Resumen

A constructive method applicable to the solution of a wide class of free boundary problems is presented. A solution-dependent transformation technique is introduced. By considering a singular limit of the transformation, a related problem, to which local bifurcation theory may be applied, is derived. By inverting the (near singular) mapping between the two problems, an expression for solutions of the original problem is obtained. The method is illustrated by the study of a singularly perturbed elliptic equation. Approximate solutions are constructed and the validity of the approximations established by means of the Contraction Mapping Theorem.

Resumen
We study a Hele-Shaw problem with a mushy region obtained as a mesa type limit of one-phase Stefan problems in exterior domains. We deal with both Neumann and Dirichlet data and show pointwise convergence of the Stefan solutions to the Hele-Shaw solution. We make no assumptions on the geometry, topology, or connectivity of the injection slot.