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, trace 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
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 closed-book one. The problems in
the final exam will come solely from my examples and the homework
problems.