Topics in PDE's and Quasiconformal Mappings

本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)偏微分方程的正则性方面的工作所引起的一些问题。在(a)项下，我想知道对于某些带阻力项的抛物型和椭圆型偏微分方程，Dirichlet问题何时有解。考虑到这些偏微分方程的狄利克雷问题有一个解，我想知道相应的测度何时具有加倍性质等基本性质。对于(b)，我想构造更多的域的例子，这些域与球是拟共形的，并且边界上的调和测度和n - 1维豪斯多夫测度相等。最后，在(c)项下，我想知道用于证明为抛物型p拉普拉斯模型建模的系统解的反向Holder不等式的技术是否可以用于其他PDE，如Navier Stokes方程。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)它是具有良好的性质还是有规律的？对于(a)和(b)，人们通常关心的是存在域内解的所谓边界值或边界条件。所谓的过定边值问题没有解，而像Dirichlet和Neumann问题这样的经典问题，如果给定域的边界和边界条件足够好（光滑），就有解。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，在过去的四分之一世纪里，光滑域上拉普拉斯定理的许多经典定理已经被证明在一类叫做李普希茨域或锯齿域的粗糙域上成立。我和我的合作者已经获得了热方程的利普希茨域的模拟。另一个研究途径是考虑问题(a)-(c)在半空间粗糙抛物PDE。我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。

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).