Quasiconformal Analysis and the p-Laplacian

本项目研究拟共形分析和非线性位势理论中的基本问题。具体主题包括拟正则映射的分支、拟共形雅可比和p-调和函数的边界行为。拟正则映射是解析函数从复平面到欧氏空间的几何推广。光滑拟正则映射的分支性质问题是几何函数论和拓扑学的交叉点，它与紧流形上雪花嵌入的存在性、拟共形分解和扩张以及拟共形结构的加细有着密切的联系。拟共形雅可比问题研究了从指定的体积比重建拟共形映射的可能性和方法。拟共形映射和拟正则映射都具有有界失真的特征，并且都是与n-拉普拉斯相关的方程的解。当p不同于2时，由于p-Laplace方程的非线性和简并性，其解的性质在很大程度上仍然是一个谜。最近首席研究员和她的合作者的一个例子表明，p-拉普拉斯的解可能表现出比沃尔夫和刘易斯之前所展示的更糟糕的行为。在这个项目中，主要的研究者将继续研究p-调和测度的支撑的维度，相关的Fatou集的大小，以及当边界函数呈现快速增加的频率时p-调和方程的解的增长。概率中的方法将被用来处理一些分析困难。这个项目集合了数学的几个领域，如果成功，将为解决几何分析和势论中的难题提供新的工具。没有光滑结构的物体在物理和生物学中自然出现。拟共形映射具有有界失真，非常适合于研究非光滑结构。最近，在应用准共形映射来研究大脑皮质表面的图像和确定身体的电导率方面，有了令人兴奋的发现。P-Laplacian既存在于流体力学中的多孔介质方程中，也存在于非线性弹性力学中。由于这个原因，它在许多物理问题上都有应用。这个方案中的项目涉及有界失真函数的内在性质和非线性方程的解。这些发现可能会在上述应用中发挥重要作用。

This project investigates fundamental questions in quasiconformal analysis and nonlinear potential theory. Specific topics include the branching of quasiregular maps, quasiconformal Jacobians, and the boundary behavior of p-harmonic functions. Quasiregular maps are geometrical generalizations of analytic functions from the complex plane to Euclidean spaces. Problems on branching properties of smooth quasiregular mappings lie at the intersection of geometric function theory and topology, and, they have close connections to the existence of snowflake embeddings, quasiconformal decomposition and extension, and refinement of quasiconformal structures on compact manifolds. The quasiconformal Jacobian problem studies the possibility of, and the method for, reconstructing quasiconformal maps from assigned volume ratios. Both quasiconformal and quasiregular maps have the characteristic of bounded distortion and are solutions to equations related to the n-Laplacian. When p is different from 2, because of the nonlinearity and the degeneracy of the p-Laplace equation the nature of its solutions is still largely a mystery. A recent example of the principal investigator and her collaborators suggests that solutions to the p-Laplacian may exhibit even worse behavior than previously shown by Wolff and Lewis. In this project, the principal investigator will continue to investigate the dimension of the support of the p-harmonic measure, the size of the associated Fatou sets, and the growth of solutions to the p-harmonic equation when the boundary functions exhibit rapidly increasing frequencies. Methods from probability will be used to handle some of the analytical difficulties. This project brings together several areas of mathematics, and if successful, will provide new tools for attacking difficult problems in geometric analysis and potential theory.Objects that do not have smooth structure appear naturally in physics and biology. Quasiconformal mappings, having bounded distortion, are very suitable for studying nonsmooth structures. Recently, there have been exciting discoveries in applying quasiconformal mappings to study images of brain cortical surfaces and to determine the conductivity of a body. The p-Laplacian occurs both in the porous medium equation from fluid dynamics and in nonlinear elasticity. For this reason it has applications to many physical problems. The project in this proposal deals with intrinsic properties of functions of bounded distortion and solutions of nonlinear equations. The findings will likely play an important role in the aforementioned applications.