Topics in PDE's and Quasiconformal Mappings

偏微分方程和拟共形映射主题

• 批准号: 9876881
• 负责人: John Lewis
• 金额: $ 7.5万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9876881&HistoricalAwards=false
• 关键词: Topics; PDE; Quasiconformal; Mappings

DMS-9876881LewisThe aim of this project is to investigate further some problemsoriginating from my work on (a) parabolic and elliptic measure,(b) quasiconformal mappings, and (c) regularity of PDE's. Under(a) I would like to know when the Dirichlet Problem has a solutionfor certain parabolic and elliptic PDE's with drag term. Given thatthe Dirichlet problem for these PDE's has a solution, I would like to know when the corresponding measures possess basic properties such as a doubling property. As for (b), I would like to construct more examples of domains which are quasiconformal to a sphere and for which harmonic measure and n - 1 dimensional Hausdorff measure on the boundary are equal. Finally under (c) I would like to know if the techniques used to prove reverse Holder inequalities for solutions to systems modeled on for the parabolic p Laplacian can be used onother PDE's such as the Navier Stokes equation. Many physical problems can be described in the language ofpartial differential equations (PDE's). Well known examples of suchequations arising in the 19 th century are Laplace's equation,the heat equation, the wave equation, Maxwell's equations, andthe Navier- Stokes' equation. Without question knowledge derivedfrom 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 is (a) does there exist a solution, (b) is it unique and (c) does it possess nice properites or is it regular? As for (a) and (b) one is often concerned with so called boundary values or boundary conditions for a solution in the domain of existence. So called overdetermined boundaryvalue problems have no solution whereas such classical problemsas the Dirichlet and Neumann problems have solutions if theboundary of the given domain and the boundary conditions are sufficiently nice (smooth). My work is concerned with how muchone can relax these assumptions and still get meaningful theorems. For example, during the last quarter century, many classicaltheorems for the Laplacian in smooth domains have been shown to hold in a class of rough domains called Lipschitz or sawtoothdomains. My coauthors and I have obtained the analogue of Lipschitz domains for the heat equation. Another avenue of investigation has been to consider questions (a)-(c) in a half space for rough parabolic PDE's. My work provides a model for certain free boundary problems such as ice melting (the Stefan problem).

DMS-9876881Lewis 该项目的目的是进一步研究我在 (a) 抛物线和椭圆测度、(b) 拟共形映射和 (c) 偏微分方程正则性方面的工作中产生的一些问题。在(a)下，我想知道狄利克雷问题何时对某些带有阻力项的抛物线和椭圆偏微分方程有解决方案。鉴于这些偏微分方程的狄利克雷问题有解，我想知道相应的测度何时具有基本属性，例如加倍属性。至于 (b)，我想构造更多与球体拟共形的域的示例，并且边界上的调和测度和 n - 1 维 Hausdorff 测度相等。最后，在 (c) 下，我想知道用于证明抛物线 p 拉普拉斯建模系统解的逆霍尔德不等式的技术是否可以用于其他偏微分方程，例如纳维斯托克斯方程。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？ 至于（a）和（b），人们通常关心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，而狄利克雷和诺依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，在过去的四分之一个世纪中，平滑域中拉普拉斯算子的许多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。 我和我的合著者已经获得了热方程的 Lipschitz 域的模拟。 另一种研究途径是在粗抛物线偏微分方程的半空间中考虑问题 (a)-(c)。 我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。