Mathematical Sciences: Quasiconformal Mappings and NonlinearPotential Theory

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Quasiconformal mappings evolved from studies of plane mappings whose infinitesimal distortions remained bounded within fixed limits. Such transformations were considered as the natural mathematical generalization of a conformal map (which infinitesimally maps circles onto circles). The theory has grown in several directions and dimensions, making important contact with nonlinear potential theory and Teichmuller theory of Riemann surfaces. This project is concerned with problems relating quasiconformal maps and the geometric behavior of solutions to degenerate elliptic partial differential equations. The work derives from a recent discovery that quasiconformal homeomorphisms mapping onto a ball have a stability property: in certain subsets of the domain, their distortion is globally controlled without regard to the geometry of the domain. Little is understood concerning domains which may be mapped onto a ball; work will be done to characterize such domains. This phenomenon was first observed in conformal maps where several deep results has subsequently been produced. A second line of investigation concerns the properties of supersolutions of the p-Laplace equation. These functions form the basis for a nonlinear potential theory. Two particular goals the study of possible fine topologies available which yield the best continuity results for such functions and to seek a boundary Harnack principle for domains other than a ball.