Maximal Subellipticity

椭圆偏微分方程（PDE）在数学和科学的许多领域发挥着核心作用。椭圆偏微分方程的一个典型例子是控制稳态温度分布的拉普拉斯方程。椭圆方程如此有用的一个原因是，对于非常一般的椭圆偏微分方程，即使在众所周知的完全非线性方程的困难设置中，精确的结果也是众所周知的。在椭圆设置之外，目前的技术通常需要使用所考虑的方程的特殊性质，而抽象的一般结果很少。极大次椭圆方程理论是对椭圆方程的广泛推广，起源于20世纪60年代和70年代。在此期间，许多数学家将椭圆理论的结果应用于线性极大次椭圆方程的各种特殊情况。本课题将全面发展线性最大次椭圆偏微分方程的正则性理论，并解决完全非线性最大次椭圆偏微分方程的一般问题。这将为在工作中遇到偏微分方程的数学家和科学家提供一个比椭圆理论更通用的工具箱。该项目将为研究生提供研究机会。本课题将分三个主要步骤发展极大次椭圆偏微分方程的理论。首先，研究了一类具有光滑系数的一般线性极大次椭圆偏微分算子。以前已经考虑过特殊情况，但这将是第一个完全一般的关于这些算子的理论。将使用的一个关键工具是底层的Carnot-Caratheodory几何图形以及相关的缩放地图。该项目的下一阶段将是Besov和triiebel - lizorkin函数空间理论的发展，这些函数空间最大限度地适应亚椭圆算子。椭圆算子的一个重要性质是，模光滑函数在许多经典函数空间上是左可逆的。前面提到的Besov和triiebel - lizorkin空间将这一事实推广到最大的亚椭圆设置。这些函数空间的特殊情况包括Sobolev和Zygmund-Holder空间，它们适用于最大次椭圆算子。该项目的第三阶段将涉及全非线性偏微分方程的研究。上述的函数空间理论和线性算子将用于理解完全非线性极大次椭圆方程的内部正则性。在椭圆型设置之外，完全非线性方程通常很难研究。这个项目的结果将为未来在最大次椭圆设置下的完全非线性方程的研究提供一个框架。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Elliptic partial differential equations (PDE) play a central role in many areas of mathematics and science. A canonical example of an elliptic PDE is Laplace’s equation, which governs steady-state temperature distributions. One reason that elliptic equations are so useful is that precise results are known for very general elliptic PDE, even in the notoriously difficult setting of fully nonlinear equations. Outside of the elliptic setting, current techniques usually require the use of special properties of the equation under consideration, and abstract general results are rare. The theory of maximally subelliptic equations, a far-reaching generalization of elliptic equations, originated in the 1960s and 1970s. In the intervening years, many mathematicians have adapted results from the elliptic theory to various special cases of linear maximally subelliptic equations. This project will develop the regularity theory of linear maximally subelliptic PDE in full generality, and moreover will address the general situation of fully nonlinear maximally subelliptic PDE. This will provide a toolbox, more general than the usual one from the elliptic theory, for mathematicians and scientists who encounter such partial differential equations in their work. The project will provide research opportunities for graduate students.This project will develop the theory of maximally subelliptic partial differential equations in three main steps. The first is a study of general linear maximally subelliptic partial differential operators with smooth coefficients. Special cases have previously been considered, but this will be the first such theory of these operators in full generality. A key tool which will be used is the underlying Carnot-Caratheodory geometry along with associated scaling maps. The next stage of the project will be a development of the theory of Besov and Triebel-Lizorkin function spaces adapted to maximally subelliptic operators. An important property of elliptic operators is that, modulo smooth functions, they are left invertible on many classical function spaces. The aforementioned Besov and Triebel-Lizorkin spaces will generalize this fact to the maximally subelliptic setting. Special cases of these function spaces include both Sobolev and Zygmund-Holder spaces adapted to a maximally subelliptic operator. The third stage of the project will involve a study of fully nonlinear partial differential equations. The theory of function spaces and linear operators as described above will be used to understand the interior regularity of fully nonlinear maximally subelliptic equations. Outside of the elliptic setting, fully nonlinear equations are often difficult to study. The results of this project will provide a framework for future study of fully nonlinear equations in the maximally subelliptic setting.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.