Questions Concerning Parabolic Measure, Uniform Rectifiability and the Kato Square Root Problem

关于抛物线测度、均匀可整流性和加藤平方根问题的问题

• 批准号: 0139748
• 负责人: John Lewis
• 金额: $ 13.33万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2005-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0139748&HistoricalAwards=false
• 关键词: Questions; Concerning; Parabolic; Measure; Uniform

March 6, 2002PI: John L. LewisDMS-0139748Abstract:The aim of this project is to investigate further some problems originating from my work on (a) caloric measure in parabolic flat domains, (b) inverse problems and (c) Kato type problems. Under (a) I would like to know to what extent such concepts as uniform rectifiability, Reifenberg flatness, and asymptotic optimal doubling which have been extensively studied in regard to Laplace's equation, can be generalized to the heat equation. As regards (b), I would like to investigate whether very weak overdetermined boundary conditions for solutions to certain p-Laplacian type equations in a domain D imply that the boundary of D satisfies a regularity condition similar to uniform rectifiability. Finally under (c) I would like to know if an extrapolation technique, used by the author and co-authors on some parabolic measure and Kato problems, could be applied to other Kato type problems. Many physical problems can be described in the language of partial differential equations (PDE's). Well known examples of such equations arising in the 19 th century are Laplace's equation, the heat equation, the wave equation, Maxwell's equations, and the Navier- Stokes' equation. Without question knowledge derived from a theoretical study of these equations led to many fundamental technological advances during the 19 th and 20 th centuries. Three questions often asked by those who study PDE's are (a) does there exist a solution, (b) is it unique and (c) does it possess nice properties or is it regular? As concerns (a) and (b) one is often concerned with so called boundary values or boundary conditions for a solution in its domain of existence. So called overdetermined boundary value problems have no solution whereas such classical problems as the Dirichlet and Neumann problems have solutions if the boundary of the given domain and the boundary conditions are sufficiently nice (smooth). My work is concerned with how much one can relax these assumptions and still get meaningful theorems. For example, my co-authors and I have obtained nearly optimal results, which show that certain boundary value problems for Laplace's equation can only be solved if the given domain is a ball. As another example of my work, classical theorems for the Laplacian in smooth domains have been shown to hold in a class of rough domains called Lipschitz or sawtooth domains. More recent work has generalized these results to nongraph domains satisfying `uniform rectifiability' assumptions. My co-authors and I have obtained the analogue of Lipschitz and uniformly rectifiable domains for the heat equation. Our work provides a model for certain free boundary problems such as ice melting (the Stefan problem).

2002年3月6日：John L. Lewisdms-0139748 Abstract：该项目的目的是研究进一步的问题，这些问题源于我对（a）抛物线寄生膜平坦域中的热量测量，（b）逆问题和（c）Kato型问题。在（a）之下，我想知道在多大程度上可以将有关拉普拉斯方程的广泛研究，可以推广到热方程式。关于（b），我想调查域D中某些p拉普拉斯型方程的解决方案的非常弱的过度确定边界条件，这意味着D的边界满足了类似于统一的可重新可相关性的规律性条件。 最终，在（c）下，我想知道是否可以将推断技术和合着者在某些抛物线措施和加藤问题上使用，可以应用于其他Kato型问题。 可以用部分微分方程（PDE）的语言描述许多物理问题。在19世纪出现的这种方程式的众所周知的例子是拉普拉斯方程，热方程，波动方程，麦克斯韦方程和纳维尔·斯托克斯方程。从这些方程式的理论研究中得出的知识毫无疑问，导致了19世​​纪和20世纪的许多基本技术进步。那些研究PDE的人经常问的三个问题（a）是否存在解决方案，（b）它是独一无二的，并且（c）它具有不错的特性还是常规？ 作为关注（a）和（b）的问题，人们经常关注所谓的边界值或解决方案中的边界条件。所谓的过度确定的边界价值问题没有解决方案，而如果给定域的边界和边界条件的边界足够好（平滑），那么dirichlet和neumann问题等经典问题就具有解决方案。我的工作关注的是，有多少人可以放松这些假设并仍然获得有意义的定理。例如，我和我的合着者获得了几乎最佳的结果，这表明只有在给定域是球的情况下，才能解决Laplace方程的某些边界值问题。作为我作品的另一个例子，已证明在光滑域中的拉普拉斯式的经典定理可以在一类称为Lipschitz或锯齿状域的粗糙域中。 最近的工作将这些结果概括为满足“统一重新讨论”假设的Nongraph领域。 我和我的共同作者获得了Lipschitz的类似物，并为热方程式获得了均匀的可重新处理域。我们的工作为某些自由边界问题（例如冰熔化（Stefan问题））提供了模型。