Maximal Subellipticity

最大次椭圆度

基本信息

批准号：
2153069
负责人：
Brian Street
金额：
$ 34.47万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153069&HistoricalAwards=false
关键词：
Maximal Subellipticity

项目摘要

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.

椭圆形部分微分方程（PDE）在许多数学和科学领域都起着核心作用。椭圆PDE的一个规范示例是Laplace的方程，它控制稳态温度分布。椭圆方程是如此有用的原因之一是，即使在完全非线性方程的众所周知的困难环境中，精确的结果也以非常通用的椭圆PDE而闻名。在椭圆设置之外，当前技术通常需要使用所考虑的方程式的特殊属性，而抽象的一般结果很少。最大的升压方程理论是椭圆方程的深远概括，起源于1960年代和1970年代。在随后的几年中，许多数学家将椭圆理论的结果调整为线性最大sublipic方程的各种特殊情况。该项目将在完全普遍的情况下发展线性最大sublipic PDE的规律性理论，此外，将解决完全非线性最大亚假桥PDE的一般情况。对于椭圆理论中遇到这种部分微分方程的数学家和科学家来说，这将提供一个工具箱，比椭圆理论的通常的工具箱更通用。该项目将为研究生提供研究机会。该项目将在三个主要步骤中开发最大亚抗性偏微分方程的理论。首先是对具有光滑系数的一般线性最大次数差分运算符的研究。以前已经考虑过特殊情况，但这将是这些运营商完全普遍的第一个这样的理论。将使用的关键工具是基础的Carnot-Caratheodory几何形状以及相关的缩放图。该项目的下一个阶段将是BESOV和Triebel-Lizorkin函数空间的发展，适用于最大的亚足型操作员。椭圆运算符的一个重要属性是，Modulo平滑函数在许多经典函数空间上都是可逆的。 Priore提到的BESOV和Triebel-lizorkin空间将把这一事实推广到最大的亚洲道路设置。这些功能空间的特殊情况包括Sobolev和Zygmund-Holder空间，适用于最大的亚颌下算子。该项目的第三阶段将涉及对完全非线性偏微分方程的研究。如上所述，功能空间和线性运算符的理论将用于了解完全非线性最大马上方程的内部规则性。在椭圆环境之外，完全非线性方程通常很难研究。该项目的结果将为未来在最大抗下层环境中进行完全非线性方程的研究提供一个框架。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响评估标准来评估值得支持。