Boundary regularity for fractional porous medium-type equations in bounded domains
We prove boundary Hölder regularity for fractional porous medium-type equations in bounded smooth domains with zero exterior data. For $0<σ<1$, $n>2σ$, and $m>1$, we show that nonnegative weak dual solutions $v$ of $\partial_t v+(-Δ)^σ(v^m)=0$ satisfy \[ \frac{v}{d^{σ/m}}\in C_{x,t}^{α,\frac{αm}{σ(m+1)}} \] up to the boundary and away from the initial time, for some $α\in(0,σ)$, where $d$ denotes the distance to the boundary. More generally, we establish boundary Hölder estimates for $u/d^σ$ for nonnegative viscosity solutions of $\partial_t u=u^γLu$, with $0<γ<1$ and initial data comparable to $d^σ$. Here $L$ is a symmetric stable operator of order $2σ$ whose measurable angular density is bounded above and below by positive constants. Our proof combines comparison principles and boundary barriers with a blow-up and compactness argument based on a Liouville theorem for ancient solutions in a half-space.