Topics in PDE's and Quasiconformal Mappings

本项目的目的是进一步研究（a）抛物和椭圆测度，（B）拟共形映射，（c）偏微分方程的正则性等工作中的一些问题。在（a）中，我想知道狄利克雷问题对于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的狄利克雷问题有一个解决方案，我想知道当相应的措施拥有基本的属性，如加倍财产。对于（B），我想构造更多的关于球面的拟共形域的例子，并且对于这些域，调和测度和边界上的n - 1维Hausdorff测度相等。最后，在（c）中，我想知道用于证明抛物型p拉普拉斯方程系统解的反向保持器不等式的技术是否可以用于其他偏微分方程，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。在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世纪导致了许多基本的技术进步。那些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）它是唯一的，（c）它具有好的性质还是正则的？ 至于（a）和（B），人们通常关心的是存在域中解的所谓边界值或边界条件。所谓超定边值问题无解，而经典问题如Dirichlet和Neumann问题有解，只要给定区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，仍然得到有意义的定理。例如，在过去的四分之一世纪，许多经典定理的拉普拉斯在光滑域已被证明持有一类粗糙域称为Lipschitz或锯齿域。 我和我的合著者已经得到了热方程的Lipschitz域的模拟。 另一种研究途径是在半空间中考虑粗糙抛物型偏微分方程的问题（a）-（c）。 我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）。

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).