A Two-Sided Contracting Stefan Problem

We study a novel two-sided Stefan problem-motivated by the study of certain 2D interfaces-in which boundaries at both sides of the sample encroach into the bulk with rate equal to the boundary value of the gradient. Here the density is in [0, 1] and takes the two extreme values at the two free boundaries. It is noted that the problem is borderline ill-posed: densities in excess of unity liable to cause catastrophic behavior. We provide a general proof of existence and uniqueness for these systems under the condition that the initial data is in [0, 1] and with some mild conditions near the boundaries. Applications to 2D shapes are provided, in particular motion by weighted mean curvature for the relevant interfaces is established.