Mathematical Sciences: Regularity Problems in Nonlinear Potential Theory and Quasiregular Mappings

点击翻译按钮获取中文摘要

This project continues mathematical research aimed at exploiting newly discovered relationships between the classical theory of quasiregular mappings and nonlinear elliptic systems of variational equations. Quasiregular mappings first appeared as a modest branch of complex function theory (quasiconformal mapping). The mappings are characterized geometrically by the property that they carry infinitesimal spheres into infinitesimal ellipsoids. In the second half of this century it was discovered that these mappings proved to be fundamental objects in geometry, such a Teichmuller theory, as well as to other areas of analysis. The present work focuses on how the Donaldson - Sullivan work on quasiconformal four-manifolds has led to new techniques applicable to questions in nonlinear partial differential equations. One of the first objectives of this work is to understand what, if any, distinctions arise in the study of quasiregular mappings defined on even and odd dimensional spaces. Some of the deeper new discoveries are only known in the even dimensional case. Arguments based on Hodge decompositions and Cacciappoli type inequalities may yield additional information about odd dimensions. A second application of quasiregular mappings relates to vector-valued analogues of singular integral transformations consisting of matrices whose elements are Riesz transforms. It is believed that the p-norms of these operators do not depend on the dimension of the underlying space. Although it may be impossible to achieve exact values for the norms, work will be done in showing that the bounds are dimension-free. To achieve this, a new technique called the complex method of rotation has been introduced which applies to a broad class of integral operators in estimating their mapping norms. Finally work of a more geometric nature will continue on the questions of the maximum dimension of removable sets for quasiregular mapping and how Hausdorff dimension is distorted under such maps.