Maximal Subellipticity

椭圆型偏微分方程(PDE)在数学和科学的许多领域中扮演着重要的角色。椭圆型偏微分方程组的一个典型例子是拉普拉斯方程，它控制着稳态温度分布。椭圆型方程如此有用的一个原因是，对于非常一般的椭圆型偏微分方程组，即使在出了名的困难的完全非线性方程的设置下，也能得到精确的结果。在椭圆背景之外，目前的技术通常需要使用所考虑的方程的特殊性质，抽象的一般结果很少。极大次椭圆型方程理论起源于20世纪60年代和70年代，是椭圆型方程的一个深远的推广。在其间的几年里，许多数学家将椭圆理论的结果应用于线性极大次椭圆型方程的各种特殊情况。本课题将全面发展线性极大次椭圆偏微分方程正则性理论，并进一步研究完全非线性极大次椭圆偏微分方程解的一般情况。这将为在工作中遇到这种偏微分方程的数学家和科学家提供一个比椭圆理论中通常的工具箱更通用的工具箱。该项目将为研究生提供研究机会。该项目将分三个主要步骤发展极大次椭圆型偏微分方程理论。第一部分研究了具有光滑系数的一般线性极大次椭圆偏微分算子。以前曾考虑过特殊情况，但这将是这些算子的第一个完全一般化的理论。将使用的一个关键工具是底层的卡诺-Caratheodory几何体以及相关的缩放贴图。该项目的下一阶段将是适用于极大次椭圆算子的Besov和Triebel-Lizorkin函数空间理论的发展。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。前述的Besov和Triebel-Lizorkin空间将把这一事实推广到极大次椭圆环境。这些函数空间的特例包括适用于极大次椭圆算子的Soblev空间和Zygmund-Holder空间。该项目的第三阶段将涉及对完全非线性偏微分方程组的研究。如上所述的函数空间和线性算子理论将被用来理解完全非线性极大次椭圆型方程的内部正则性。在椭圆环境之外，完全的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背景下研究完全非线性方程提供一个框架。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.