Regularity and uniqueness results for a phase change problem in binary alloys

An isothermal model describing the separation of the components of a binary metallic alloy is considered. A process of phase transition is also assumed to occur in the solder; hence, the state of the material is described by two order parameters, i.e., the concentration c of the first component and the phase field p. Existence of a solution to the related initial and boundary value problem has been proved in a former paper, where, anyway, uniqueness was obtained only in a very special case. Here some further regularity and uniqueness results are shown in a more general setting by use of an a priori estimates - compactness argument. A key point of the proofs is the analysis of the fine continuity properties of the inverse map of the solution-dependent elliptic operator characterizing one of the equations of the system