9300966 Jensen This project will examine mathematical problems arising from both the calculus of variations and control theory. A common feature of the problems is their affinity for the framework of viscosity solutions. The development of new techniques and methods necessary to resolve these problems will both advance the theory of viscosity solutions and provide new examples of their applicability. Viscosity solutions of differential equations were introduced in 1981 to characterize uniquely the value functions of optimal control and differential games. Part of the project will continue the study of such applications. Two other problem areas come from the calculus of variations. The first is based on the strategy of finding functions with prescribed boundary values on a domain which has minimal L-1 norm. These problems have been successfully studied using viscosity solutions. The current work will use viscosity solutions to analyze specific solutions of problems of motion by mean curvature to help understand the level sets of these functions. The second line of investigation is to study the same sets of functions as before with the goal of minimizing the maximal norm. This gives rise to fourth order homogeneous partial differential equations. Some work has already been done on the analysis of such questions but much more remains. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***
