CAREER: Analytic and Geometric Aspects of Partial Differential Equations

职业：偏微分方程的解析和几何方面

• 批准号: 0239771
• 负责人: Donatella Danielli
• 金额: $ 40万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0239771&HistoricalAwards=false
• 关键词: CAREER; Analytic; Geometric; Aspects; Partial

PI: Donatella Danielli, Purdue UniversityDMS-0239771 Abstract:********************************************The research part of this proposal presents a collection of problems motivated by the study of elliptic and parabolic free boundary problems, calculus of variations, and geometric measure theory. The P.I intends to study free boundary problems of interest in flame propagation, and related to Lord Rayleigh's conjecture that among all clamped plates of a given area, the circular one gives the lowest principal frequency. One of the main objectives of the proposed research is to prove regularity properties of the free boundary. Another area of interest is the optimal regularity of the solution and of the free boundary in the subelliptic obstacle problem. The necessary tools from harmonic analysis and PDEs for the study of these problems will be developed concurrently. The P.I. has also a program aimed at developing the regularity theory of minimal surfaces in Carnot groups. Such program entails the study of several basic questions. Among these, we mention the existence and characterization of traces on lower dimensional manifolds of Sobolev or BV functions. This issue is instrumental also in the study of the Neumann problem for sub-Laplacians. In connection with questions arising in geometry, the P.I. intends to develop a regularity theory for subelliptic fully nonlinear equations modeled on the classicalMonge-Ampere operator. This program involves establishing an appropriate version of the celebrated Alexandrov-Bakelman-Pucci maximum principle, which in turn requires the investigation of a suitable notion of convexity. The P.I. is also interested in studying the method of ``moving spheres" for so-called Weingarten hypersurfaces, and in its use to prove symmetry properties of solutions to fully nonlinear equations. The P.I. proposes to integrate this research plan with several educational activities. In particular, we mention the organization of an annual Summer Symposium at Purdue University. The P.I. will supervise undergraduate research projects as part of Purdue's REU program. At the K-12 level, the P.I. hopes to hook receptive young minds organizing fun, hands-on mathematics workshops at the local science museum, as well as in the framework of Expanding Your Horizons conferences. To increase the representation of women in the scientific community, the P.I. will also continue mentoring women in science.Free boundary problems 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 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 non-homogeneous media. The P.I. has also a research program 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 PDEs involving a system of non-commuting vector fields. The problems described in the proposal not only arise in a variety of mathematical context (e.g. optimal control theory, mathematical finance, and geometry), but are also of interest in other fields such as mechanical engineering and robotics. The P.I. 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 activities for graduate, undergraduate, and K-12 students.

PI：Donatella Danielli，普渡大学DMS-0239771 摘要：******************************************** *****本提案的研究部分提出了一系列由椭圆和抛物线自由边界问题、变分法和几何测度理论研究引发的问题。 P.I 打算研究火焰传播中感兴趣的自由边界问题，并与瑞利勋爵的猜想相关，即在给定区域的所有夹紧板中，圆形板给出最低的主频率。该研究的主要目标之一是证明自由边界的规律性。另一个令人感兴趣的领域是亚椭圆障碍问题中解和自由边界的最优规律性。研究这些问题所需的调和分析和偏微分方程工具将同时开发。 P.I.还有一个旨在发展卡诺群中最小曲面的正则理论的计划。该计划需要研究几个基本问​​题。其中，我们提到了 Sobolev 或 BV 函数的低维流形上迹的存在和表征。这个问题对于亚拉普拉斯算子的诺伊曼问题的研究也很有帮助。对于几何中出现的问题，P.I.打算开发一种基于经典 Monge-Ampere 算子建模的亚椭圆完全非线性方程的正则理论。该计划涉及建立著名的 Alexandrov-Bakelman-Pucci 最大原理的适当版本，这反过来又需要研究适当的凸性概念。 P.I.还对研究所谓的 Weingarten 超曲面的“移动球体”方法以及用其证明完全非线性方程解的对称性感兴趣。P.I. 提议将该研究计划与多项教育活动结合起来。特别是，我们提到普渡大学每年组织一次夏季研讨会，作为普渡大学 REU 项目的一部分，P.I.希望通过在当地科学博物馆以及“拓展视野”会议的框架内组织有趣的实践数学研讨会来吸引乐于接受的年轻人。为了增加女性在科学界的代表性，P.I.当守恒量或关系在所考虑的变量的某些值上不连续变化时，自由边界问题自然会出现在物理学和工程中，例如，自由边界表现为流体和空气之间的界面，或水和空气之间的界面。冰。 拟议项目之一旨在研究已燃-未燃混合物中自由边界的规律性特性。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。 P.I.还有一个研究项目涉及变分法、偏微分方程和几何测度理论。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。提案中描述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），而且在机械工程和机器人技术等其他领域也很有趣。 P.I.致力于通过为研究生、本科生和 K-12 学生组织各种教育活动，培养未来几代数学家，并提高女性在科学界的代表性。