Analysis and Geometry of Nonlinear PDEs

近年来，亚黎曼空间的分析和几何问题越来越受到人们的关注。亚黎曼环境的典型例子是所谓的卡诺群，它在分析中的基本作用是由e.m.斯坦因首先强调的。它们现在不仅在数学领域如半椭圆偏微分方程、调和分析和CR几何函数理论中占据中心地位，而且在应用科学（如数学金融、机械工程、大脑神经生理学）中也占据中心地位。亚黎曼空间最显著的特征是度量结构可以被看作是一个受限的几何，其中运动只能沿着一组规定的方向，从一点到另一点改变。首席研究员有一个长期的项目，旨在探索这些结构的几何和分析性质。更具体地说，她建议继续研究Bernstein问题和Carnot群中最小曲面的正则性，研究亚椭圆边值问题，并发展Monge-Ampere型全非线性方程的正则性理论。本项目的另一个兴趣领域是研究火焰传播理论中自然产生的椭圆和抛物自由边界问题。首席研究员还打算研究一类最小化问题，其中相关的泛函是在Alt和Caffarelli引入的泛函的基础上建模的。提出的研究的主要目的之一是证明自由边界的正则性。谐波分析和偏微分方程研究这些问题的必要工具将同时发展。最后，受欧几里得最小曲面理论和自由边界理论之间惊人的相似性的激励，PI计划将她不同的研究方向合并到一个尚未开发的领域，即卡诺群中的自由边界问题（障碍和阿尔卡法雷利型）的研究。首席研究员将把她的研究计划与一些教育、指导和外展活动结合起来。本项目将在变分学、偏微分方程和几何测度理论的交叉领域开展研究。本课程的重点是研究涉及非交换向量场系统的变分不等式和偏微分方程解的解析和几何性质。所考虑的问题不仅出现在各种数学背景中（例如，最优控制理论，数学金融和几何），而且还引起了其他领域的兴趣，如机械工程，机器人技术和神经生理学。另一个提议的研究领域涉及自由边界问题，当一个守恒量或关系在考虑的变量的某些值上不连续地变化时，自然会出现在物理和工程中。例如，自由边界表现为流体与空气之间或水与冰之间的界面。该项目的一部分目的是研究燃烧-未燃烧混合物中自由边界的规律性。这项研究的结果将有助于更好地理解模型，改进模拟方法，并最终精确描述火焰如何在非均匀介质中传播。这个项目的几个要素在应用科学中找到了它们的动机。另一方面，这些问题的解决方案涉及来自不同分析和几何领域的思想的相互作用。可以想象，所有这些不同的领域都将从这种协同作用中受益。首席研究员致力于培养未来一代的数学家，并通过为研究生、本科生和K-12学生组织各种教育和指导活动来增加女性在科学界的代表性。

In recent years, the analysis and geometry of sub-Riemannian spaces has received increasing attention. The quintessential examples of sub-Riemannian settings are the so-called Carnot groups, whose fundamental role in analysis was first highlighted by E. M. Stein. They now occupy a central position not only in such mathematical areas as hypoelliptic partial differential equations, harmonic analysis, and CR geometric function theory, but also in the applied sciences (e.g., mathematical finance, mechanical engineering, neurophysiology of the brain). The most distinctive feature of sub-Riemannian spaces is that the metric structure can be viewed as a constrained geometry, where motion is only possible along a prescribed set of directions, changing from point to point. The principal investigator has a long-term project aimed at exploring geometric and analytic properties of these structures. More specifically, she proposes to continue her study of the Bernstein problem and of the regularity of minimal surfaces in Carnot groups, to investigate subelliptic boundary value problems, and to develop a regularity theory for fully nonlinear equations of Monge-Ampere type. Another area of interest in this project is the investigation of elliptic and parabolic free boundary problems naturally arising in the theory of flame propagation. The principal investigator also intends to study a class of minimization problems, in which the relevant functional is modeled after the one introduced by Alt and Caffarelli. One of the main objectives of the proposed research is to prove regularity properties of the free boundary. The necessary tools from harmonic analysis and partial differential equations for the study of these problems will be developed concurrently. Finally, motivated by the striking analogy between the theories of minimal surfaces and of free boundaries in the Euclidean setting, the PI plans to merge her different lines of research into a yet quite unexplored area, namely, the study of free boundary problems (both of obstacle and Alt-Caffarelli type) in Carnot groups. The principal investigator will integrate her research plan with several educational, mentoring, and outreach activities. This project will conduct research that lies at the interface of calculus of variations, partial differential equations, and geometric measure theory. The focus is on the study of analytic and geometric properties of solutions to variational inequalities and partial differential equations involving a system of noncommuting vector fields. The problems under consideration not only arise in a variety of mathematical contexts (e.g., optimal control theory, mathematical finance, and geometry), but also are of interest in other fields such as mechanical engineering, robotics, and neurophysiology. Another proposed research area concerns free boundary problems, which naturally arise in physics and engineering when a conserved quantity or relation changes discontinuously across some value of the variables under consideration. The free boundary appears, for instance, as the interface between a fluid and the air, or between water and ice. Part of the project aims at studying regularity properties of the free boundary in burnt-unburnt mixtures. The results of this investigation will lead to a better understanding of the models, to the improvement of simulation methods, and ultimately to a precise description of how flames propagate in nonhomogeneous media. Several elements of this project find their motivations in the applied sciences. On the other hand, the solutions to these probelms involve an interplay of ideas from different areas of analysis and geometry. It is conceivable that all these different fields will benefit from this synergy. The principal investigator is committed to the training of future generations of mathematicians, and to increasing the representation of women in the scientific community, via the organization of a variety of educational and mentoring activities for graduate, undergraduate, and K-12 students.