Partial Differential Equations and Fourier Analysis

这个项目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。 PI旨在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，并找到凸域水平集形状的最佳界限，即使该域变得又长又薄。 另一个目标是解决数理经济学中的一个问题，即找到价格独立的社会福利函数的所有可能的函数形式（使用外代数和超定系统的方法）。 第三个目标是利用傅里叶变换来描述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，则描述建筑物表面上的周期性图案的失真。PI计划描述物理和工程中出现的几个问题中的两相平衡。 重点将放在两种物质之间的边界上，如冰和水。 其它例子包括两种液体之间的边界和船尾流的轮廓。 在另一个方向上，PI将试图证实J.Rauch首先提出的一个原理，即随着时间的增加，隔热房间中的最高温度点倾向于墙壁。 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相同温度表面的形状。 在第三个方向，PI将试图找到所有可能的价格独立的社会福利函数。 这是当前的兴趣，因为这些功能被用来比较国家和描述的趋势，更大的收入不平等在20世纪90年代在美国。S.以及相反方向的早期趋势。问题是，在这样的比较中，价格不能保持不变。 PI预计，价格独立性对允许的函数形式施加了严重的限制，这反过来又限制了进行比较的方式。 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。

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.