Analytic and geometric properties of variational inequalities and PDE

近年来，次黎曼空间的分析和几何学受到越来越多的关注。亚黎曼背景的典型例子是所谓的卡诺群，它在分析中的基本作用首先由E。M.斯坦他们现在占据中心地位，不仅在研究亚椭圆偏微分方程，调和分析，几何函数理论，而且在应用科学，如数学金融，机械工程，和神经生理学的大脑。次黎曼空间最显著的特征是，度量结构可以被看作是一个受约束的几何，其中运动只能沿着一组指定的方向沿着进行，从点到点变化。 首席研究员有一个长期项目，旨在研究这些结构的几何和分析特性。更具体地说，她建议继续她的研究的伯恩斯坦问题和规律性的最小表面的卡诺集团，调查次椭圆边值问题，并制定了规律性理论完全非线性方程的蒙赫安培型。 在这个项目中感兴趣的另一个领域是椭圆和抛物线的自由边界问题，自然产生的火焰传播理论的调查。主要研究者还打算研究一类最小化问题，其中相关的功能是仿照一个由Alt和Caffarelli介绍。此外，她有兴趣探索变分不等式的椭圆和抛物型的障碍局限于躺在低维流形。所提出的研究的主要目标之一是证明自由边界的正则性。从调和分析和偏微分方程理论的研究这些问题的必要工具将同时开发。最后，激发了惊人的相似理论之间的最小曲面和自由边界的欧几里德设置，主要研究计划合并她的不同线的研究到一个尚未探索的领域，即研究自由边界问题（障碍和阿尔特-卡法雷利型）在卡诺集团。首席研究员有一个研究计划，该计划位于数学领域的界面，称为变分法，偏微分方程和几何测量理论。重点是研究所谓的变分不等式和偏微分方程的解决方案的分析和几何性质，涉及一个系统的“noncommuting”向量场。所提出的问题不仅出现在各种数学背景下（例如，最优控制理论、数学金融学和几何学），但在其他领域也有兴趣，如机械工程、机器人和神经生理学。该项目的第二个重点是自由边界问题，它在物理和工程中的表面，在这种情况下，守恒量或关系在考虑的变量的某些值上不连续地变化。例如，自由边界表现为流体和空气之间的界面，或水和冰之间的界面。 其中一个项目旨在研究燃烧-未燃烧混合物中自由边界的正则性。这项调查的结果将导致更好地理解模型，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。 如前所述，这个项目的几个部分在应用科学中找到了动力。另一方面，他们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受益于这种协同作用。首席研究员致力于培养未来几代数学家，并通过为非终身教职的教师和研究生，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 increased 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 the study of hypoelliptic partial differential equations, harmonic analysis, and geometric function theory, but also in the applied sciences such as mathematical finance, mechanical engineering, and the 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 possible only along a prescribed set of directions, changing from point to point. The principal investigator has a long-term project aimed at investigating 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 that arise naturally 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. In addition, she is interested in exploring variational inequalities of elliptic and parabolic type with obstacles confined to lie in lower dimensional manifolds. 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 the theory of partial differential equations for the study of such 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 principal investigator 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 has a research program that lies at the interface of the areas of mathematics known as the calculus of variations, partial differential equations, and geometric measure theory. The focus is on the study of analytic and geometric properties of solutions to so-called variational inequalities and partial differential equations involving a system of "noncommuting" vector fields. The proposed problems not only turn up in a variety of mathematical contexts (e.g., optimal control theory, mathematical finance, and geometry) but are also of interest in other fields such as mechanical engineering, robotics, and neurophysiology. A second focus of the project concerns free boundary problems, which surface in physics and engineering in situations where 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. One of the proposed projects 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. As mentioned earlier, several parts of this project find their motivation in the applied sciences. On the other hand, their solutions 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 untenured faculty and graduate, undergraduate, and K-12 students.