Calculus of Variations in L-infinity and Related Nonlinear Partial Differential Equations

Loyola University Of Chicago, Chicago IL

Investigators

Abstract

DMS-0200169: Calculus of Variations in L-infinity and Nonlinear PDE's PI: Robert R. Jensen, Co-PI: Emmanuel N. Barron Abstract: The study of nonlinear essential supremum functionals is the focus of this project. In variational problems with such functionals the natural questions of interest are existence of an absolute (or local) minimizer, i.e., a function which minimizes the functional on every subdomain, necessary conditions for the minimizer leading to very nonlinear differential equations, and regularity of the minimizer. In contrast to the classical calculus of variations, even under very strong assumptions, regularity beyond continuous differentiability is not expected, but even this much smoothness is unknown. Virtually every question posed in classical variational analysis can be posed for supremum functionals-- constraints, relaxation, homogenization, duality--and this can be done for both scalar and vector valued problems where the results are more difficult. This subject has placed a new focus on viscosity solutions for fully nonlinear equations, and a new emphasis on an old area of convex analysis, quasiconvexity. The motivation for the study of such problems comes from consideration of physical problems in which the use of the typical energy norm is not adequate, that is, one must design for the worst case. Supremum norm functionals lack strong differentiability and the subsequent difficulties associated with this lack must be dealt with using all forms of nonlinear analysis. The use of variational analysis in engineering, physics, medicine, economics, and other fields is well established. Extreme value engineering is an area in which one seeks to consider the worst case in order to properly design or build some mechanism. In this project these two disciplines are merged to apply variational analysis in order to accomodate a worst case analysis. In certain areas, such as medicine, or structural engineering, it is clear that the worst case analysis is the only realistic design. For example, an oncological treatment cannot seek to minimize the average tumor load, but must minimize the maximum tumor load. A bridge should not be designed to minimize average stresses but maximum pointwise stresses. Many control mechanisms implemented only when a maximum indicator is triggered. These are all problems which are applications of the project considered and in which many new techniques must be developed. The basic and fundamental results of this area of variational analysis, including existence and the determination of criteria sufficient to determine the optimal function, will be studied. The analysis leads to the study of nonlinear partial differential equations and systems of such equations.