Instructor: Charles L. Epstein

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### The Course

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 very useful.

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 example:

• Partial Differential Equations: Basic Material (vol. 1) by Michael E. Taylor
• 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

### Tentative Syllabus

The first 4 topics are pretty well set, but 5-7 are subject to change.
1. Quick Review of Ordinary Differential Equations
2. The Physical Origins of Partial Differential Equations
3. Basic Fourier Analysis
4. The Equations of Mathematical Physics on Euclidean Space
1. 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
2. The Heat Equation: The Maximum Principle, Cauchy's problem
3. The Wave Equation: Energy estimates, finite propagation speed, the Cauchy problem, the Radon transform
5. Sobolev spaces in bounded domains
6. Boundary value problems for Laplace's equation
7. Fundamental solutions and boundary integral methods for Laplace's equation

• 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: cle@math.upenn.edu.  Send e-mail if you have a question or need to contact me.

### Announcements

• The first lecture will be on September 1, 2020 at 12:00. I'll post a zoom-link soon.

### Problem Sets

1. Problem set 1 is due September 15, 2020.