Analytic and geometric properties of variational inequalities and PDE

变分不等式和偏微分方程的解析和几何性质

• 批准号: 1101246
• 负责人: Donatella Danielli
• 金额: $ 22.48万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-15 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101246&HistoricalAwards=false
• 关键词: Analytic; geometric; properties; variational; inequalities

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.

近年来，次黎曼空间的分析和几何受到越来越多的关注。次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.Stein首先强调的。它们现在不仅在亚椭圆型偏微分方程、调和分析和几何函数理论的研究中占据中心地位，而且在数学金融、机械工程和大脑的神经生理学等应用科学中也占有中心地位。次黎曼空间最明显的特征是，度量结构可以被视为一个受约束的几何，其中运动只能沿着一组指定的方向，从一个点到另一个点变化。首席研究员有一个长期项目，旨在调查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问题和卡诺群中极小曲面的正则性，研究亚椭圆边值问题，并发展Monge-Ampere型完全非线性方程的正则性理论。这个项目的另一个感兴趣的领域是研究火焰传播理论中自然产生的椭圆和抛物线自由边界问题。主要研究者还打算研究一类极小化问题，其中相关泛函是以Alt和Caffarelli提出的泛函为模型的。此外，她对探索椭圆型和抛物型变分不等式感兴趣，障碍被限制在低维流形上。提出的研究的主要目的之一是证明自由边界的正则性。调和分析和偏微分方程组理论研究这类问题所需的工具将同时发展。最后，在欧几里得背景下极小曲面理论和自由边界理论的惊人相似的激励下，首席研究员计划将她的不同研究方向合并到一个尚未探索的领域，即卡诺群中的自由边界问题(障碍和Alt-Caffarelli类型)的研究。首席研究人员有一个研究计划，它位于数学领域的交界处，被称为变分法、偏微分方程式和几何测度论。重点研究涉及“非对易”向量场系统的所谓变分不等式和偏微分方程解的解析性质和几何性质。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、数学金融和几何)，而且还涉及其他领域，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，在物理和工程中，当守恒量或关系在所考虑的变量的某些值上不连续变化的情况下，自由边界问题浮出水面。例如，自由边界表现为流体与空气或水与冰之间的界面。其中一个拟议的项目旨在研究燃烧-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在非均匀介质中的传播方式。正如前面提到的，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想法的相互作用。可以想象，所有这些不同的领域都将从这种协同中受益。首席研究人员致力于培训未来几代数学家，并通过为终身教员和研究生、本科生和K-12学生组织各种教育和辅导活动，增加妇女在科学界的代表性。