A MULTIPHASE MULLINS-SEKERKA SYSTEM - MATCHED ASYMPTOTIC EXPANSIONS AND AN IMPLICIT TIME DISCRETIZATION FOR THE GEOMETRIC EVOLUTION PROBLEM

We propose a generalisation of the Mullins-Sekerka problem to model ph ase separation in multi-component systems. The model includes equilibr ium equations in bulk, the Gibbs-Thomson relation on the interfaces, Y oung's law at triple junctions, together with a dynamic law of Stefan type. Using formal asymptotic expansions, we establish the relationshi p to a transition layer model known as the Cahn-Hilliard system. We in troduce a notion of weak solutions for this sharp interface model base d on integration by parts on manifolds, together with measure theoreti cal tools. Through an implicit time discretisation, we construct appro ximate solutions by stepwise minimisation. Under the assumption that t here is no loss of area as the time step tends to zero, we show the ex istence of a weak solution.