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

摘要：本项目的目的是进一步研究(a)抛物线平面域的热量测量问题，(b)逆问题和(c) Kato型问题。在(a)项下，我想知道，在拉普拉斯方程中广泛研究的均匀可整流性、赖芬贝格平坦性和渐近最优倍性等概念，在多大程度上可以推广到热方程中。关于(b)，我想研究D域中某些p-拉普拉斯型方程解的非常弱的过定边界条件是否意味着D的边界满足类似于一致可纠偏性的正则性条件。最后，在(c)项下，我想知道作者和合著者在一些抛物线测度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的问题。许多物理问题可以用偏微分方程的语言来描述。19世纪出现的这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世纪许多基本的技术进步。研究偏微分方程的人经常问的三个问题是：(a)是否存在一个解，(b)它是唯一的，(c)它具有良好的性质还是规则的？关于(a)和(b)，人们通常关心的是解在其存在域内的所谓边界值或边界条件。所谓的过定边值问题没有解，而Dirichlet和Neumann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，我和我的合著者已经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能在给定域是球时才能解决。作为我工作的另一个例子，光滑域上拉普拉斯定理的经典定理已经被证明在一类叫做利普希茨域或锯齿域的粗糙域上成立。最近的工作将这些结果推广到满足“一致可纠偏性”假设的非图域。我和我的合作者已经获得了热方程的利普希茨模拟和均匀可整流域。我们的工作为某些自由边界问题（如冰融化问题）提供了一个模型。

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).