The modern approach to the study of partial differential equations  (PDEs) involves a different point
of view regarding the notion of differentiability that leads  one to study  generalized or weak solutions.
The proper setting for this study is certain function spaces that arise in a very natural way from  the
PDEs of interest. For this reason,  a course in classical PDEs is not an essential prerequisite.

In Math 221, we will present an overview  of many important topics in the  theory of partial differential
equations. These topics have a wide range of applications to areas of pure and applied mathematics,
geometry,  engineering, and physics.

Course work will include regular homework assignments. In consultation with the instructor,  students
will  select  a course  project suitable to their own interests. This may involve reading a research article
 or text and giving a short presentation.

Prerequisites:  A previous course in partial differential equations (such as Math 219)  is useful, but not essential.
  Contact the instructor if you are interested in this course or have questions about your background.

Course: Math 221 Modern Partial Differential Equations    CRN: 86173
Date:  Monday &  Friday                                                       Time:   11:10  - 12:25 PM

Text:Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19,
  AMS,   Providence, RI, 1998.

Tentative Topics List

• Introduction - Strategies for studying PDEs, classical solutions, weak solutions, regularity.
• Linear PDEs (Transport equation, Laplace's equation, heat equation, wave equation)
• Topics in functional analysis - Holder spaces, Hilbert spaces, Sobolev spaces, approximation by smooth functions, compactness.
• Second order elliptic PDEs - Lax-Milgram Theorem, energy estimates, maximum principles,

 eigenvalues of elliptic operators.
•  Nonlinear PDEs - calculus of variations (existence of minimizers, coercively, lower semi-continuity), montonicity methods, fixed point methods, bifurcation theory.