9404773 Chen The primary objective of this proposal is to study certain free boundary problems which describe geometric motions of hypersurfaces, to analyze the asymptotic behavior of certain evolution equations and systems describing interfacial phenomena, to connect the solutions of the free boundary problems with the asymptotic behaviors of evolution equations or systems, and to develop numerical schemes calculating the solutions to the free boundary problems. In the study of free boundary problems, the research will place particular emphasis on a Hele-Shaw free boundary problem , which is to find the motion of a hypersurface such that the normal velocity of the motion is equal to the jump across the hypersurface of the normal derivative of a function that is harmonic in the domain complementary to the hypersurface and whose boundary value on the hypersurface is equal to the mean curvature of the hypersurface. In the study of the asymptotic behavior, the research will aim at the Cahn- Hilliard equation modeling the phase separation and phase coarsening phenomena in a binary alloy, and at the phase- field model describing the temperature and liquid-solid phase changes in a solid/liquidation process. This project deals with differential equations of applied mathematics. Techniques of functional analysis and geometry will be used to analyze these equations. Since the problems studied here are very basic in mathematical and physical sciences, the methods developed in this project can be applied to many other currently active research areas and consequently advance the knowledge of interfacial phenomena. In particular, applications of this research can be made to advanced materials research. ***
