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<σ<1, n>2σ, and m>1, we show that nonnegative weak dual solutions v of ∂t v+(−Δ)σ(vm)=0 satisfy v/dσ/m∈ Cx,tα,αm/(σ(m+1)) up to the boundary and away from the initial time, for some α∈(0,σ), where d denotes the distance to the boundary. More generally, we establish boundary Hölder estimates for u/dσ for nonnegative viscosity solutions of ∂t u=uγLu, with 0<γ<1 and initial data comparable to dσ. 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.
Related
- Nonlocal Boundary Conditions for Truncated-Fractional Differential Equations Posed on Bounded Domains
- Optimal regularity and fine asymptotics for very fast diffusion equations in bounded domains
- Maximal regularity for the Stokes operator in bounded C1 domains
- Sharp regularity and small ball probabilities for the stochastic heat equation on bounded domains
- Fractional hypocoercivity in bounded domains in the anomalous diffusion limit
BibTeX
@misc{huo2026boundary,
title = {{Boundary regularity for fractional porous medium-type equations in bounded domains}},
author = {Mengyu Huo and Tianling Jin and Jingang Xiong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16113},
eprint = {2610.16113},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16113},
}
RIS
TY - UNPB
TI - Boundary regularity for fractional porous medium-type equations in bounded domains
AU - Mengyu Huo
AU - Tianling Jin
AU - Jingang Xiong
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16113
UR - https://arxiv.org/abs/2610.16113
AB - 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 ∂_t v+(−Δ)^σ(v^m)=0 satisfy v/d^σ/m∈ C_x,t^α,αm/(σ(m+1)) up to the boundary and away from the initial time, for some α∈(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 ∂_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.
ER -