This page was last updated January 29, 2005
Official description
Partial Differential Equations. Advanced theory including existence proofs and distributional approach. PREREQ: MTH 513. Normally offered alternate years.
Actual description
A bit of Functional Analysis is assumed in Mth 627, but very little. Whatever Functional Analysis is required will be reviewed. In particular, Mth 614 is not required.
The first part of the course is an introduction to the theory of distributions. We will study quite a few examples including finite parts of divergent integrals, analytic distribution valued functions, and fundamental solutions of partial differential operators.
One of the nicer parts of Schwartz' (1915-2002) theory of distributions (1940’s) is the theory of the Fourier transform of tempered distributions, a beautiful unification, and more, of earlier piecemeal theories - see the October 2003 issue of the AMS Notices for some discussion of Schwartz' work. This theory also includes the theory of Fourier series in a natural way since periodic distributions are tempered. Schwartz' approach to the Fourier transform also provides a nice basis for the theory of the Laplace transform. We will discuss these ideas well.
As an application of all this theory we will prove the existence of a fundamental solution for any linear constant coefficient partial differential operator (Malgrange and Ehrenpreis). This result was first obtained in the 1950's and so is now approaching "classical" status.
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