Mathematical Sciences: Quasiconformal and Quasiregular Mappings and Elliptic Partial Differential Equations

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Postdoctoral funding through this award provides support for mathematical research on problems concerning boundary values of quasiconformal mappings and weighted Sobolev inequalities associated with quasiregular mappings. The problems addressed in the project contribute to the understanding of the interplay between analytic and geometric aspects in the analysis of these mappings. By quasiconformal mappings one understands univalent mappings between Euclidean spaces of the same dimension in which the distortion or dilatation of the mapping is bounded. Quasiregular mappings satisfy the same conditions but are not required to be univalent. Both reflect a natural outgrowth of the study of analytic functions of a complex variable. Work will be done examining the derivative of quasiconformal mappings. It had been thought that these functions always had powers which were locally integrable. This is not the case. However, a recently advanced notion of average derivative suggests that it is this function which will ultimately have the right power integrals (for conformal maps this average and conventional derivatives agree). Also to be examined are the radial boundary values of average derivatives in terms of the distance to the boundary. Other work focuses on changes in elliptic equations when composed with quasiconformal and quasiregular maps and on Sobolev mappings with integrable dilatation. In the latter, the fundamental question is one of showing that if powers of a derivative of an arbitrary map are bounded by the Jacobian times a bounded function then the mapping is open and discrete. The results has been validated in the two-dimensional case.