Math 557/558/559
Introduction to Partial Differential Equations
Hart Smith (557/558), Gunther Uhlman (559)
Autumn/Winter/Spring 2002-2003, Monday/Wednesday/Friday 12:30-1:20
This course will be an introduction to the theory
of partial differential equations. In Math 557 we will cover
the three basic constant coefficient equations: Laplace's equation,
the heat equation, and the wave equation. This is essentially the
material of Chapters 2 through 5 of Folland's text. Math 558 will
be devoted to Sobolev spaces and the theory of pseudodifferential
operators, in somewhat more detail than covered in Folland.
This may be thought of as an introduction to microlocal analysis,
which has become an increasingly important tool in many branches of pde's.
In 559 we will consider hyperbolic equations with variable
coefficients. A basic example is the acoustic wave equation, which
describes the propagation of sound waves with variable speed. We will
describe the energy method to show existence and uniqueness for the
initial value problem. We will also go through the progressive wave
expansion and the Lax parametrix construction, which give detailed
information about the singularities of the fundamental solution. The last
topic will be the wave front set of a distribution and Hörmander's
theorem on propagation of singularities for hyperbolic equations.
Texts:
- Autumn & Winter:
- Folland, Introduction to Partial Differential Equations, 2nd
edition.
- Spring
- Notes will be provided.
Prerequisites: Real Analysis (524-5-6) or Linear Analysis (554-5-6).
We will start with a review of distributions and the Fourier transform
for those who have not seen this material.