Free boundaries and extremal inequalities

自由边界和极端不平等

• 批准号: 1500771
• 负责人: David Jerison
• 金额: $ 38.51万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500771&HistoricalAwards=false
• 关键词: Free; boundaries; extremal; inequalities

A free boundary is an interface between two materials like oil and water. Another example is the curve outlining the wake of a boat. Yet another is the interface between plasma and ordinary matter in a fusion reactor. Remarkably, the same mathematics of free boundaries that describes these physical phenomena can be used to design optimal shapes. For example, when one wants to enclose an oven or pipe with insulating material, there is a shape that is optimal in the sense that the sum of the cost of insulation and the cost due to heat loss is minimized. Moreover, even farther from physics, one can seek optimal ways to divide data sets into yes/no regions according to rules that minimize the errors, that is, false positive or false negative identifications or diagnoses. The overarching goal of this project is to reduce the complexity of the problem of searching for these optimal shapes. The expectation is that there are broad classes of situations in which the optimal divider (free boundary) resembles a straight line or plane at an appropriate scale. In those cases, one can be confident of finding a near optimal shape quickly. In addition to conducting his own research, the PI has served and will serve as faculty advisor for research projects by dozens of undergraduates and high school students in programs at MIT. Moreover, he has posted videos and lecture notes of a widely viewed single variable calculus course on MIT's Open Courseware site. He is currently working on an on-line course to be disseminated by MITx.Free boundaries arise as the interface between materials in which the materials retain some energy. Typically, space is divided into level sets of some quantity like temperature or pressure. In contrast, the interface represented by a minimal surface lives in an ambient space that is empty. Despite this difference between these two types of interfaces, there are profound connections between them. The main goal of this project is to show that interfaces and level sets of least energy for a wide variety of problems are as simple as possible. The PI proposes that the level sets of optimizers resemble parallel planes in that these surfaces are connected and cleanly separated. Usually, methods from the more developed theory of minimal surfaces have guided the study of free boundaries, but here ideas from the theory of free boundaries will guide the study of minimal surfaces. The proposal also gives a pathway to proving analogous simple behavior of level sets of the least energy Neumann eigenfunction for a convex symmetric domain. This would yield an important case of the longstanding ``hot spots'' conjecture of J. Rauch. A second project is to identify the cases of equality in the celebrated Alexandrov-Fenchel inequalities in convex geometry. The PI will use a geometric approach based on establishing new properties, of independent interest, of the Brenier (optimal transportation) mapping. A third project is aimed at developing a highly accurate description of localization of eigenfunctions and quantum tunneling, relevant to the design of LEDs.

自由边界是两种物质（如油和水）之间的界面。 另一个例子是描绘船尾流的曲线。另一个是聚变反应堆中等离子体和普通物质之间的界面。 值得注意的是，描述这些物理现象的自由边界数学可以用来设计最佳形状。 例如，当人们想要用隔热材料封闭烤箱或管道时，在隔热成本和由于热损失引起的成本的总和最小化的意义上，存在最佳的形状。此外，即使远离物理学，人们也可以根据最小化错误（即假阳性或假阴性识别或诊断）的规则，寻求将数据集划分为是/否区域的最佳方法。 该项目的首要目标是降低搜索这些最佳形状的问题的复杂性。 期望的是，有广泛的情况下，最佳分割线（自由边界）类似于一条直线或平面在适当的比例。 在这些情况下，可以有信心快速找到接近最佳的形状。 除了进行自己的研究外，PI还担任并将担任麻省理工学院数十名本科生和高中生的研究项目的教师顾问。 此外，他还在麻省理工学院的开放式课程网站上发布了一门被广泛观看的单变量微积分课程的视频和讲义。他目前正在研究一个由MITx传播的在线课程。自由边界出现在材料之间的界面上，其中材料保留了一些能量。 通常，空间被划分为温度或压力等某个量的水平集。 相反，由最小表面表示的界面生活在空的环境空间中。 尽管这两种类型的接口之间存在差异，但它们之间存在深刻的联系。 这个项目的主要目标是表明，接口和水平集的最低能源的各种各样的问题是尽可能简单。PI提出优化器的水平集类似于平行平面，因为这些表面是连接的并且干净地分离。通常，从更发达的极小曲面理论的方法指导了自由边界的研究，但在这里，从自由边界理论的想法将指导极小曲面的研究。该方法也为证明凸对称区域上最小能量Neumann本征函数水平集的类似简单行为提供了一条途径。 这将产生一个重要的情况下，长期的“热点”猜想的J。 第二个项目是确定平等的情况下，著名的亚历山德罗夫-Fenchel不等式凸几何。 PI将使用基于建立新属性的几何方法，独立的兴趣，Brenier（最佳运输）映射。第三个项目的目的是开发一个高度准确的描述本地化的本征函数和量子隧穿，相关的LED的设计。