# Introduction to 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: Tu, Th 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 April 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!)

 01/02/2011: Introduction; linear transport equation 03/02/2011: Laplace equation, fundamental solution 08/02/2011: Poisson equation, mean value formulas 10/02/2011: Harmonic functions: maximum principle, uniqueness for solutions to the Poisson equation, regularity 15/02/2011: Harmonic functions: analyticity 17/02/2011: Harmonic functions: Liouville's theorem, Harnack's inequality 22/02/2011: Dirichlet problem, Green's functions 24/02/2011: Green's function for the half-space; energy methods 01/03/2011: Heat equation: fundamental solution, solution formulas 03/03/2011: Mean value formulas for the heat equation, maximum principle 08/03/2011: Maximum principle on unbounded domains 10/03/2011: Regularity 15/03/2011: Energy methods, backward uniqueness 17/03/2011: Wave equation: D'Alembert's formula, energy methods 22/03/2011: Midterm review 24/03/2011: Midterm Exam 29/03/2011: Midterm return and discussion 31/03/2011: Introduction to scalar conservation laws (begin) 05/04/2011: 07/04/2011: 12/04/2011: 14/04/2011: 26/04/2011: 28/04/2011: 03/05/2011: 05/05/2011: Presentation 10/05/2011: Presentation 12/05/2011: Review for final exam TBA: Final Exam