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-9876881 Lewis的目的是从我对（a）抛物线和椭圆度措施的工作中进一步研究一些问题，（b）准文化映射，以及（c）PDE的规律性。在（a）下，我想知道何时Dirichlet问题具有解决方案的解决方案，其中某些抛物线和椭圆形PDE带有拖放术语。鉴于这些PDE的Dirichlet问题具有解决方案，我想知道何时相应的度量具有基本属性，例如倍增属性。至于（b），我想构建更多的域的示例，这些域是与球体的符合形式的域，并且在边界上的谐波测量和n -1维豪斯多夫度量相等。最终，在（c）下，我想知道是否可以证明对抛物线pde pde的系统（例如Navier Stokes方程）使用的解决方案的解决方案的解决方案是否反向持有人不等式。可以用各种微分方程（PDE）的语言描述许多物理问题。 19世纪出现的这种方程的众所周知的例子是拉普拉斯的方程，热方程，波动方程，麦克斯韦方程和纳维尔·斯托克斯方程。毫无疑问，知识从这些方程式的理论研究得出，导致了19世​​纪和20世纪的许多基本技术进步。研究PDE的人（a）经常问的三个问题（a）是否存在解决方案，（b）它是独一无二的吗？ 至于（a）和（b）通常与存在域中的解决方案的所谓边界值或边界条件有关。所谓的过度确定的边界值问题没有解决方案，而这种经典问题则是迪利奇和诺伊曼问题如果给定域的包裹和边界条件足够好（平滑），则具有解决方案。我的工作关心的是有多少可以放松这些假设并仍然获得有意义的定理。例如，在上个四分之一世纪，在光滑领域中的Laplacian的许多古典理论已显示在一类称为Lipschitz或Sawtoothdomains的粗糙领域中。 我和我的合着者为热方程式获得了Lipschitz域的类似物。 调查的另一种途径是考虑（a） - （c）在半半空间中用于粗糙的抛物线PDE的问题。 我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。