Solutions to the Stefan problem with Gibbs-Thomson law by a local minimization

We propose a new approximation scheme for the Stefan problem with Gibbs-Thomson law. Within a time-discretization given by Luckhaus we choose approximate phase functions by a local instead of a global minimization. Driving the approximations to a limit we have to deal with a possible loss of surface area of the phase interfaces. The idea is to investigate the convergence of the surface measures. We formulate a generalization of the Gibbs-Thomson law and prove long-time existence of solutions using a convergence result due to Schätzle.