The main goal of this project is to describe equilibrium solutions to boundary value problems for linear and nonlinear elliptic partial differential equations. The PI intends to prove existence and regularity theorems for two-phase free boundary problems. The PI will also describe level sets of eigenfunctions, especially nodal sets, maxima, and minima, and find optimal bounds on the shape of level sets for convex domains even if the domain becomes long and thin. Another goal is to solve a problem in mathematical economics of finding all possible functional forms of a price-independent social welfare function (using methods of exterior algebra and overdetermined systems). A third goal is to characterize the camera image of a building using the Fourier transform, that is, to characterize the distortion of the periodic pattern on the face of the building if the building is not exactly parallel to the camera.
The PI plans to describe the equilibrium between two phases in several problems arising in physics and engineering. The focus will be on the boundary between two materials such as ice and water. Other examples include the boundary between two liquids and the profile of the wake of a boat. In another direction, the PI will attempt to confirm a principle first proposed by J. Rauch that the point of maximum temperature in an insulated room tends towards a wall as time increases. More generally, the PI will examine how the geometry of a fixed boundary (as in the walls of a room) affects the shape of surfaces of equal temperature at or near the equilibrium steady state temperature. In a third direction, the PI will try to find all possible price-independent social welfare functions. This is of current interest because such functions are used to compare countries and to describe the trend towards greater income inequality in the 1990s in the U. S. as well as earlier trends in the opposite direction. The problem is that prices cannot be held fixed in such comparisons. The PI expects that price independence puts serious limits on the functional forms allowed, which in turn puts limits on the way comparisons can be made. Finally, the PI proposes a project intended to help solve the problem in computer vision of recognizing the orientation of a building from a photograph.
