Calculus of Variations in L-infinity and Nonlinear PDE's

9972043 Barron本项目拟研究的领域涉及多维和向量值的优化问题的模拟，具有本质上的成本泛函。特别是，计划考虑这样的泛函具有极小元的充要条件的确定。该项目还将研究涉及这些泛函的齐化问题和相关集合函数的表示定理。涉及这些泛函和极小化约束的变分问题也将被考虑。研究中的问题导致了非常非线性的方程。因此，在这个项目中使用的技术将是涉及动力系统的非线性分析、偏微分方程组的粘性解、变分不等式和非光滑分析。本项目的重点是系统的优化，其中一个人试图最小化可能发生的最坏情况。这种方法适用的例子包括以下重要问题：(1)确定承重梁的形状，以使梁长度上的最大应变和应力最小化。过去人们设计的光束都是以总平均能量最小为目标的，但仍有可能出现裂纹。(2)寻找一种治疗癌症的化疗方案，寻求将最大肿瘤负荷降至最低。这比最小化平均肿瘤负荷要现实得多。(3)使反射雷达波束中的最大噪声最小。(4)确定将引导轨迹尽可能接近目标的控制。(5)找到一个能在一段时间内最大限度地增加财富的投资组合。这些都是重大而有趣的问题的例子，在这些问题中，适当的观点是最坏的情况。它们出现在医学、工程学、经济学和许多其他领域。此外，对最坏情况的分析为设计师提供了一个判断所有竞争设计的基准。

9972043BarronThe areas of proposed study of this project involve the multidimensional and vector valued analogue of optimization problems with essential supremum cost functional. In particular it is planned to consider the determination of the necessary and sufficient conditions under which such functionals will possess a minimizer. The project will also study homogenization problems involving these functionals and representation theorems for the associated set functions. Variational problems involving these functionals with constraints on the minimizers will also be considered. The problems under study lead to very nonlinear equations. The techniques to be used in this project will therefore be nonlinear analysis involving dynamical systems, viscosity solutions for partial differential equations, variational inequalities, and nonsmooth analysis.The focus of this project is optimization of systems where one seeks to minimize the worst that can occur. Examples where this is appropriate include the following important problems: (1) Determine the shape of a load bearing beam so as to minimize the maximum strains and stresses over the length of the beam. In the past one designs the beam minimizing the total average energy, but cracks may still occur. (2) Find a chemotherapy regimen for treatment of cancer seeking to minimize the maximum tumor load. This is much more realistic than minimizing the average tumor load. (3) Minimize the maximum noise in a reflected radar beam. (4) Determine a control which will steer a trajectory as close as possible to a target. (5) Find a portfolio that will maximize the minimum wealth over some time period. These are all examples of significant and interesting problems in which the appropriate point of view is the worst case scenario. They arise in medicine, engineering, economics, and many other fields. Furthermore, the analysis of the worst case gives the designer a benchmark on which to judge all competing designs.