9401355 Jerison The primary goal of this mathematical research is to obtain existence, regularity and estimation on the size and shape of solutions of elliptic partial differential equations. The solutions are variational solutions, that is, minimizers or higher critical points of energy or some other cost function. The methods employed are symmetry and invariance in the form of Fourier analysis, rescaling and iteration, curvature and convexity, maximum principle and comparison functions, Green's formula and more elaborate integration by parts formulas, asymptotic analysis and perturbations and trial functions for energy estimation. The first problem is to estimate eigenvalues of eigenfunctions defined on convex domains or on positively curved surfaces. Good estimation necessitates locating the nodal line, that is, the arc where the eigenfunction vanishes. The problem of locating the nodal line is viewed as a two-phase free boundary problem. In addition, it work will be done on the evolution to equilibrium for two-phase free boundary problems. A second problem to be addressed is a classical allocation problem first proposed by Monge in the 18th century. The problem can be viewed as a linear programming problem. It can be formulated as one of deciding how to retrain a group of workers to fit new jobs at minimal cost. The approach to be used is geometric and related to the study of volume-preserving maps of N-space and the theory of the Monge Ampere equation, a fully nonlinear partial differential equation. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***
