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

DMS-0200169：L-无穷大和非线性偏微分方程的变分法。詹森，共同PI：伊曼纽尔N。Barron摘要：非线性本质上确界泛函的研究是本项目的重点。在变分问题与这样的泛函thenatural感兴趣的问题是存在一个绝对（或局部）极小，即，最小化泛函在每个子域上的函数，极小化导致非常非线性微分方程的必要条件，以及极小化的正则性。与经典的变分法相反，即使在非常强的假设下，也不期望超过连续可微性的规律性，但即使如此多的光滑性也是未知的。实际上，在经典变分分析中提出的每一个问题都可以针对上确界泛函--约束、松弛、均匀化、对偶--提出，这对于结果更困难的标量和向量值问题都可以做到。这门学科的研究重点是完全非线性方程的粘性解，以及凸分析的一个老领域--拟凸性。研究这类问题的动力来自于对物理问题的解释，其中典型能量范数的使用是不够的，即必须为最坏情况设计。上确界范数泛函缺乏强可微性，与此相关的潜在困难必须用各种形式的非线性分析来处理。变分分析在工程、物理、医学、经济、和其他领域都很成熟。极端价值工程是一个寻求考虑最坏情况以正确设计或构建某种机制的领域。在这个项目中，这两个学科合并应用变分分析，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。桥梁的设计不应使平均应力最小，而应使点方向应力最大。许多控制机制仅在触发最大值指示器时实施。这些都是问题，这是项目的应用考虑，其中许多新技术必须开发。基本和fundamental结果这一领域的变分分析，包括存在和确定的标准足以确定最佳功能，将进行研究。分析导致研究非线性偏微分方程和系统的suchequations。

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.