In this project the principal investigator and a graduate student will work on several interesting problems involving partial differential equations that arise in various applications. Among other things, they will study ill-posed problems, St. Venant and Phragmen-Lindelof problems, stability questions for nonlinear systems and overdetermined problems. This work will be characterized by a deft use of analytical techniques such as a' priori estimates and maximum principles to derive sharp bounds on physically relevant quantities that are described by ill-posed problems. Important physical phenomena are described by what are called "ill-posed" problems, to which the tried and true methods of mathematics often do not apply. These are problems that can have more than one solution or no solution at all and whose solutions, if they exist, exhibit strange behavior. Most problems in partial differential equations are "well-posed", by which mathematicians mean that there is a unique solution that depends continuously on the data of the problem. In this project the principal investigator and a graduate student will apply mathematical methods to study the behavior of solutions of ill- posed problems that come from physics and engineering.
