# 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: We 10:00-11:00 in Research I, 107 Grader: Zymantas Darbenas Email: z.darbenas@jacobs-university.de

Time and Place:
 Lectures: Tu 9:45, Th 8:15 in West 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: 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!)

 05/02/2013: Introduction; linear transport equation 07/02/2013: Laplace equation, fundamental solution 12/02/2013: Poisson equation, mean value formulas 14/02/2013: Harmonic functions: maximum principle, uniqueness for solutions to the Poisson equation, regularity 19/02/2013: Harmonic functions: analyticity 21/02/2013: Harmonic functions: Liouville's theorem, Harnack's inequality 26/02/2013: Dirichlet problem, Green's functions 28/02/2013: Green's function for the half-space; energy methods 05/03/2013: Heat equation: fundamental solution, solution formulas 07/03/2013: Mean value formulas for the heat equation, maximum principle 12/03/2013: Maximum principle on unbounded domains 14/03/2013: Regularity 19/03/2013: Midterm review 21/03/2013: Midterm Exam 02/04/2013: Energy methods, backward uniqueness 04/04/2013: Wave equation: D'Alembert's formula, energy methods 09/04/2013: First order PDEs, method of characteristics 11/04/2013: Boundary conditions, local existence theory, examples 16/04/2013: Scalar conservation laws, Hamilton Jacobi equations, connections to Hamilton's equation of motion in classical mechanics and the Hamilton variational principle; Legendre transform 18/04/2013: Legendre transform, Hopf-Lax formula 23/04/2013: Weak solutions for Hamilton Jacobi equations, uniqueness, semiconcavity; uniqueness theorem without proof 25/04/2013: Integral solutions for conservation laws 30/04/2013: Shocks, rarefaction waves, entropy condition 02/05/2013: No class 07/05/2013: Lax-Oleinik solutions. Afternoon Presentations: 15:45-17:00 in Research I, 107 14/05/2013: Presentations 16/05/2013: Review for final exam 30/05/2013: Final Exam, 9:00-11:00 in the Research II Lecture Hall