Mathematical Sciences: Regularity Problems for Variational Integrals and Quasiregular Mappings

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There are two main themes to be followed in this project. One concerns questions of regularity involving quasi-regular mappings. These are mappings of the plane or higher dimensional space in which distortion of small sets such as discs is controlled in a measurable way. The maps are not required to be univalent (quasi-conformal) and they are characterized as solutions of systems of elliptic partial differential equations (of degenerate type). One of the primary concerns in the study of quasi-regular mappings is that of finding estimates on the growth of such mappings as well as characterizing extremal mappings subject to given boundary conditions. In two dimensions, one is eventually led to the complex Hilbert transform. Various regularity results depend on estimates of the norm of this transform, specifically the p-th power norm. Work will be done on sharpening estimates for the norm. A second line of investigation will focus on variational energy integrals of classes of functions whose stationary points are quasi-conformal. In particular, work will be done in estimating energy integrals over subsets of a given domain in terms of the energy integral over the full domain, multiplied by a power of the measure of the subset. The best value of the power remains to be determined. At this time, the only known results are in the case where the domain is a solid disc or ball.