Mathematical Sciences: "Variational Analysis in L-Infinity".

9532030性能泛函是可容许函数及其导数的上确界的变分问题是本课题的重点。正在考虑的具体项目包括：(1)从给定的Lipschitz常数的Lipschitz函数族中确定与超范数中的给定函数最接近的函数；(2)确定多维超范数泛函的松弛；(3)找到超范数中齐化泛函的极限(如果它存在的话)；(4)将Young测度理论推广到超范数泛函；(5)确定L无穷大中受约束的变分问题的相关欧拉-阿隆松条件；(6)刻画L无限中的拉夫伦提夫现象；(7)分析了成本为超范数时的形状优化问题；(8)将鲁棒H_无穷控制理论推广到最大成本泛函。最优控制、微分对策和Hamilton-Jacobi-Bellman理论中的辅助问题也是本研究的一部分。工程、经济、医学和其他领域的许多问题都要求在设计变量的各种约束条件下使某些性能指标最小化。为了简化分析，对于特定应用来说，最合适的性能标准通常被另一个更适合数学分析但对所考虑的问题不是最优的性能标准所取代。例如，在结构构件的形状设计中，设计者可能主要感兴趣的是最小化构件中的最大应力，但实际上可能会基于最小化这些应力的平均值来选择设计参数。同样，在设计空间飞行器的发射计划时，可能应该最小化飞行器上的最大加速度及其随之而来的力，而不是平均年龄的加速度和力。在药物或放射治疗中，治疗试图最小化的是肿瘤负荷的最大水平，而不是平均肿瘤负荷。通常，用一个基于该密度最大值的性能标准替换一个基于某个密度平均值的性能标准会极大地使问题的分析复杂化。该项目的目标是提供工具、方法和理论，以便能够直接处理此类问题。这项研究的结果将是改善许多重要应用领域的性能，例如上述领域。***

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. ***