CAREER: Analytic and Geometric Aspects of Partial Differential Equations

主要研究者：Donatella Danielli，普渡大学DMS-0239771摘要：* 本提案的研究部分提出了一个集合的问题，椭圆和抛物自由边界问题，变分法和几何测量理论的研究动机。我打算研究自由边界问题感兴趣的火焰传播，并与瑞利勋爵的猜想，在所有固定板的一个给定的区域，圆形的一个给出了最低的主频率。所提出的研究的主要目标之一是证明自由边界的正则性。另一个感兴趣的领域是最佳的正则性的解决方案和自由边界的次椭圆障碍问题。从谐波分析和偏微分方程的研究这些问题的必要工具将同时开发。私家侦探也有一个计划，旨在发展规则理论的极小曲面卡诺集团。这样的计划需要研究几个基本问题。其中，我们提到的存在性和低维流形上的Sobolev或BV功能的痕迹的特征。这个问题是工具也在研究的诺依曼问题的次拉普拉斯。在与几何中出现的问题，PI。拟发展一个以经典Monge-Ampere算子为模型的次椭圆型完全非线性方程的正则性理论。这个程序涉及建立一个适当的版本著名的Alexandrov-Bakelman-Pucci最大值原则，这反过来又需要调查一个合适的概念凸性。私家侦探他还对研究所谓的Weingarten超曲面的“移动球”方法，以及用它来证明完全非线性方程解的对称性很感兴趣。私家侦探建议将这项研究计划与几项教育活动结合起来。特别是，我们提到在普渡大学举办的年度夏季研讨会。私家侦探将监督本科生的研究项目，作为普渡大学的REU计划的一部分。在K-12级别，PI。希望能在当地科学博物馆组织有趣的动手数学研讨会，以及在扩大你的视野会议的框架内，吸引接受的年轻人。为了增加妇女在科学界的代表性，P.I.在物理学和工程学中，当一个守恒量或关系在所考虑的变量的某个值上不连续地变化时，自然会出现自由边界问题。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。私家侦探也有一个研究计划，位于变分法，偏微分方程和几何测量理论的接口。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。该提案中描述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），而且在其他领域也很感兴趣，如机械工程和机器人。私家侦探致力于培养未来几代数学家，并通过为研究生，本科生和K-12学生组织各种教育活动，增加女性在科学界的代表性。

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.