Mathematical Sciences: The Degree of Regularity of Quasiregular Mapping and Minima of Related NondifferentiableFunctions in the Calculus of Variations

这项工作将集中在欧几里德空间的拟共形和拟正则映射上。这些映射-(区别在于拟共形映射是单叶的)-出现在许多分析和几何环境中。它们在我们对偏微分方程类的理解中起着重要作用。要研究的问题是密切相关的，其动机既有理论上的兴趣，也有它们对椭圆型方程正则性理论和奇异积分估计的适用性。拟正则映射是n维Beltrami偏微分方程组的解，它可能有间断项。这项工作的目标之一将是在这些映射的合适类别中找到总能量的最小化。与这种最小化有关的变分问题出现在非线性弹性静力学中。人们已经认识到，度量拟正则映射的正则度以及研究不可微泛函的MIMIMA的正确方法是通过Holder指数或其导数的可积性指数。这项工作的第二个目标将是确定最佳指数，或者至少描述允许指数的数量特征。相关工作将寻求寻找二维希尔伯特变换的精确估计，并检查变换的幂范数的渐近行为。

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.