Topics in PDE's and Quasiconformal Mappings

DMS-9876881刘易斯这个项目的目的是进一步研究我在(A)抛物型和椭圆型测度，(B)拟共形映照和(C)偏微分方程正则性方面的工作中的一些问题.在(A)项下，我想知道Dirichlet问题对某些带阻力项的抛物型和椭圆型偏微分方程何时有解.鉴于这些偏微分方程解的Dirichlet问题是有解的，我想知道相应的度量何时具有基本性质，如加倍性质。关于(B)，我想构造更多关于拟共形于球面且其边界上的调和测度与n-1维Hausdorff测度相等的区域的例子。最后，在(C)项下，我想知道用来证明抛物型拉普拉斯方程组解的逆Holder不等式的技巧是否可以用于其他偏微分方程组，如Navier Stokes方程。许多物理问题都可以用偏微分方程组(PDE)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在19世纪和20世纪带来了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)是否存在解，(B)它是否唯一，(C)它是否具有良好的性质或它是否规则？至于(A)和(B)，人们常常关心存在域中的解的所谓边值或边界条件。所谓的超定边值问题没有解，而Dirichlet和Neumann问题等经典问题有解，如果给定域的边界和边界条件足够好(光滑)。我的工作关注的是，人们可以在多大程度上放松这些假设，同时仍然得到有意义的定理。例如，在过去的四分之一世纪中，许多关于光滑区域上的拉普拉斯的经典定理已被证明适用于一类称为Lipschitz或锯齿区域的粗糙区域。我和我的合作者已经得到了热方程的Lipschitz域的类比。研究的另一个途径是在半空间中考虑粗糙抛物型偏微分方程组的问题(A)-(C)。我的工作为某些自由边界问题提供了一个模型，如冰融化(Stefan问题)。

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).