Convergence of the two-phase Stefan problem to the one-phase problem

We study the limit of the one-dimensional Stefan problem as the diffusivity coefficient of the solid phase approaches zero. We derive a weak formulation of the equilibrium condition for the resulting one-phase problem that allows jumps of the temperature across the interface. The weak formulation consists of a regularity condition that only enforces the usual equilibrium condition to hold from the liquid phase. At the end we briefly discuss the radial problem in higher space dimensions. The main tools in order to prove the convergence are the uniform bounds on the total variation of the free boundary that are derived using a regularized problem, where the equilibrium condition is substituted by a dynamical condition.