Mathematics 775
Partial Differentail  Equations II
Syllabus

Spring, 2010

Instructor: Prof. Xuefeng Wang
Office: Gibson Hall 305
Office Hour: MWF 11:00-11:50 or by appointment
Phone: 862-3451
email: xdw@math.tulane.edu

Course Description


This is the second semester of the year-long course on basic PDE theories.  The course  will cover the following topics:

 L^2 regularity theory for second order elliptic equations. 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.

The Goal of Course: to introduce to the student the basic theories of partial differential equations; to prepare the student for qualifying and oral exams , and ultimately, for research in PDEs and related fields.

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