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I am looking for a weak-formulation that would give me an existence and uniqueness of a solution of a Stefan-type problem. It is basically a 2-phase Stefan problem in 2D except that the free-boundary does not have the nice "solution $u=0$ " criteria.

Instead, it has a robin boundary condition $$u+\partial_{\nu}u = g,$$ where $g$ is a given function and $\partial_{\nu}$ denotes the derivative in direction of the normal vector to the free boundary, at the free boundary (not fixed boundary).

(The above robin condition at the free boundary is assigned in addition to the typical condition on the velocity of the free boundary: $v = \partial_{\nu} u^+ - \partial_{\nu}u^-$.)

Does anyone know if this is an open problem? Or does anyone know a work related to this? I will greatly appreciate any reference to a 2-phase Stefan-type problem that is not the classical Stefan problem. Thank you!

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