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

University Of California-San Diego, La Jolla CA

Investigators

Abstract

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.