On weak solutions to the Stefan problem with Gibbs-Thomson correction. (English) Zbl 1212.35514

Summary: The paper investigates the well posedness of the quasi-stationary Stefan problem with the Gibbs-Thomson correction. The main result proves the existence of unique weak solutions provided the initial surface belongs to the \(W_p^{2-3/p}\)-Sobolev-Slobodeckij class for \(p>n+3\), only. The proof is based on Schauder-type estimates in \(L_p\)-type spaces for a linearization of the original system in a rigid domain.


35R35 Free boundary problems for PDEs
35K55 Nonlinear parabolic equations
35K99 Parabolic equations and parabolic systems
80A22 Stefan problems, phase changes, etc.
74N20 Dynamics of phase boundaries in solids