The proposed research deals with two important areas of analysis: quasiconformal deformation of fractals 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 will answer 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.
