Mathematical Sciences: Partial Differential Equations and Classical Analysis

本项目将主要侧重于在 椭圆型偏微分方程理论 他们是 在某种程度上是由拟正则性的考虑引起的 n维欧氏空间中的映射 这些被认为 作为对经典分析的高维数的自然扩展， 功能理论 此外，还得到了拟线性椭圆型方程的解 用偏微分方程代替调和函数。 的 n维拉普拉斯的共形不变扩张 这就是所谓的N-Laplacian。 本文将在有界弱 具有有界梯度的拟线性方程的解 平均振荡 即使在线性情况下，这个问题仍然是 打开. 方程边值振动性的估计 将寻求梯度。 另一条调查线将集中在是否存在 分支拟正则映射 当地图足够 光滑，没有分支发生。 最小平滑假设 不知道。 最后，对抛物型发散型进行了研究 用于确定保持器连续性是否可以预期的方程 有界弱解 这项工作涉及到保形几何的研究， 潜力理论

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.