Optimal regularity for degenerate parabolic equations on a flat boundary

We establish the optimal regularity of viscosity solutions to u t − x n γ Δ u = f , which arises in the regularity theory of the porous medium equation. Specifically, we prove that under the zero Dirichlet boundary condition on { x n = 0 } , the optimal regularity of u up to the flat boundary { x n = 0 } is C 1 , 1 − γ . Moreover, for the homogeneous equations, we establish that the optimal regularity of u is C 2 , 1 − γ in the spatial variables, and that x n − γ u is smooth in the variables x ′ and t .