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

9972043BarronThe建议的研究领域的这个项目涉及多维和向量值模拟的优化问题的基本上确界成本功能。特别是它计划考虑确定的必要和充分条件下，这些泛函将拥有一个极小。该项目还将研究涉及这些泛函和相关集函数的表示定理的均匀化问题。变分问题，涉及这些泛函的约束极小也将被考虑。所研究的问题导致非常非线性的方程。因此，在这个项目中使用的技术将是非线性分析，包括动力系统，粘性解偏微分方程，变分不等式，和nonsmooth analysis.Focus的这个项目是系统的优化，其中一个寻求最大限度地减少可能发生的最坏情况。这是适当的例子包括以下重要问题：（1）确定承载梁的形状，以使梁长度上的最大应变和应力最小化。在过去，人们设计的梁最小化的总平均能量，但裂纹仍然可能发生。(2)寻找一种化疗方案来治疗癌症，寻求最大限度地减少肿瘤负荷。这比最小化平均肿瘤负荷更现实。(3)使反射雷达波束中的最大噪声最小化。(4)确定一个控制器，使轨迹尽可能接近目标。(5)找到一个投资组合，将最大化的最小财富在一段时间内。这些都是重要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基准，以判断所有竞争设计。

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.