Quasiconformal Deformation of Self-similar Sets and Fatou Theorems for p-Laplacian

所提出的研究涉及两个重要的分析领域：拟共形变形副产物和p-Laplace方程的解。近几年来，对分形学的研究一直很活跃。拟共形映射是介于同胚和微分同胚之间的一类中间映射，是变形分形的自然候选者。在数学上，分形图表现为迭代函数系统的不变集。给出了迭代函数系统中这些不变集是拟对称等价的条件；将这种拟对称同胚推广到全局拟共形映射的条件；降低自相似分形图的Hausdorff维的方法；以及更一般集上的相关问题的答案。PI与J·泰森合作，已经对Sierpinski垫片--一类特殊的分形学--得到了确凿的结果。在非线性弹性力学中，p-Laplace方程的解使总能量最小。当p不同于2时，由于解的非线性和简并性，解的边界行为在很大程度上仍然是一个谜。在80年代后期，S、沃尔夫和J·刘易斯给出了意想不到的例子，表明当p不等于2时，解的行为与当p=2时不同，即Fatou定理失败。最近，PI与R.Kaufman合作，将概率和离散的思想融入到研究程序中，并取得了进展，表明在半平面的情况下，对于p的某些值，边界行为比以前已知的更差。PI建议研究全p范围内奇怪的边界行为；研究p-Laplace算子在半平面边界上的调和测度的性质；此外，她还将继续她在树环境中的工作，许多概率概念都是在树的背景下产生的。由于从蕨类植物到星系，自然界中到处都有分形图，因此找出两个看似不同的分形图是如何通过变换联系在一起的，以及在变换过程中分形图的维度是如何变化的，这一点很重要。提案第一部分的成功将回答其中一些问题，这些问题应该是生物学家、物理学家和数学家感兴趣的。许多基于物理模型的微分方程是真正的非线性方程；而且通常不可能找到显式解。要了解解的性质，需要依靠偏微分方程解的估计。在这个项目中，引入了离散分析和概率论的思想来处理一些困难。该项目的成功将为研究一大类非线性偏微分方程组的边界行为提供新的工具，并将在非线性流体力学、流体流动、弹性力学和动力系统中有实际的应用。此外，一些关于树和分形学的问题自然会导致适合学生研究的问题，PI计划指导这些问题。

The proposed research deals with two important areas of analysis: quasiconformal deformation offractals and solutions of p-Laplace equations. Analysis on fractals has been actively pursued in recent years. Quasiconformal mappings, an intermediate class between homeomorphisms and diffeomorphisms, are natural candidates for deforming fractals. Mathematically, fractals appear as invariant sets of iterated function systems. This project provides conditions on the iterated function system under which these invariant sets are quasisymmetrically equivalent; conditions for extending such quasisymmetric homeomorphisms to global quasiconformal mappings; methods of lowering the Hausdorff dimension of self-similar fractals; and answers to related questions on more general sets. The PI, in collaboration with J. Tyson, has obtained conclusive results for Sierpinski gaskets, a special class of fractals. In nonlinear elasticity, solutions of p-Laplace equation minimize the total energy. When p is different from 2, due to the nonlinearity and degeneracy, the boundary behavior of solutions is still largely a mystery. In the late 80's, T. Wolff and J. Lewis produced unexpected examples showing, when p was not equal to 2, solutions behave differently from the case when p= 2, i.e., Fatou's theorem fails. Recently the PI, in collaboration with R. Kaufman, incorporated probabilistic and discrete ideas into the research program, and made progress by showing that in the case of the half plane the boundary behavior is worse than previously known for certain values of p.The PI proposes to investigate the curious boundary behavior for the full range of p; to study properties of harmonic measure for the p-Laplacian on boundary of the half plane; and also to continue her work in the tree setting, where many of the probabilistic ideas originated.As fractals appear everywhere in nature from ferns to galaxies, it is important to find out how two seemingly different fractals are related through transformations, and how the dimension of a fractal changes during the transformation. The success of the first part of proposal willanswer some of these questions, which should be of interest to biologists, physicists and mathematicians. Many differential equations based on physical models are genuinely nonlinear; and finding explicit solutions is usually impossible. To learn the nature of the solutions, one relies on estimates from partial differential equations. In this project, ideas from discrete analysis and probability are brought in to handle some of the difficulties. The success of this project will provide new tools to study the boundary behavior of a large class of nonlinear partial differential equations and will have practical applications to nonlinear hydrodynamics, fluid flow, elasticity and dynamical systems. Moreover, some questions on trees and on fractals lead naturally to problems suitable for student research, which the PI plans to direct.