Work to be carried out on this project will focus on quasiconformal and quasiregular mapping of Euclidean space. These mappings - (the distinction being that quasiconformal maps are univalent) - arise in a number of analytical and geometric contexts. They play important roles in our understanding of classes of partial differential equations. The problems to be investigated are closely related, motivated by both theoretical interest as well as by their applicability to regularity theory for elliptic equations and estimates of singular integrals. Quasiregular mappings are known to be solutions of n- dimensional Beltrami systems of partial differential equations, which may have discontinuous terms. One of the objectives of this work will be to find minimizers for the total energy among suitable classes of these mappings. The variational problems associated with this minimization arise in nonlinear elastostatics. It has been recognized that the correct way of measuring the degree of regularity of a quasiregular mapping as well as studying the mimima of non-differentiable functionals is through the Holder exponent or by the exponent of integrability of their derivatives. A second objective of this work will be to identify the best exponents or at least describe the quantitative character of the allowable exponents. Related work will seek to find sharp estimates for two-dimensional Hilbert transforms and to examine the asymptotic behavior of the power norms of the transform.
