Global Methods in the Analysis on Singular Spaces and Partial Differential Equations

本课题致力于研究非紧流形上微分算子的分析和谱理论。研究者特别感兴趣的是推广紧致流形的椭圆理论的经典结果，包括指数理论。关于一般的非紧流形，人们所能说的很少，但Bismut，Beunning，Cordes，Mazzeo，Melrose，Meuller，Shubin等人的结果已经挑出了一类更适合研究的流形：在无穷远处具有一致结构的流形。这些流形的局部理论(正则性、局部存在性)与紧致流形相同，因此我们的方法必然是全局的。因此，除了上述作者使用的偏微分方程组和微分几何的方法外，算子代数的方法在研究具有无穷大均匀结构的非紧致流形中发挥着越来越重要的作用，这从Connes，V.F.R.Jones，Lauter，Monthorbert，Skandalis，Taylor和研究人员的工作中可以看出。在散射论，微分几何，表示理论，数学物理和应用数学的某些问题中自然地出现了在无穷大具有均匀结构的流形。从长远来看，拟议的研究结果将适用于所有这些领域。提出的方法主要属于分析方法，特别是偏微分方程组、算子代数、谱理论和K-理论。一个主要的技术工具将由在无穷远处具有统一结构的非紧致流形上的微分算子生成的代数来提供。通过使用Sobolev空间，我们可以将许多基本问题归结为关于有界算子的代数的问题。这一建议的一个新特点是研究在无穷远处具有一致结构且具有Lipschitz边界的流形的边值问题，就像Mitrea和Taylor在这种紧致区域上的工作一样。

AbstractNistorThis project is devoted to the analysis and spectral theory of differential operators on non-compact manifolds. The investigator is especially interested in generalizing the classical results of elliptic theory for compact manifolds, including index theory. Little can be said about all non-compact manifolds in general, but results of Bismut, Beunning, Cordes, Mazzeo, Melrose, Meuller, Shubin, and others have singledout a class of manifolds that is more amenable to study: the class of manifolds with a uniform structure at infinity. The local theory (regularity, local existence) for these manifolds is the same as for compact manifolds, so our methods will necessarily be global. Thus, in addition to the methods of Partial differential equations and Differential geometry used by the above mentioned authors, methods from Operator algebras have come to play an increasingly important role in the study of non-compact manifolds witha uniform structure at infinity, as is seen from the work of Connes, V.F.R. Jones, Lauter, Monthubert, Skandalis, Taylor, and the investigator.Manifolds with a uniform structure at infinity appear naturally in Scattering theory, Differential geometry, Representation theory, Mathematical physics, and certain problems of Applied mathematics. The results of the proposed research will have, in the long run, applications to all these domains. The main methods that are proposed belong to Analysis, especially Partial differential equations, Operator algebras, Spectral theory, and K-theory. A main technical tool will be provided by algebras generated by differentialoperators on non-compact manifolds with a uniform structure at infinity. By using Sobolev spaces, one can reduce many of our basic questions to questions about algebras of bounded operators. A novel feature of this proposal is the study of boundary value problems for manifolds with a uniform structure at infinity that have Lipschitz boundaries, as in the work of Mitrea and Taylor on such compact domains.