Optimal regularity for degenerate parabolic equations on a flat boundary
Hyungsung Yun
TL;DR
This work establishes optimal boundary regularity for viscosity solutions of the degenerate parabolic equation $u_t - x_n^γ Δu = f$ with $0<γ<1$, connecting to porous medium equation regularity in both bounded and flat-boundary settings. The authors develop a two-fold strategy: an inhomogeneous analysis using a parabolic-scaling distance and boundary blow-up with compactness to obtain $C^{1,1-γ}$ up to the boundary, and a homogeneous analysis leveraging time-differentiation structure to convert to elliptic problems and apply Schauder theory, yielding $C^{2,1-γ}$ regularity in tangential directions along with smoothness of $x_n^{-γ}u$ in $(x',t)$. These results tighten previous global Lipschitz and $C^{2,α}$ bounds to the sharp $C^{1,1-γ}$ and $C^{2,1-γ}$ regularities, under precise data conditions. The methods combine compactness, boundary blow-up arguments, and elliptic reduction on time slices, providing a framework for higher tangential regularity and robust a priori estimates near the flat boundary. Overall, the paper advances the understanding of boundary behavior in degenerate parabolic problems with implications for the porous medium equation in bounded domains.
Abstract
We establish the optimal regularity of viscosity solutions to \begin{equation*} u_t - x_n^γΔu = 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 $C^{1,1-γ}$. 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 $t$.