Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation

Abstract

A new construction scheme for a time-discrete version of the Stefan problem with Gibbs-Thomson law is introduced. Extending a scheme due to Luckhaus our approach uses a local minimisation of certain penalised functionals instead of minimising these functionals globally. The main difference is that local minimisation allows for surface loss of approximate phase interfaces in the limit. The theory of varifolds is used to obtain the convergence of approximate Gibbs-Thomson equations. A particular situation exhibits that local minimisation provides more physically appealing solutions than those constructed by global minimisation.