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

提出的研究涉及两个重要的分析领域：分形的拟共形变形和p-拉普拉斯方程的解。分形分析是近年来研究的热点。拟共形映射是介于同胚和差胚之间的一种中间类，是变形分形的自然候选者。在数学上，分形表现为迭代函数系统的不变集合。本文给出了迭代函数系统中这些不变集是拟对称等价的条件；将拟对称同胚扩展到全局拟共形映射的条件降低自相似分形的Hausdorff维数的方法；并在更一般的集合上回答相关问题。PI与J. Tyson合作，获得了Sierpinski垫片（一种特殊的分形）的结论性结果。在非线性弹性力学中，p-拉普拉斯方程的解使总能量最小。当p不等于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年代末，T. Wolff和J. Lewis提出了意想不到的例子，表明当p不等于2时，解的行为与p= 2时不同，即法图定理失效。最近，PI与R. Kaufman合作，将概率和离散思想纳入研究计划，并取得了进展，表明在半平面的情况下，某些p值的边界行为比以前已知的更差。PI建议研究整个p范围内的奇怪边界行为；研究了半平面边界上p-拉普拉斯算子的调和测度的性质；同时继续她在树形环境下的研究，这也是许多概率理论的起源。由于分形在自然界中随处可见，从蕨类植物到星系，因此了解两个看似不同的分形是如何通过变换联系在一起的，以及在变换过程中分形的维数是如何变化的是很重要的。提案第一部分的成功将回答其中的一些问题，这些问题应该引起生物学家、物理学家和数学家的兴趣。许多基于物理模型的微分方程是真正非线性的；而找到明确的解决方案通常是不可能的。为了了解解的性质，我们依赖于偏微分方程的估计。在这个项目中，离散分析和概率论的思想被引入来处理一些困难。该项目的成功将为研究一类非线性偏微分方程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、弹性和动力系统中具有实际应用。此外，关于树和分形的一些问题自然会引出适合PI计划指导的学生研究的问题。

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.