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

本项目致力于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椭圆理论的经典结果的紧流形，包括指数理论。很少可以说所有的非紧流形一般，但结果的Bismut，Beunning，科德斯，Mazzeo，梅尔罗斯，Meuller，舒宾，和其他人已经挑选出一类流形，是更适合研究：一类流形的一致结构在无穷远。这些流形的局部理论（正则性，局部存在性）与紧致流形相同，所以我们的方法必然是全局的。因此，除了上述作者所使用的偏微分方程和微分几何的方法之外，算子代数的方法在研究无穷远处具有一致结构的非紧流形中也起着越来越重要的作用，这可以从Connes，V.F.R.琼斯，劳特，蒙休伯特，Skandalis，泰勒，和调查。流形与一致的结构在无穷大自然出现在散射理论，微分几何，表示论，数学物理，和某些问题的应用数学。从长远来看，拟议研究的结果将应用于所有这些领域。提出的主要方法属于分析，特别是偏微分方程，算子代数，谱理论和K-理论。在无穷远处具有一致结构的非紧流形上的微分算子生成的代数将提供一个主要的技术工具。通过使用Sobolev空间，人们可以将我们的许多基本问题简化为关于有界算子代数的问题。一个新的特点，这一建议是研究边界值问题的流形具有一致的结构在无穷大的Lipschitz边界，在工作中的Mitrea和泰勒等紧凑的领域。

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.