Math 6326
Partial
Differential Equations,
Fall 2017.
Lectures will be
TuTh 2.304pm in AH301.
Office hours are TuTh
12pm in PGH696
or by appointment.
Prerequisite for this
class is the Introduction
to Real Analysis sequence
M43314332 and a good
knowledge of classical
multivariable
calculus. A certain amount
of material from
M63206321 will also be
used  especially the
properties of real Hilbert
spaces as covered in
M6360.
The course will start with a
quick revision of
multidimensional calculus
and the definitions of weak
derivatives and their
properties.
Hilbert Sobolev spaces
will be defined and
properties of bilinear forms
on these spaces will be
introduced.
Variational methods
and weak solutions of
boundary value problems will
be described and analyzed.
Results about eigenvalue
problems for elliptic
operators and trace theorems
for Sobolev functions will
be proved.
Throughout issues related to
the Laplace and Poisson
equations are used for
motivating the general
approach. These issues are
central to providing a
mathematical basis for much
of physics and engineering,
so problems from
applications will also be
used as examples.
There is no prescribed text
for the course and the
instructor will
provide notes on much
of the material. The first
semester will cover much of
the material in the first
three chapters of "Hilbert
Space Methods in Partial
Differential Equations" by
Ralph E. Showalter. The
treatment will be quite
different to that of this
text. The text is available
free on the internet or from
Dover Publications.
Other excellent texts
that treat some of the
material that will be
covered include
Robert McOwen, "Partial
Differential Equations,
Methods and Applications",
Prentice Hall, chapters 4 
8.
L.C. Evans, "Partial
Differential Equations",
AMS, chapters 5 and 6.
Haim Brezis,
"Functional Analysis,
Sobolev Spaces and Partial
Differential Equations",
Springer
At the beginning some
homework will be assigned;
later in the semester
students will be asked to
participate in some
(group) projects. There will
not be any exams for the
course.
If you have any
questions, please call me
at 7137433475 or send
email to auchmuty@uh.edu.
