The investigator will study the theoretical and numerical analysis of nonlinear partial differential equations (PDEs), especially those of mixed type and generalized Monge-Ampere (MA) equations. These equations arise in several applications. The first application is that of optical system deign and analysis. The second is the design of fuel-efficient axisymmetric bodies and the study of axisymmetric jet flows. Ultimately in both of the above applications, one must solve a nonlinear Monge-Ampere equation which may be of elliptic, hyperbolic, or mixed type. As with most completely nonlinear PDEs, one does not expect to find existence or uniqueness results for these equations except in special cases. However, in order for work on these applications to maintain interest outside the mathematics community, one must demonstrate solidly how the techniques in modeling optical systems or airplane wings can be use to design such objects effectively. Therefore, some questions which must be considered are: -What are well-posed boundary value problems? -Are there cases of interest for which one can establish existence or uniqueness results? -What numerical schemes would provide the best solutions to the MA equations? -What can we say about these numerical solutions mathematically? While the MA equations describing the two applications are considerably different, progress in answering these questions for one MA equation may lead to the answers for other equations. The investigators planning activities will include several visits to the University of Delaware to meet with Professors L. Pamela Cook and Gilberto Schleiniger who will advise her as she seeks answers to the questions posed above.
