Quasiconformal Analysis and the p-Laplacian

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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.