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

DMS 0400639 M密执安大学奇异积分、光滑空间和椭圆与抛物边值问题的最优估计 调和分析的最新发展促进了对大类偏微分方程的深入理解，模拟了各种物理问题，包括弹性力学、流体力学和电磁学。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性如何影响解决方案的平滑性。当所讨论的域是Lipschitz时，即，满足统一的（内部/外部）条件。非正式地说，Lipschitz域构成了可以发展丰富的函数理论的最一般的一类，在能力和范围上与上半空间相关联。Besov和Triebel-Lizorkin空间的尺度提供了一个自然的函数框架，在这个框架内可以分析光滑性问题。它们包括Sobolev和哈代类，以及在经典Lebesgue标度下难以利用（甚至完全丧失）的特殊位特征。本文提出的主要问题之一是Besov和Triebel-Lizorkin空间上Lipschitz域（具有Dirichlet和Neumann边界条件）上绿色势的正则性。PI建议在椭圆和抛物设置下，通过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.