CAREER: Analytic and Geometric Aspects of Partial Differential Equations

Pi：Donatella Danielli，普渡大学DMS-0239771 Abstract：********************************************The研究本提案的一部分提出了一系列问题，这些问题的动机是研究椭圆和抛物线自由边界问题、变分法和几何测量理论。P.I打算研究火焰传播中感兴趣的自由边界问题，并与Rayleigh勋爵的猜想有关，即在给定区域的所有固定板中，圆形板给出的主频率最低。提出的研究的主要目的之一是证明自由边界的正则性。另一个有趣的领域是次椭圆障碍问题解的最佳正则性和自由边界的最优正则性。将同时开发研究这些问题所需的谐波分析工具和偏微分方程组。P.I.还有一个旨在发展卡诺群中极小曲面的正则性理论的程序。这样的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刻画。这个问题也有助于研究次拉普拉斯学派的诺依曼问题。结合几何学中出现的问题，P.I.打算在经典的Monge-Ampere算子的基础上发展一种次椭圆完全非线性方程的正则性理论。这一计划涉及建立著名的亚历山大-巴克尔曼-普奇最大值原理的适当版本，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“移动球面”方法感兴趣，并用它来证明完全非线性方程解的对称性。P.I.建议将这一研究计划与几项教育活动相结合。特别是，我们提到在普渡大学组织一年一度的夏季研讨会。作为普渡大学REU项目的一部分，P.I.将监督本科生的研究项目。在K-12级别，私人情报局希望吸引乐于接受的年轻人，在当地的科学博物馆组织有趣的动手数学研讨会，以及在扩大你的地平线会议的框架内组织。为了增加女性在科学界的代表性，P.I.还将继续指导科学领域的女性。当守恒量或关系在所考虑的变量的某些值之间不连续地变化时，在物理和工程中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。P.I.也有一个研究项目，它位于变分、偏微分方程组和几何测度论的交界处。重点研究了涉及非对易向量场系统的变分不等式和偏微分方程组的解的解析性质和几何性质。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)，而且还涉及其他领域，如机械工程和机器人技术。P.I.致力于培训未来几代数学家，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.