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.

由于牛顿和其他人对微积分的发展，我们能够通过使用微积分语言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，这些方程用于制定基本定律。从经典力学中的牛顿定律和电磁学中的麦克斯韦方程到广义相对论中的爱因斯坦场方程和量子力学中的薛定谔方程，并模拟最多样化的现象（在工程、化学、生物学、许多重要的应用都涉及微分方程，其解是在流形上定义的函数；粗略地说，流形是曲面，因此，流形上的函数空间的研究在应用中至关重要。该项目的一个主要部分是集中于对流形上的某些函数空间（称为索博列夫空间）的属性进行更完整的数学理解。此外，微分方程通常无法使用分析技术来求解，因此需要进行严格的设计和求解。分析这些方程的近似解的算法的各个方面是至关重要的，也是该项目的第二个主要部分。如果我们的目标得以实现，该项目的结果将对数学和物理学等领域产生广泛的影响。预计将在加州大学圣地亚哥分校就该项目的主题对至少一名研究生进行广义相对论、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培训。该项目涉及索博列夫的性质。函数空间，微分形式，以及流形上向量束的更一般部分，特别关注非光滑流形，我们的主要应用是任意维超曲面和更一般流形上的偏微分方程 (PDE) 的通用 Petrov-Galerkin 数值方法，以及贯穿我们工作的重要技术工具将是有限元外微积分 (FEEC) 框架，这种函数空间在偏微分方程的数值处理中以两种不同的方式自然出现：首先，边值问题的研究。 Rn 中 Lipschitz 域上的 (BVP) 微分形式导致涉及 Lipschitz 边界流形的非光滑微分形式 其次，对三角表面上的 PDE 进行仔细分析，该三角表面是通过光滑表面的离散化并用近似流形替换它而获得的。尽管有关非光滑（主要是紧致）流形上的索博列夫空间的性质的结果散布在文献中，但完整且连贯的。该项目的主要目标是研究非光滑流形上的偏微分方程的理论和数值分析所需的索博列夫空间的性质，并建立目前文献中缺少的结果。众所周知，在 BVP 的研究中，人们很快就会遇到分数阶 Sobolev 空间，即使在 Rn 的域上，它也表现出令人惊讶的行为。该项目的挑战性特征之一是探索分数阶 Sobolev 的性质的程度。 Rn 中的域上的空间将转移到由于超曲面三角剖分而获得的开流形和 Lipschitz 流形上微分形式的 Sobolev 空间。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值进行评估，被认为值得支持以及更广泛的影响审查标准。