Mathematics 753
Partial Differentail Equations
Syllabus
Fall 2007
Instructor: Prof. Xuefeng Wang
Office: Gibson Hall 305
Office Hour: MWF 10:00-10:50 or
by appointment
Phone: 862-3451
email: xdw@math.tulane.edu
Course Description
Students are
expected to have already known the motivation
and background of some of the basic PDE's (heat, wave and
Laplace equations); they are also supposed to feel comfortable
with Analysis courses (the undergraduate level Real Analysis,
the graduate level Lebesgue Theory and baby Functional
Analysis, though more linear and nonlinear functional analysis will be
taught here when needed ).
This is the first semester of the year-long course on basic PDE
theories. The two-semester course will cover the following topics:
Classical weak
and strong maximum principles for 2nd order elliptic and parabolic
equations, Hopf boundary point lemma, and their applications. Sobolev
spaces, weak derivatives, approximation, density theorem, Sobolev
inequalities, Kondrachov compact imbedding. L^2 theory for second order
elliptic equations, existence via Lax-Milgram Theorem, Fredholm
alternative, a brief introduction to L^2 estimates, Harnack inequality,
eigenexpansion. L^2 theory for second order parabolic and hyperbolic
equations, existence via Galerkin method, uniqueness and regularity via
energy method. Semigroup theory applied to second order parabolic and
hyperbolic equations. A brief introduction to elliptic and parabolic
regularity theory, the L^p and Schauder estimates. Nonlinear
elliptic equations, variational methods, method of upper and lower
solutions, fixed point method, bifurcation method. Nonlinear parabolic
equations, global existence, stability of steady states, traveling wave
solutions. Conservation laws, Rankine-Hugoniot jump condition,
uniqueness issue, entropy condition, Riemann problem for Burger's
equation, p-systems.
Textbook
Partial Differential Equations, by L. C. Evans
Reference Books
- Elliptic Partial
Differential Equations of Second Order, by D. Gilbarg and
N. Trudinger
- Elliptic and Parabolic Equations,
by Wu, Zhuoqun;
Yin, Jingxue; Wang, Chunpeng
- Partial Differential Equations,
by R. Mcowen
- An introduction to
Partial Differential Equations, by M. Renardy and R. Rogers
- Geometric Theory of Semilinear
Parabolic Equations, by Dan Henry
- Nonlinear Analysis on
Manifolds. Monge-Ampere Equations, by T. Aubin
- Shock Waves and
Reaction-Diffusion Equations, by J. Smoller
Course Grade
The semester letter grade will be given based on your performance
in homework(60%) and the in-class final exam(40%).
Discussions with classmates and me on homework problems are
allowed; rephrasing other people's solutions in your own words is
allowed. The in-class final exam will be a close-book one. The problems
in the final exam will come solely
from my examples and the homework problems.