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

由于牛顿等人对微积分的发展，我们能够通过使用微积分的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式的形式。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式和量子力学中的薛定谔方程，这些方程被用来阐明基本的自然定律，并用来模拟最多样化的现象(在工程、化学、生物、天文学和许多其他领域)。许多重要的应用涉及到微分方程，其解是定义在流形上的函数；粗略地说，流形是曲面。因此，流形上的函数空间的研究在应用数学中是非常重要的，本项目的主要部分集中在发展对流形上的某些函数空间的性质的更完整的数学理解，即流形上的Sobolev空间。此外，微分方程通常不能用解析技术求解，因此设计和严格分析算法的各个方面来近似这些方程的解是至关重要的，也是本项目的第二个主要部分。如果我们的目标得以实现，这个项目的成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、数学和计算薄膜力学以及其他科学和工程领域。加州大学圣迭戈分校至少有一名研究生将接受有关该项目主题的培训。本项目主要研究流形上函数、微分形式以及更一般的向量丛的截面的Sobolev空间的性质，特别关注非光滑流形。我们的主要应用是求解任意维超曲面和更一般流形上偏微分方程(PDE)的一般Petrov-Galerkin数值方法，而我们整个工作的一个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组的数值处理中以两种不同的方式自然地产生：第一，研究Rn中Lipschitz域上涉及微分形式的边值问题(BVP)导致在Lipschitz边界流形上的非光滑微分形式。其次，对三角曲面上的偏微分方程组进行了细致的分析，其中涉及到Lipschitz流形上的Soblev空间。尽管文献中关于非光滑(主要是紧)流形上的Sobolev空间的性质已经有了一些结果，但对这类空间的性质缺乏完整而连贯的严谨研究。这个项目的一个主要目的是研究非光滑流形上偏微分方程解的理论和数值分析所需的Soblev空间的性质，并建立目前文献中所缺少的结果。众所周知，在边值问题的研究中，人们很快就会遇到分数阶Soblev空间，即使在Rn中的区域上也表现出令人惊讶的行为。这个项目的挑战性特征之一将是探索Rn中区域上的分数阶Sobolev空间的性质将在多大程度上转移到由于超曲面三角剖分而得到的开放流形和Lipschitz流形上的微分形式的Sobolev空间上。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.