Spring Semester 2013

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:  We 10:00-11:00 in Research I, 107
Grader:Zymantas Darbenas

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

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


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

Last modified: 2013/02/05
This page: http://math.jacobs-university.de/oliver/teaching/jacobs/spring2013/math362/index.html
Marcel Oliver (m.oliver@jacobs-university.de)