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

2002 年 3 月 6 日PI：John L. LewisDMS-0139748 摘要：该项目的目的是进一步研究源自我在 (a) 抛物线平面域中的热量测量、(b) 逆问题和 (c) 加藤型问题的工作中产生的一些问题。在（a）下，我想知道在拉普拉斯方程中广泛研究的均匀可整流性、赖芬伯格平坦度和渐近最优加倍等概念在多大程度上可以推广到热方程。关于（b），我想研究域 D 中某些 p-拉普拉斯型方程的解的非常弱的超定边界条件是否意味着 D 的边界满足类似于均匀可整流性的正则性条件。 最后，在（c）下，我想知道作者和合著者在某些抛物线测量和加藤问题上使用的外推技术是否可以应用于其他加藤类型问题。 许多物理问题都可以用偏微分方程（PDE）的语言来描述。 19 世纪出现的此类方程的著名例子有拉普拉斯方程、热方程、波动方程、麦克斯韦方程和纳维-斯托克斯方程。毫无疑问，从这些方程的理论研究中获得的知识导致了 19 世纪和 20 世纪的许多基础技术进步。研究偏微分方程的人经常问的三个问题是（a）是否存在解，（b）它是否唯一，（c）它是否具有良好的性质还是有规律的？ 对于（a）和（b），人们通常关心其存在域中解的所谓边界值或边界条件。所谓的超定边值问题没有解，而狄利克雷和诺伊曼问题等经典问题如果给定域的边界和边界条件足够好（光滑）则有解。我的工作关注的是人们可以在多大程度上放松这些假设并仍然得到有意义的定理。例如，我和我的合著者获得了近乎最优的结果，这表明拉普拉斯方程的某些边值问题只有在给定域是球的情况下才能求解。作为我工作的另一个例子，平滑域中拉普拉斯的经典定理已被证明在一类称为 Lipschitz 或锯齿域的粗糙域中成立。 最近的工作已将这些结果推广到满足“一致可校正性”假设的非图域。 我和我的合著者已经获得了 Lipschitz 的类似物和热方程的一致可校正域。我们的工作为某些自由边界问题（例如冰融化（Stefan 问题））提供了模型。

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).