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

拟议的研究涉及两个重要的分析领域：拟共形变形的分形和p-Laplace方程的解决方案。近年来，人们一直在积极研究分形分析。拟共形映射是介于同胚和复同胚之间的一类映射，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。该项目提供的条件下，这些不变集是拟对称等价的迭代函数系统;条件扩展等拟对称同胚的全球拟共形映射;降低Hausdorff维数的自相似分形的方法;并回答相关问题更一般的集。PI与J. Tyson合作，已经获得了Sierpinski垫片的结论性结果，这是一类特殊的分形。在非线性弹性力学中，p-Laplace方程的解使总能量最小。当p不等于2时，由于非线性和退化，解的边界行为在很大程度上仍然是一个谜。80年代末，T。Wolff和J.刘易斯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时的情况不同，即，法图定理失败了。最近，PI与R.考夫曼，将概率和离散的想法纳入研究计划，并取得了进展，表明在半平面的情况下，边界行为比以前已知的某些值的p。PI建议调查好奇的边界行为的全范围的p;研究性质的调和措施的p-拉普拉斯算子的边界上的半平面;由于分形在自然界中无处不在，从蕨类植物到星系，因此重要的是要找出两个看似不同的分形是如何通过变换联系起来的，以及分形的维数如何在变换过程中变化。该计划第一部分的成功将回答其中的一些问题，这应该是生物学家，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都是真正的非线性方程，并且通常不可能找到显式解。为了了解解的性质，人们依赖于偏微分方程的估计。在这个项目中，从离散分析和概率的想法被带到处理的一些困难。该项目的成功将为研究一大类非线性偏微分方程的边界行为提供新的工具，并将在非线性流体动力学、流体流动、弹性力学和动力系统等方面有实际应用。此外，一些关于树和分形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.