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

由于牛顿和其他人的微积分的发展，我们能够通过使用微积分语言构建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。这些方程被用来表述自然界的基本定律，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，并用于模拟最多样化的现象（在工程，化学，生物学，天文学和许多其他领域）。许多重要的应用涉及微分方程，其解是定义在流形上的函数;粗略地说，流形是曲面。由于这个原因，流形上的函数空间的研究在应用数学中是至关重要的，这个项目的主要部分是专注于发展一个更完整的数学理解某些函数空间的属性称为流形上的Sobolev空间。此外，微分方程通常不能使用解析技术求解，因此设计和严格分析近似解这些方程的算法的各个方面是至关重要的，也是本项目的第二个主要部分。如果我们的目标得以实现，这个项目的结果将对数学和物理领域产生广泛的影响，如广义相对论的数学理论，数值相对论，数学和计算膜力学，以及其他科学和工程领域。该项目涉及函数的Sobolev空间的性质，微分形式，以及流形上向量丛的更一般的截面，特别关注非光滑流形。我们的主要应用是一般的彼得罗夫-伽辽金数值方法的偏微分方程（PDE）的超曲面的任意尺寸和更一般的流形，和一个重要的技术工具，在我们的工作将是有限元外微积分（FEEC）框架。这样的函数空间自然出现在数值处理的偏微分方程在两个不同的方式：第一，边值问题（BVP）的研究涉及微分形式的Lipschitz域在Rn导致非光滑微分形式的Lipschitz边界流形。其次，详细分析了三角化曲面上的偏微分方程，三角化曲面是通过将光滑曲面离散化并用近似流形代替而得到的，涉及到Lipschitz流形上的Sobolev空间。虽然在非光滑（主要是紧的）流形上的Sobolev空间的性质的结果分散在整个文献中，但对这种空间的性质的完整和连贯的严格研究是缺失的。 该项目的主要目标是研究Sobolev空间的性质，这些性质是非光滑流形上PDE的理论和数值分析所需的，并建立目前文献中缺少的结果。众所周知，在边值问题的研究中，人们很快就会遇到分数阶Sobolev空间，即使在Rn中的域上也表现出令人惊讶的行为。该项目的一个具有挑战性的特点将是探索分数的性质在多大程度上-Rn中域上的一阶Sobolev空间将转化为开流形上和Lipschitz流形上的微分形式的Sobolev空间，该空间是超曲面三角剖分的结果。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准。

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.