Boundary regularity for fractional porous medium-type equations in bounded domains

Abstract

We prove boundary Hölder regularity for fractional porous medium-type equations in bounded smooth domains with zero exterior data. For $0<\sigma <1$, $n>2\sigma$, and $m>1$, we show that nonnegative weak dual solutions $v$ of ${\partial }_{t}v+(-\Delta {)}^{\sigma }(v^m)=0$ 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 $\alpha \in (0,\sigma )$, where $d$ denotes the distance to the boundary. More generally, we establish boundary Hölder estimates for $u/d^{\sigma }$ for nonnegative viscosity solutions of ${\partial }_{t}u=u^{\gamma }Lu$, with $0<\gamma <1$ and initial data comparable to $d^{\sigma }$. Here $L$ is a symmetric stable operator of order $2\sigma$ 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.

arXiv comment59 pages