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

9532030巴伦，詹森和刘 其中性能泛函是容许函数及其导数的上确界的变分问题是本项目的重点。 具体的研究工作包括：（1）从具有给定Lipschitz常数的Lipschitz函数类中确定在超范数下最接近给定函数的函数;（2）确定多维超范数泛函的松弛;（3）求在超范数下齐次泛函的极限（如果它存在的话）;（4）将Young测度理论推广到超范数泛函上，（5）确定了L-无穷域上约束变分问题的Euler-Aronsson条件，（6）刻画了L-无穷域上的Lavrentiev现象，（7）分析了代价包含超范数时的形状优化问题，（8）给出了L-无穷域上约束变分问题的Euler-Aronsson条件，（9）给出了L-无穷域上约束变分问题的Euler-Aronsson条件，（10）给出了L-无穷域上约束变分问题的Euler-Aronsson条件，（11）给出了L-无穷域上约束变分问题的Euler-Aronsson条件，（12）给出了L-无穷域上约束变分问题的Euler-Aronsson条件，（13）给出了L-无穷域上约束变分问题的Euler-Aronsson条件，（14）给出了L-无穷域上约束变分问题的Euler-Aronsson条件，（15）给出了L（8）将鲁棒和H_∞控制理论推广到最大性能泛函。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。 在工程、经济、医学和其他领域中的许多问题都要求在设计变量受到各种约束的情况下使某些性能指标最小化。 为了简化分析，对于特定应用最合适的性能标准通常被另一个更适合数学分析但对于所考虑的问题不是最优的标准所取代。 例如，在结构构件的形状设计中，设计者可能主要关注使构件中的最大应力最小化，但实际上可能基于使这些应力的平均值最小化来选择设计参数。 同样，在设计航天器的升空计划时，应该最小化的可能是最大加速度及其对航天器的伴随力，而不是平均加速度和力。 在药物或放射治疗中，它是治疗试图最小化的肿瘤负荷的最大水平，而不是平均肿瘤负荷。 一般来说，将基于某个密度的平均值的性能标准替换为基于该密度的最大值的性能标准会使问题的分析变得非常复杂。 该项目的目标是提供工具，方法和理论，将允许直接处理这些问题。 研究的结果将是在许多重要的应用领域，如上述的性能提高。 ***

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. ***