamental solution 10/09/2008: Poisson equation, mean value formulas 12/09/2008: Harmonic functions: maximum principle, uniqueness for solutions to the Poisson equation, regularity 17/09/2008: Harmonic functions: analyticity 19/09/2008: Harmonic functions: Liouville's theorem, Harnack's inequality 24/09/2008: Dirichlet problem, Green's functions 26/09/2008: Green's function for the half-space; energy methods 01/10/2008: Heat equation: fundamental solution, solution formulas 08/10/2008: Mean value formulas for the heat equation, maximum principle 10/10/2008: Maximum principle on unbounded domains 15/10/2008: Regularity 17/10/2008: Energy methods, backward uniqueness 22/10/2008: Wave equation: D'Alembert's formula, energy methods 24/10/2008: Midterm review 29/10/2008: Midterm Exam 31/10/2008: Midterm return and discussion 05/11/2008: Tools from Functional Analysis 07/11/2008: Reaction-Diffusion equations (overview) 12/11/2008: Proof of existence and uniqueness of solutions to the Fisher-Kolmogorov equation I 14/11/2008: Proof of existence and uniqueness of solutions to the Fisher-Kolmogorov equation II 19/11/2008: Proof of existence and uniqueness of solutions to the Fisher-Kolmogorov equation III 21/11/2008: Introduction to scalar conservation laws I 26/11/2008: Introduction to scalar conservation laws II 28/11/2008: Presentation 03/12/2008: Presentation 05/12/2008: Review for final exam 11/12/2008: Final Exam, 13:00-15:00 in East Hall 4