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：本项目的目的是进一步研究源于我的工作的一些问题：（a）抛物平坦区域中的热量测量，（B）反问题和（c）Kato型问题。在（a）中，我想知道在多大程度上，诸如一致可求长性、莱芬贝格平坦性和渐近最优加倍等概念，可以推广到热方程，这些概念在拉普拉斯方程中已经得到了广泛的研究。关于（B），我想研究区域D中某些p-Laplacian型方程解的非常弱的超定边界条件是否意味着D的边界满足类似于一致可求长的正则性条件。 最后，根据（c），我想知道，如果外推技术，使用的作者和合著者对一些抛物措施和加藤问题，可以适用于其他加藤型问题。 许多物理问题可以用偏微分方程（PDE）的语言来描述。在19世纪出现的这种方程的著名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-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世纪的许多基本技术进步。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）解是否唯一，（c）解是否具有好的性质，或者解是否是正则的？ 关于（a）和（B），人们经常关心的是解在其存在域中的所谓边界值或边界条件。所谓的超定边值问题没有解，而经典问题如Dirichlet和Neumann问题有解，如果给定区域的边界和边界条件足够好（光滑）。我的工作是关注人们可以在多大程度上放松这些假设，仍然得到有意义的定理。例如，我的合著者和我已经获得了接近最优的结果，这表明，某些边值问题的拉普拉斯方程只能解决，如果给定的区域是一个球。作为我工作的另一个例子，光滑域中拉普拉斯算子的经典定理已经被证明在一类称为Lipschitz或Lipschitz域的粗糙域中成立。 最近的工作已经将这些结果推广到非图域，满足“一致可求正性”的假设。 我的合著者和我已经获得了类似的Lipschitz和一致求长域的热方程。我们的工作提供了一个模型，某些自由边界问题，如冰融化（斯特凡问题）。

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).