# Existence and uniqueness of an elliptic equation coupled with a parabolic equation (mean curvature flow)

Given a parabolic equation on a simply connected smooth domain $\Omega(t)$ with boundary $\Gamma(t)$ $$\Delta u = 1 \quad on \quad \Omega(t) \\ \nabla u \cdot n + u = g \quad on \quad \Gamma(t)$$ (where $g$ is a constant), coupled with a parabolic equation that controls the evolution of $\Gamma(t)$ (in this case mean curvature flow, but really any parabolic equation will do) $$V = \kappa + u \quad on \quad \Gamma(t)$$ (where $V$ is the normal velocity, and $\kappa$ is the mean curvature).

Do you know of any good references for proving existence and uniqueness? This is somewhat similar to Hele-Shaw flows, but I can't find anything similar enough that I can get it working for this system.

• Related (but not answered) mathoverflow.net/questions/100767/… – Paul Bryan Jul 13 '17 at 9:22
• Related, but quite different, as they have two parabolic equations. The extra time derivative makes things a bit easier. The only methods I have found that might prove my problem involve regularizing it with a time derivative. – Josiki Jul 14 '17 at 6:44