Math 557/558/559
Introduction to Partial Differential Equations

Gunther Uhlmann

Autumn/Winter/Spring 2004-2005, Monday/Wednesday 2:30-3:45
This is an introductory course to partial differential equations, and no previous knowledge of the subject will be assumed. Partial differential equations model many physical situations, and their study has critical importance in applications. It also has connections with many fields of mathematics like differential geometry, complex analysis, Lie groups etc.

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 Dirichlet problem for Laplace's equation and more general elliptic equations.

In the second quarter, we will start by proving elliptic regularity. To this effect, we will develop the theory of pseudodifferential operators and give several applications, including the propagation of singularities for hyperbolic equations.

During the third quarter, we will study in more detail hyperbolic equations. We will construct the Lax parametrix and discuss also the Hart Smith parametrix for the wave equation with less regular coefficients.

Prerequisites: The Linear Analysis (Math 554/5/6) or Real Analysis course(Math 524/5/6). The class will start with an extensive review of the Fourier transform and distribution theory.

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, second edition, 2003.
  3. M. Taylor,Partial Differential Equations Vols. I and II, Springer, 1996.