9532030 Barron, Jensen and Liu Calculus of variations problems in which the performance functional is the supremum of the admissible functions and their derivatives are the focus of this project. The specific projects under consideration include the following: (1) determination of the function from the class of Lipschitz functions with prescribed Lipschitz constant that is closest to a given function in the sup norm; (2) determination of the relaxation of multidimensional sup norm functionals; (3) finding the limit, if it exists in any sense, of homogenized functionals in the sup norm; (4) extending the theory of Young measures to sup norm functionals; (5) determination of the associated Euler-Aronsson conditions for variational problems in L-infinity subject to constraints; (6) characterizing the Lavrentiev phenomenon in L-infinity; (7) analysis of the problem of shape optimization when the cost involves the sup norm; (8) extending the theory of robust and H-infinity control to maximum cost functionals. Auxiliary problems in optimal control, differential games, and Hamilton-Jacobi-Bellman theory are also part of this investigation. Many problems in engineering, economics, medicine, and in other fields require the minimization of some performance criterion subject to various constraints on the design variables. In order to simplify the analysis, the most appropriate performance criterion for a particular application is often replaced by another that is more amenable to mathematical analysis but is suboptimal for the problem under consideration. For example, in the design of the shape of a structural member, the designer may be primarily interested in minimizing the largest stresses in the member, but in fact may select design parameters based on minimizing the average of such stresses. Similarly, In designing a liftoff plan for a space vehicle, it may be the maximum acceleration and its attendant forces on the vehicle that should be minimized, not the aver age acceleration and forces. In drug or radiation therapy it is the maximum level of the tumor load that the therapy is attempting to minimize, not the average tumor load. In general, replacing a performance criterion based on an average of some density with one based on the maximum of that density enormously complicates the analysis of the problem. This project has the goal of providing the tools, methodology, and theory that will allow the direct treatment of such problems. Consequences of the research will be improved performance in many important applications areas such as those described above. ***
