Collaborative Research: Construction and Properties of Sobolev Spaces of Differential Forms on Smooth and Lipschitz Manifolds with Applications to FEEC

合作研究：光滑流形和 Lipschitz 流形上微分形式 Sobolev 空间的构造和性质及其在 FEEC 中的应用

• 批准号: 2309780
• 负责人: Michael Holst
• 金额: $ 16.74万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2309780&HistoricalAwards=false
• 关键词: Collaborative; Research; Construction; Properties; Sobolev

Thanks to the development of calculus due to Newton and others, we are able to understand the physical world around us by constructing sentences using the language of calculus; these sentences often take the form of differential equations. These equations are used to formulate the fundamental laws of nature, from Newton’s law in classical mechanics and Maxwell’s equations in electromagnetism to Einstein’s field equations in general relativity and Schrodinger equation in quantum mechanics, and to model the most diverse phenomena (in engineering, chemistry, biology, astronomy, and numerous other fields). Many important applications involve differential equations whose solutions are functions that are defined on manifolds; roughly speaking, a manifold is curved surface. For this reason, the study of function spaces on manifolds is of paramount importance in applied mathematics, and a major part of this project is focused on developing a more complete mathematical understanding of properties of certain function spaces known as Sobolev spaces on manifolds. Additionally, differential equations usually cannot be solved using analytic techniques, and therefore designing and rigorously analyzing various aspects of algorithms for approximating solutions to these equations is of central importance and is a second major part of this project. If our goals are achieved, the results of this project will have a broad impact on areas of mathematics and physics such as the mathematical theory of general relativity, numerical relativity, mathematical and computational membrane mechanics, and other areas of science and engineering. Training of at least one graduate student at UCSD on the topics of the project is expected.This project is concerned with the properties of Sobolev spaces of functions, differential forms, and more generally sections of vector bundles on manifolds, with particular focus on nonsmooth manifolds. Our primary application is to general Petrov-Galerkin numerical methods for partial differential equations (PDE) on hypersurfaces of arbitrary dimension and on more general manifolds, and an important technical tool throughout our work will be the Finite Element Exterior Calculus (FEEC) framework. Such function spaces arise naturally in numerical treatment of PDE in two distinct ways: First, the study of boundary value problems (BVP) involving differential forms on Lipschitz domains in Rn leads to nonsmooth differential forms on the Lipschitz boundary manifold. Second, a careful analysis of PDE on triangulated surfaces, which are obtained by discretization of a smooth surface and replacing it with an approximate manifold, involves Sobolev spaces on Lipschitz manifolds. Although there are results on the properties of Sobolev spaces on nonsmooth (primarily compact) manifolds scattered throughout the literature, a complete and coherent rigorous study of the properties of such spaces is missing. A primary goal of this project is to study the properties of Sobolev spaces needed for theoretical and numerical analysis of PDE on nonsmooth manifolds, and establish results that are currently missing in the literature. It is well-known that in the study of BVP, one quickly encounters fractional-order Sobolev spaces that exhibit surprising behavior even on domains in Rn. One of the challenging features of this project will be to explore the extent to which properties of fractional-order Sobolev spaces on domains in Rn will transfer to Sobolev spaces of differential forms on open manifolds and on Lipschitz manifolds obtained as a result of the triangulation of hypersurfaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

感谢您由于牛顿和其他人而发展的微积分，我们能够通过使用微积分来构造句子来理解我们周围的物理世界；这些句子通常采用微分方程的形式。这些方程式用于构成自然的基本定律，从经典力学的牛顿定律和麦克斯韦的电子磁学方程到量子力学中的一般可靠性和施罗德林格方程的爱因斯坦野外方程，以及在工程，化学，化学，生物学，生物学，天文学和许多其他领域中建模最多样化的现象。许多重要的应用涉及微分方程，其解决方案是在流形中定义的函数。粗略地说，歧管是弯曲的表面。因此，对流形的功能空间的研究在应用数学中至关重要，并且该项目的主要部分侧重于对某些功能空间的属性更完整的数学理解，在流形上被称为Sobolev空间。此外，通常无法使用分析技术来解决微分方程，因此设计和严格分析算法的各个方面以将解决方案近似于这些方程是至关重要的，并且是该项目的第二个主要部分。如果实现了我们的目标，那么该项目的结果将对数学和物理学领域产生广泛的影响，例如一般相对论，数值相对论，数学和计算膜力学以及其他科学和工程领域的数学理论。预计将对UCSD的至少一名研究生进行了有关该项目主题的培训。该项目涉及功能，不同形式的Sobolev空间的性质，以及更普遍的矢量捆绑包部分，尤其侧重于非平滑歧管。我们的主要应用是在任意维度和更通用的歧管上进行部分微分方程（PDE）的一般Petrov-Galerkin数值方法（PDE），整个工作中的重要技术工具将是有限的元素外部计算（FEEC）框架。这种功能空间自然出现在PDE的数值处理中，以两种不同的方式进行：首先，边界价值问题的研究（BVP）涉及RN中Lipschitz域上的差异形式，导致Lipschitz边界歧管上的非平滑差分形式。其次，对三角形表面上的PDE进行了仔细的分析，该表面是通过平滑表面离散并用大约歧管替换的，涉及Lipschitz歧管上的Sobolev空间。尽管Sobolev空间在非平滑（主要是紧凑）散布在整个文献中的非平滑歧管上的性能有一些结果，但缺少对此类空间特性的完整而连贯的严格研究。该项目的主要目的是研究对非平滑歧管上PDE的理论和数值分析所需的Sobolev空间的特性，并确定文献中目前缺少的结果。众所周知，在BVP的研究中，一个迅速遇到的分数Sobolev空间，即使在RN中的域上也暴露了惊喜行为。该项目的挑战者特征之一将是探索RN中索波列夫空间的分数sobolev空间的特性，将转移到开放式流形和Lipschitz流形上的sobolev空间，并在lipschitz流形上转移到sobolev spaces，这是通过评估nsf的构建奖的三角策略而获得的。更广泛的影响审查标准。