In this talk, we discuss the all time regularity of the free-boundary problem associated to the deformation of a compact weakly convex surface $\Sigma$ in ${\Real }^3$, with a flat side, by its Gaussian Curvature. We show that under certain necessary regularity and non-degeneracy initial conditions the interface separating the flat from the strictly convex side, remains smooth on $0 < t < T_c$, up to the vanishing time $T_c$ of the flat side. In addition, we will discuss Harnack estimate for degenerate elliptic and parabolic equations required to show the result above.
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