Math 557/558/559: Introduction to Partial Differential Equations

Gunther Uhlmann

Autumn/Winter/Spring 1998-1999, MWF 11:30-12:20
This is an introductory course on partial differential equations; no previous knowledge of the subject will be assumed. Partial differential equations model many physical situations, and their study has critical importance in applications. They also have connections with many fields of mathematics, such as differential geometry, complex analysis, and Lie groups.

During the first quarter we will study in detail the main linear equations, namely the wave equation, Laplace's equation, the heat equation, and the Schrödinger equation. Then we will start considering the variable-coefficient case, in particular the initial value problem for hyperbolic systems using the energy method.

In the second quarter we will start by proving elliptic regularity. To this end we will develop the theory of pseudodifferential operators. We will also consider the Dirichlet problem for a large class of elliptic equations using several methods, including variational analysis.

Several topics are possible for discussion in the third quarter. These will be chosen depending on the interests of the audience. These include nonlinear hyperbolic equations, inverse problems, eigenvalue problems, and other topics.

Prerequisites

The Linear Analysis course or a good working knowledge of the Fourier transform and elementary distribution theory. Some of these topics will be reviewed in class during the first week of the fall quarter.

References

The following texts are recommended:

  1. G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, 1995.
  2. R. McOwen Partial Differential Equations, Prentice Hall, 1995.