Variational Methods and Dynamics

变分方法和动力学

• 批准号: 1160939
• 负责人: Wilfrid Gangbo
• 金额: $ 21.3万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1160939&HistoricalAwards=false
• 关键词: Variational; Methods; Dynamics

This project focuses on problems in the calculus of variations and partial differential equations. A theory, now called direct methods of the calculus of variations, was developed by Charles Morrey in the middle of the last century wherein he introduced various fundamental concepts of convexity. This theory has wide application to many fields, including John Ball's groundbreaking work in elasticity theory. Recently, John Ball published a list of outstanding problems in the calculus of variations whose solutions will not only advance that field but will also have major impacts in elasticity theory. In recent work, the principal investigator has started to develop tools to handle specific concrete problems from that list. This will, he hopes, lead to general theories. The project will extend this study to include important classes of variational problems that have defied existing theory, such as the stored energy functional of Ogden material, which appears in elasticity. Another class of problems to be studied arises in mass transportation theory. It was first formulated by Monge in 1771 as a simple geometry problem and later extended by Kantorovich to more general cost functions. It consists in finding the optimal (minimal work) method for transporting a pile of earth at one location to an excavation site at another location. These classical results have been extended to more general cost functions by the principal investigator, his students, postdocs, collaborators and many others, with special attention to the Wasserstein (or Kantorovich) metric. Building on these results, the project will investigate so-called axi-symmetric flows, which preliminary computations indicate are Hamiltonian systems on the Wasserstein space. The principal investigator has established a link between the study of these flows and a poorly understood (challenging) class of parameter-dependent Monge-Ampere equations. He will study the existence of paths satisfying a certain basic stability criterion and connecting two prescribed symplectic forms. These problems, interesting in their own right, include searching for geodesics in the set of symplectic forms.Tproject will have a number of important scientific and educational components. First, the theoretical advances will directly aid other scientific research areas, including meteorology, elasticity theory, and other applied sciences, that use variational methods in their algorithms. Second, the principal investigator will continue to disseminate the research by giving courses and lectures in international conferences, workshops, and seminars. Finally, in addition to continuing to mentor undergraduates and Ph.D. students at his home institution, he will extend his active support of underrepresented groups in mathematics, in part by mentoring students and faculty at Spelman College, a local HBCU. Current projects involve Spelman's Professor Yewande Olubummo, who together with the principal investigator plans to develop tools for use in the study of shape recognition.

这个项目的重点是变分法和偏微分方程的问题。一个理论，现在被称为直接方法的变分法，是由查尔斯Morrey在中间的最后一个世纪，他介绍了各种基本概念的凸性。这个理论在许多领域都有广泛的应用，包括约翰·鲍尔在弹性理论方面的开创性工作。最近，约翰·鲍尔（John Ball）发表了一系列变分法中的突出问题，这些问题的解决方案不仅将推动该领域的发展，而且将对弹性理论产生重大影响。 在最近的工作中，首席研究员已开始开发工具，以处理该清单中的具体问题。他希望，这将导致一般理论。 该项目将扩展这项研究，以包括重要类别的变分问题，挑战现有的理论，如奥格登材料的储能功能，出现在弹性。另一类需要研究的问题出现在物质运输理论中。 它首先制定了蒙日在1771年作为一个简单的几何问题，后来扩展到康托洛维奇更一般的成本函数。它包括寻找最佳（最小工作）方法，将一个位置的一堆土运输到另一个位置的挖掘现场。这些经典的结果已被扩展到更一般的成本函数的主要研究者，他的学生，博士后，合作者和许多其他人，特别注意瓦瑟斯坦（或康托洛维奇）度量。 在这些结果的基础上，该项目将研究所谓的轴对称流，初步计算表明这是Wasserstein空间上的Hamilton系统。 主要研究者已经建立了这些流动的研究和一个鲜为人知的（具有挑战性的）类参数依赖的蒙赫-安培方程之间的联系。他将研究存在的道路满足一定的基本稳定性标准和连接两个规定的辛形式。这些问题本身就很有趣，包括在辛形式集中寻找测地线。Tproject将有许多重要的科学和教育组成部分。 首先，理论上的进步将直接帮助其他科学研究领域，包括气象学，弹性理论和其他应用科学，在其算法中使用变分方法。第二，首席研究员将继续通过在国际会议、讲习班和研讨会上开设课程和讲座来传播研究成果。最后，除了继续指导本科生和博士生，在他的家乡机构的学生，他将扩大他的数学代表性不足的群体的积极支持，部分通过指导学生和教师在斯佩尔曼学院，当地HBCU。目前的项目涉及Spelman的教授Yewande Olubummo，他与首席研究员一起计划开发用于形状识别研究的工具。