Calculus of Variations in L-infinity and Related Nonlinear Partial Differential Equations

DMS-0200169：L中的变分-无穷与非线性偏微分方程集：Robert R.Jensen，Co-PI：Emmanuel N.Barron摘要：非线性本质确界泛函的研究是本课题的重点。在含有这种泛函的变分问题中，我们关心的自然问题是绝对(或局部)极小值的存在性，即每个子域上的函数最小化的函数，极小值导致非常非线性的微分方程组的必要条件，以及极小值的正则性。与经典的变分法相反，即使在非常强的假设下，超越连续可微性的正则性也是不可望的，但即使是如此光滑性也是未知的。实际上，经典变分分析中提出的每一个问题都可以用来提出上确界泛函--约束、松弛、齐次化、对偶--而这既可以用来解决标量问题，也可以用来解决结果比较困难的矢量值问题。这门学科将新的焦点放在完全非线性方程的粘性解上，并将新的重点放在一个旧的凸分析领域--拟凸性。研究这类问题的动机来自于对物理问题的考虑，在这些物理问题中，典型的能量范数的使用是不够的，即必须为最坏的情况设计。上范数泛函缺乏很强的可微性，与此相关的后续困难必须使用所有形式的非线性分析来解决。变分分析在工程、物理、医学、经济和其他领域的应用已经得到了广泛的应用。极端价值工程是一个寻求考虑最坏情况以适当设计或建立某种机制的领域。在这个项目中，这两个学科被合并以应用变分分析，以适应最坏的情况分析。在某些领域，如医学或结构工程，很明显，最坏的情况分析是唯一现实的设计。例如，肿瘤治疗不能寻求将平均肿瘤负荷降至最低，但必须将最大肿瘤负荷降至最低。桥的设计不应使平均应力最小，而应使最大点向应力最大。许多控制机制只有在触发最大指示器时才实施。这些都是所考虑的项目的应用问题，其中需要开发许多新技术。将研究这一变分分析领域的基本和基本结果，包括存在和确定足以确定最优函数的标准。这一分析导致了对非线性偏微分方程组和此类方程组的研究。

DMS-0200169: Calculus of Variations in L-infinity and Nonlinear PDE'sPI: Robert R. Jensen, Co-PI: Emmanuel N. BarronAbstract: The study of nonlinear essential supremum functionals is the focusof this project. In variational problems with such functionals thenatural questions of interest are existence of an absolute (orlocal) minimizer, i.e., a function which minimizes the functionalon every subdomain, necessary conditions for the minimizer leadingto very nonlinear differential equations, and regularity of theminimizer. In contrast to the classical calculus of variations,even under very strong assumptions, regularity beyond continuousdifferentiability is not expected, but even this much smoothnessis unknown. Virtually every question posed in classicalvariational analysis can be posed for supremum functionals--constraints, relaxation, homogenization, duality--and this can bedone for both scalar and vector valued problems where the resultsare more difficult. This subject has placed a new focus onviscosity solutions for fully nonlinear equations, and a newemphasis on an old area of convex analysis, quasiconvexity. Themotivation for the study of such problems comes from considerationof physical problems in which the use of the typical energy normis not adequate, that is, one must design for the worst case.Supremum norm functionals lack strong differentiability and thesubsequent difficulties associated with this lack must be dealtwith using all forms of nonlinear analysis.The use of variational analysis in engineering, physics, medicine,economics, and other fields is well established. Extreme valueengineering is an area in which one seeks to consider the worstcase in order to properly design or build some mechanism. In thisproject these two disciplines are merged to apply variationalanalysis in order to accomodate a worst case analysis. In certainareas, such as medicine, or structural engineering, it is clearthat the worst case analysis is the only realistic design. Forexample, an oncological treatment cannot seek to minimize theaverage tumor load, but must minimize the maximum tumor load. Abridge should not be designed to minimize average stresses butmaximum pointwise stresses. Many control mechanisms implementedonly when a maximum indicator is triggered. These are all problemswhich are applications of the project considered and in which manynew techniques must be developed. The basic and fundamentalresults of this area of variational analysis, including existenceand the determination of criteria sufficient to determine theoptimal function, will be studied. The analysis leads to the studyof nonlinear partial differential equations and systems of suchequations.