Mathematical Sciences: Partial Differential Equations and Classical Analysis

本课题将主要关注椭圆型偏微分方程组理论中出现的问题。它们在某种程度上是由n维欧氏空间中拟正则映射的考虑所激发的。这些都被认为是对经典解析函数理论更高维的自然扩展。此外，拟线性椭圆型偏微分方程解取代了调和函数。拉普拉斯算子对n维的共形不变扩张称为N-拉普拉斯算子。在寻求拟线性方程有界弱解具有有界平均振动梯度的条件方面，将做一些工作。即使在线性情况下，这个问题仍然是悬而未决的。我们将寻求关于梯度边值振荡的估计。另一项研究将集中在分支拟正则映射的存在性上。当贴图足够平滑时，不会发生分支。最小光滑度假设是未知的。最后，我们将研究抛物型散度型方程，以确定是否可以期望有界弱解具Holder连续性。这项工作与共形几何和位势理论的研究有关。

This project will focus primarily on problems arising in the theory of elliptic partial differential equations. They are motivated to some extent by considerations of quasiregular mappings in n-dimensional Euclidean space. These are considered as natural extensions to higher dimension of classical analytic function theory. In addition, solutions of quasilinear elliptic partial differential equations replace harmonic functions. The conformally invariant extension to n-dimensions of the Laplace operator is the so-called N-Laplacian. Work will be done in seeking conditions on bounded weak solutions of quasilinear equations to have gradients of bounded mean oscillation. Even in the linear case this question is still open. Estimates on the oscillation of boundary values of the gradient will be sought. Another line of investigation will focus on the existence of branched quasiregular maps. When the maps are sufficiently smooth, no branching occurs. The minimal smoothness assumptions are not known. Finally, studies will be made of parabolic divergence type equations to establish whether Holder continuity can be expected of bounded weak solutions. This work is related to research in conformal geometry and potential theory.