Home page for Math 644, Partial Differential Equations, Fall 2020
Instructor: Charles L. Epstein
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This course is being given online this year, and run through
Partial Differential Equations are the language mathematics provides to
describe most models in physics, chemistry, biology, economics,
engineering, etc. Math 644 is a one semester introduction to this vast
subject. After a short dicussion of ordinary differential equations we consider
the 4 fundamental equations: the first order transport equation, and the
classical second order constant coefficient equations: the wave equation, heat
equation and Laplace equation. In addition we breifly consider the
Cauchy-Riemann equation. These equations provide the motivations and foundations for most
applications of PDE and the rationale for much of the development of the
subject over the past three centuries. In addition to elementary methods
coming from Calculus, we will employ tools from functional analysis and Fourier
analysis. A good understanding of advanced calculus is an essential
prerequisite. Some familiarity with Real Analysis and Complex Analyis will be
Coursework: Problem sets
will be assigned every other week.
The textbook is Partial Differential Equations, 2nd Ed. by
L. Craig Evans. We will follow the text fairly closely. This book is
available from the AMS online
bookstore. Additional material may be taken from the other sources, for
- Partial Differential Equations: Basic Material (vol. 1) by Michael
- Partial Differential
Equations by Fritz John
- Introduction to Partial Differential Equations
by Gerald Folland
- Partial Differential Equations by Jeffrey Rauch
- Foundations of Potential Theory by O. Kellogg
The first 4 topics are pretty well set, but 5-7 are subject to change.
- Quick Review of Ordinary Differential Equations
- The Physical Origins of Partial Differential Equations
- Basic Fourier Analysis
- The Equations of Mathematical Physics on Euclidean Space
- Laplace's equation: The Maximum Principle, Dirichlet and
Neumann problems, well and ill-posed problems, regularity in Sobolev
spaces, connections to analytic functions in dimension 2
- The Heat Equation: The Maximum Principle, Cauchy's problem
- The Wave Equation: Energy estimates, finite propagation speed,
the Cauchy problem, the Radon transform
- Sobolev spaces in bounded domains
- Boundary value problems for Laplace's equation
- Fundamental solutions and boundary integral methods for Laplace's
- The class is scheduled to meet on TTh from 12:00-1:30. I plan to give an
online zoom-lecture during the class meeting time. Links will be posted
soon. All the lectures will be recorded and available on the course Canvas
page (under construction).
- I'll have a zoom office hour is 3:30 to 5:00 on Mondays. Contact me by e-mail
if you have any questions.
- My office in the Math Department is 4E7 DRL, tel. 8-8476, though who
knows when I'll be there again...
- email: firstname.lastname@example.org.
Send e-mail if you have a question or need to contact me.
- The first lecture will be on September 1, 2020 at 12:00. I'll post a
- Problem set 1 is due September 15, 2020.
Return to cle's home page.