Partial Differential Equations and Fourier Analysis

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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.