# 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 10:00-11:00 in Research I, 107

Time and Place:
 Lectures: Mo 9:45, We 8:15 in West Hall 4

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!)

 Mon, 2. Feb. 2015: Introduction; linear transport equation Wed, 4. Feb. 2015: Laplace equation, fundamental solution Mon, 9. Feb. 2015: Poisson equation, mean value formulas Wed, 11. Feb. 2015: Harmonic functions: maximum principle, uniqueness for solutions to the Poisson equation, Mon, 16. Feb. 2015: Harmonic functions: smoothness, estimates on derivatives, comment on analyticity, Liouville theorem with consequences Wed, 18. Feb. 2015: Harnack inequality; Dirichlet problem, Green's functions Mon, 23. Feb. 2015: Green's function for the half-space (ctd.); energy methods Wed, 25. Feb. 2015: Heat equation: introduction, fundamental solution, solution formulas Mon, 2. Mar. 2015: Initial value problem for the heat equation Wed, 4. Mar. 2015: Maximum principle, uniqueness Mon, 9. Mar. 2015: Energy methods for the heat equation Wed, 11. Mar. 2015: Wave equation: introduction and d'Alembert's formula for the wave equation in one spatial dimension Mon, 16. Mar. 2015: Wave equations in several dimensions; energy methods Wed, 18. Mar. 2015: Review for midterm exam Mon, 23. Mar. 2015: Midterm exam Wed, 25. Mar. 2015: First order equations, generalized characteristics Wed, 8. Apr. 2015: Scalar conservation laws, shock condition, entropy condition for piecewise smooth solutions Mon, 13. Apr. 2015: Introduction to Hamilton-Jacobi equations, Legendre transform Wed, 15. Apr. 2015: Discussion of Midterm exam Mon, 20. Apr. 2015: No class Wed, 22. Apr. 2015: Hopf-Lax formula, semiconcavity Mon, 27. Apr. 2015: Lax-Oleinik formula for scalar conservation laws, entropy condition Wed, 29. Apr. 2015: Riemann problem; Student presentation Mon, 4. May 2015: Review for final exam Wed, 6. May 2015: Final Exam, West Hall 1, 8:15-10:15 Mon, 11. May 2015: Student presentations Wed, 13. May 2015: Student presentations Wed, 20. May 2015: Student presentations, West Hall 4, 15:00-19:00