Mathematical Sciences: Subelliptic Partial Differential Equations and Harmonic Analysis

这个项目的工作将解决几个问题 关于次椭圆型偏微分方程，特别是 解析hypoellipticity。 与《公约》有关的工作 幂零李群的表示与调和分析 也将进行。 微分算子被称为 亚椭圆的，如果对应的解的光滑性 非齐次方程与非齐次方程有关 术语（通常简称为 方程）。 通过解析，一个意思是解析解是 当非齐次项为解析项时保证。 少数情况 椭圆算子的解析亚椭圆性出现的地方是 已知的和一些它失败的地方也是已知的。 广阔的中部 地面是未触及的，代表了这一主题 research. 此外，在亚椭圆率 失败了，它仍然要确定到底是什么顺序的平滑 一个人可以期望拥有并将其与潜在的 运算符所在域的几何不变量 定义了 这项工作始于三维欧几里得空间， 但对类似的幂零问题有自然的推广 像海森堡群这样的群，以及紧致流形 有边界。 这项研究与基金会有着很深的联系 偏微分方程 从表面上看， 高度抽象，远离现实世界。 在 实践中，基本成果来自于对 在各种应用中出现的具体例子。 去给予， 先验，关于复偏解的信息 微分方程可以是巨大的价值，任何用户 数学

Work on this project will address several questions concerning subelliptic partial differential equations, especially aspects of analytic hypoellipticity. Related work on the representations of nilpotent Lie groups and harmonic analysis will also be carried out. A differential operator is said to be hypoelliptic if the smoothness of a solution of the corresponding inhomogeneous equation is related to that of the inhomgeneous term (often referred to simply as the right-hand-side of the equation). By analytic, one means that analytic solutions are guaranteed when the inhomogeneous term is analytic. A few cases where analytic hypoellipticity for elliptic operators occurs are known and a few where it fails are also known. The vast middle ground is untouched and represents the main theme of this research. Additionally, in those cases where hypoellipticity fails, it remains to determine exactly what order of smoothness one can expect to possess and to relate it to underlying geometric invariants of the domain on which the operator is defined. The work begins in three-dimensional Euclidean space, but has natural extensions to similar question on nilpotent groups like the Heisenberg group, as well as to compact manifolds with boundary. This research has many deep connections with the foundations of partial differential equations. On the surface it appears highly abstract and distant from the physical world. In practice, the fundamental results derive from the study of specific examples which occur in various applications. To give, a priori, information about the solution of a complex partial differential equation can be of immense value to any user of mathematics.