The Hele-Shaw problem as a ‘‘Mesa” limit of Stefan problems: Existence, uniqueness, and regularity of the free boundary

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.