Smoothness up to the free boundary for the $p$-Laplacian evolution equation and the $\alpha$-Gauss curvature flow

The $p$-Laplacian evolution equation and the $\alpha$-Gauss curvature flow with a flat side are degenerate parabolic equations with evolving free boundaries. We give proofs of smooth short-time existence, up to the free boundaries, using a result of the authors on linear degenerate equations on a fixed domain.