In this course we study concepts and tools which are
used in much of rigorous mathematical analysis.
The course centers around elliptic P.D.E.'s such
as Laplace's or Poisson's equation, but much of
what we study is used more generally.
We will begin with a systematic development of
the Fourier transform, distributions or generalized
functions, and their relationship. (E.g., the
transform of a constant function is a delta function.)
We review some properties of harmonic functions
and study the representation of solutions of
Laplacian u = f by singular integrals. We will see
how boundary value problems for the Laplacian
can be written in terms of singular integrals,
and find the solution by solving integral equations
on the boundary. We will learn about Sobolev spaces
of functions having a certain number of derivatives
in a weak sense in L^2. We will then study existence
and regularity of elliptic boundary value problems
in this L^2 point of view. Further topics may
be chosen as time permits.
The student should have taken a course like our
PDE I (Math 232) and measure and integration
(Math 241), or have equivalent background.
Our other graduate PDE course,
Math 281, is quite independent and is not
required. Some acquaintance with functional
analysis is helpful, especially compact operators,
but is not essential. A student who is concerned
about his/her background should speak to me.
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