MATH 8583/4, THEORY OF PARTIAL DIFFERENTIAL EQUATIONS, 2003/4
	11:15 - 12:05  MWF

INSTRUCTOR: Mikhail Safonov
VinH 231, tel. 5-8571, email: safonov@math.umn.edu


We plan to cover the basic properties of different types of equations,
with a more detailed account on second order elliptic and parabolic
equations.

We essentially use the classical maximum principle and energy estimates.
In combination with local smoothness of harmonic functions or solutions to
the heat equation, such estimates are used in the study of boundary value
problems for elliptic and parabolic equations with Holder coefficients.
This allows us to get the classical Schauder type results without using
representations of solutions in terms of solid and layer potentials.
This approach works also for fully nonlinear equations, for which no
explicit representations of solutions are available.

We will in details the qualitative properties of solutions to elliptic and
parabolic equations, which do not depend on smoothness of coefficients
(Holder regularity of solutions, Harnack inequalities, etc). These
properties are very useful in the study of behavior of solutions near the
boundary and at infinity, and in various free boundary problems.

In Spring semester, we will also consider elliptic and parabolic equations
in Sobolev classes (Calderon-Zygmund technique), and briefly hyperbolic
equations and systems in different functional classes.


PREREQUISITES: Some knowledge of Real and Functional Analysis (Lebesgue
integral, Banach and Hilbert spaces).

Lecture notes will be provided for the main part of the course.

For supplementary reading, one can use the books:

L. C. Evans, Partial Differential Equations, Graduate Studies inMathematics, Vol. 19, 1998.

D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second  Order, 
Springer, 2nd Edition, 1983 or more recent 3rd  Edition.

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Holder spaces, 
Graduate Studies in Mathematics, Vol. 12, 1996.