Partial Differential Equations and Fourier Analysis

偏微分方程和傅里叶分析

• 批准号: 0070412
• 负责人: David Jerison
• 金额: --
• 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: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070412&HistoricalAwards=false
• 关键词: Partial; Differential; Equations; Fourier; Analysis

The main goal of this project is to describe equilibrium solutions toboundary value problems for linear and nonlinear elliptic partialdifferential equations. The PI intends to prove existence andregularity theorems for two-phase free boundary problems. The PI willalso describe level sets of eigenfunctions, especially nodal sets,maxima, and minima, and find optimal bounds on the shape of level setsfor convex domains even if the domain becomes long and thin. Anothergoal is to solve a problem in mathematical economics of finding allpossible functional forms of a price-independent social welfarefunction (using methods of exterior algebra and overdeterminedsystems). A third goal is to characterize the camera image of abuilding using the Fourier transform, that is, to characterize thedistortion of the periodic pattern on the face of the building if thebuilding is not exactly parallel to the camera.The PI plans to describe the equilibrium between two phases in severalproblems arising in physics and engineering. The focus will be on theboundary between two materials such as ice and water. Other examplesinclude the boundary between two liquids and the profile of the wakeof a boat. In another direction, the PI will attempt to confirm aprinciple first proposed by J. Rauch that the point of maximumtemperature in an insulated room tends towards a wall as timeincreases. More generally, the PI will examine how the geometry of afixed boundary (as in the walls of a room) affects the shape ofsurfaces of equal temperature at or near the equilibrium steady statetemperature. In a third direction, the PI will try to find allpossible price-independent social welfare functions. This is ofcurrent interest because such functions are used to compare countriesand to describe the trend towards greater income inequality in the1990s in the U. S. as well as earlier trends in the oppositedirection. The problem is that prices cannot be held fixed in suchcomparisons. The PI expects that price independence puts seriouslimits on the functional forms allowed, which in turn puts limits onthe way comparisons can be made. Finally, the PI proposes a projectintended to help solve the problem in computer vision of recognizingthe orientation of a building from a photograph.

本课题的主要目标是描述线性和非线性椭圆型偏微分方程边值问题的平衡解。目的是证明两相自由边界问题的存在性和正则性定理。PI还将描述特征函数的水平集，特别是节点集，最大值和最小值，并在凸域的水平集形状上找到最佳边界，即使该域变得又长又薄。另一个目标是解决数学经济学中的一个问题，即找到与价格无关的社会福利函数的所有可能的函数形式（使用外部代数和超确定系统的方法）。第三个目标是使用傅里叶变换表征建筑物的摄像机图像，也就是说，如果建筑物不完全平行于摄像机，则表征建筑物表面周期性图案的畸变。PI计划在物理和工程中出现的几个问题中描述两相之间的平衡。重点将放在两种物质之间的边界上，比如冰和水。其他的例子包括两种液体之间的边界和一艘船的尾流轮廓。在另一个方向上，PI将试图证实由J. Rauch首先提出的原理，即随着时间的增加，绝缘室内的最高温度点趋向于墙壁。更一般地说，PI将检查固定边界的几何形状（如房间的墙壁）如何影响在平衡稳态温度或接近平衡稳态温度的等温表面的形状。在第三个方向上，PI将试图找到所有可能的与价格无关的社会福利函数。这是当前的兴趣，因为这些函数被用来比较国家，并描述20世纪90年代美国收入不平等加剧的趋势，以及早期相反方向的趋势。问题是，在这种比较中，价格不能保持固定。PI预计，价格独立性会严重限制所允许的功能形式，这反过来又会限制进行比较的方式。最后，PI提出了一个项目，旨在帮助解决计算机视觉中从照片中识别建筑物方向的问题。