An analysis of a phase field model of a free boundary

A mathematical analysis of a new approach to solidification problems is presented. A free boundary arising from a phase transition is assumed to have finite thickness. The physics leads to a system of nonlinear parabolic differential equations. Existence and regularity of solutions are proved. Invariant regions of the solution space lead to physical interpretations of the interface. A rigorous asymptotic analysis leads to the Gibbs-Thompson condition which relates the temperature at the interface to the surface tension and curvature.