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We study a Hele-Shaw problem with a mushy region obtained as a Mesa typelimit of one phase Stefan problems in exterior domains. We study theconvergence, determine some of the qualitative properties and regularity of theunique limiting solution, and prove regularity of the free boundary of thislimit under very general conditions on the initial data. Indeed, our resultshandle changes in topology and multiple injection slots.

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