Singular Integrals, Smoothness Spaces, and Optimal Estimates for Elliptic and Parabolic Boundary Value Problems

椭圆型和抛物型边值问题的奇异积分、光滑空间和最优估计摘要谐波分析的最新发展促进了对大类别偏微分方程的更深入理解，这些偏微分方程可以模拟各种物理问题，包括非弹性、流体力学和电磁学问题。这个建议是关于这些方程的解的正则性方面。最值得注意的是，PI打算开发新的工具，并扩大更传统方法的范围，以便能够研究在存在边界不规则性的情况下，数据的平滑度如何影响解的平滑度。当所讨论的域是Lipschitz域，即满足统一的（内部/外部）条件时，就会出现自然边界。非正式地说，Lipschitz域构成了富函数理论可以发展的最一般的类，在功率和范围上可与上半部分空间相媲美。Besov和triiebel - lizorkin空间的尺度提供了一个自然的功能框架，在这个框架内可以分析平滑问题。他们既包括索博列夫和哈代类，否则，展出的特点，通常很难利用（或甚至完全丢失），当工作在更经典的勒贝格规模。本文提出的主要问题之一是在Besov和triiebel - lizorkin空间上的Lipschitz域（具有Dirichlet和neumann边界条件）上的Green势的正则性。PI建议在椭圆和抛物线两种情况下，用奇异积分方法来研究这个问题。

DMS 0400639M MitreaUniversity of MissouriSingular Integrals, Smoothness Spaces and Optimal Estimates for Elliptic and Parabolic Boundary Value ProblemsABSTRACT Recent developments in harmonic analysis have facilitated a deeperunderstanding of large classes of partial differential equations,modeling a variety of physical problems, including those arising inelasticity, hydrodynamics and electromagnetism. This proposal isconcerned with regularity aspects of solutions to these equations. Most notably, the PI intends to develop new tools, as well as broaden the scope of more traditional methods, in order to be able to study how the smoothness of the data influences the smoothness of the solution, in the presence of boundaryirregularities. The natural borderline is the case when the domains inquestion are Lipschitz, i.e., satisfy a uniform (interior/exterior) conecondition. Informally speaking, Lipschitz domains make up the most generalclass where a rich function theory can be developed, comparable in powerand scope with that associated with the upper-half space.The scales of Besov and Triebel-Lizorkin spaces offer a natural functionalframework within which the issue of smoothness can be analyzed. Theyencompass both the Sobolev and Hardy classes and, otherwise, exhibitfeatures which are generally harder to take advantage of (or are evencompletely lost) when working on the more classical Lebesgue scale.One of the main questions addressed in this proposal is the regularityof Green potentials on Lipschitz domains (with Dirichlet and Neumannboundary conditions) on Besov and Triebel-Lizorkin spaces. The PI proposesto investigate this issue both in the elliptic and parabolic setting, viasingular integral methods.