# Partial Differential Equations

### Syllabus

Summary:
The course is a first introduction to the theory of partial differential equations in a Sobolev space setting. Topics include Sobolev spaces, second order elliptic equations, parabolic equations, semi-groups, and a selection of nonlinear problems.

This course differs from the approach taken in Math 362 which focuses on solutions in classical function spaces via Greens functions. It may therefore be taken by students who have attended Math 362, but we will again start from basic principles so that Math 362 is not a prerequisite.

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: Tu 9:45 and Th 8:15 in East Hall 8

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

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

 Homework: 30% Midterm Exam: 30% Final Exam: 40%

• 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!)

 03/09/2009: Introduction; weak derivatives 08/09/2009: Sobolev spaces Wk,p; properties of weak derivatives 10/09/2009: Convolution, mollifiers, and approximation by smooth functions 15/09/2009: Extensions 17/09/2009: Traces 22/09/2009: Sobolev inequalities 24/09/2009: Compact embeddings 29/09/2009: Second order elliptic equations: definitions and weak solutions 01/10/2009: Lax-Milgram theorem, energy estimates 06/10/2009: Fredholm alternative, existence theorems for weak solutions 08/10/2009: Interior regularity 13/10/2009: No class, Reading Day 15/10/2009: Regularity up to the boundary 20/10/2009: Maximum principles 22/10/2009: Midterm review 27/10/2009: Midterm Exam 29/10/2009: Parabolic equations (5 classes) 03/11/2009: 05/11/2009: 10/11/2009: 12/11/2009: 17/11/2009: Selected nonlinear problems (5 classes) 19/11/2009: 24/11/2009: 26/11/2009: 01/12/2009: 03/12/2009: Review for final exam 15/12/2009: Final Exam, 16:00-18:00 in East Hall 4