Spring Semester 2015

Introduction to Partial Differential Equations


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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
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


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

Last modified: 2015/02/02
This page: http://math.jacobs-university.de/oliver/teaching/jacobs/spring2015/math362/index.html
Marcel Oliver (m.oliver@jacobs-university.de)