We establish the optimal regularity of viscosity solutions to \begin{ equation*} u_t - x_n^\gamma \Delta u = f, \end{ equation*} which arises in the regularity theory for 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 $C^{ 1,1-\gamma}$. Moreover, for the homogeneous equations, we establish that the optimal regularity of $u$ is $C^{ 2,1-\gamma}$ in the spatial variables, and that $x_n^{ -\gamma}u$ is smooth in the variables $x'$ and $t$.
Categories: math.AP