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

DMS-0200169: l -∞变分与非线性偏微分方程[j]: Robert R. Jensen, Co-PI: Emmanuel N. BarronAbstract：本课题主要研究非线性本质上泛函的性质。在这类泛函的变分问题中，人们自然感兴趣的问题是绝对（或局部）最小化器的存在性，即在每个子域上最小化函数的函数，导致非常非线性微分方程的最小化器的必要条件，以及最小化器的正则性。与经典的变分学相反，即使在非常强的假设下，超出连续可微性的正则性是不被期望的，但即使如此多的平滑性也是未知的。实际上，在经典变分分析中提出的每个问题都可以针对极限泛函提出——约束、松弛、均匀化、对偶性——这可以用于结果更困难的标量和向量值问题。这门学科对完全非线性方程的粘性解有了新的关注，并对凸分析的一个老领域——拟凸性有了新的关注。研究这些问题的动机来自于对物理问题的考虑，在这些物理问题中，典型能量标准的使用是不够的，也就是说，必须为最坏的情况设计。上模泛函缺乏强的可微性，与此相关的后续问题必须使用各种形式的非线性分析来解决。变分分析在工程、物理、医学、经济学和其他领域的应用已经很成熟。极端价值工程是一个领域，在这个领域中，人们试图考虑最坏的情况，以便正确地设计或构建某些机制。在这个项目中，这两个学科被合并以应用变分分析，以适应最坏情况的分析。在某些领域，如医学或结构工程，很明显，最坏情况分析是唯一现实的设计。例如，肿瘤治疗不能寻求最小化平均肿瘤负荷，但必须最小化最大肿瘤负荷。桥架的设计不应使平均应力最小，而应使点向应力最大。许多控制机制只有在最大指示符被触发时才能实现。这些都是该项目所考虑的应用问题，必须开发许多新技术。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。这种分析导致了对非线性偏微分方程及其方程组的研究。

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.