Boundary Harnack inequalities for the fractional p-Laplacian in Reifenberg flat domains
Abstract
We prove a scale-invariant boundary Harnack inequality and Hölder continuity up to the boundary of quotients of positive fractional p-harmonic functions in Reifenberg flat domains with sufficiently small flatness constant, for 0<s<1 and 2≤ p<2/(1−s). The solutions are globally nonnegative, vanish on a common exterior boundary patch, and belong to the natural local energy and (p−1)-tail class. Their distant exterior values may be unrelated and unbounded. After normalization at a corkscrew point, the comparison and Hölder constants depend only on n,s,p. We first establish local half-space boundary comparison for every p≥2 and prove relative approximation by the half-space profile in Reifenberg domains. The compactness argument identifies escaped exterior mass and excludes its constant source term. Interior C1,α regularity then yields gradient non-degeneracy and lower increment bounds in a cone of directions. To treat the degenerate kernels produced by linearization, we develop estimates for weighted non-local forms and prove a localized Harnack inequality for bounded quotients, retaining the full exterior interactions. Ordered variational deformations and a common fractional p-harmonic majorant yield comparison for solutions with unrelated exterior data. Truncation of distant data and fractional p-harmonic replacement extend the estimates from the variational energy class to the natural tail class.
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BibTeX
@misc{nystrom2026boundary,
title = {{Boundary Harnack inequalities for the fractional $p$-Laplacian in Reifenberg flat domains}},
author = {Kaj Nyström},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21689},
eprint = {2610.21689},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21689},
}
RIS
TY - UNPB
TI - Boundary Harnack inequalities for the fractional p-Laplacian in Reifenberg flat domains
AU - Kaj Nyström
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21689
UR - https://arxiv.org/abs/2610.21689
AB - We prove a scale-invariant boundary Harnack inequality and Hölder continuity up to the boundary of quotients of positive fractional p-harmonic functions in Reifenberg flat domains with sufficiently small flatness constant, for 0<s<1 and 2≤ p<2/(1−s). The solutions are globally nonnegative, vanish on a common exterior boundary patch, and belong to the natural local energy and (p−1)-tail class. Their distant exterior values may be unrelated and unbounded. After normalization at a corkscrew point, the comparison and Hölder constants depend only on n,s,p. We first establish local half-space boundary comparison for every p≥2 and prove relative approximation by the half-space profile in Reifenberg domains. The compactness argument identifies escaped exterior mass and excludes its constant source term. Interior C^1,α regularity then yields gradient non-degeneracy and lower increment bounds in a cone of directions. To treat the degenerate kernels produced by linearization, we develop estimates for weighted non-local forms and prove a localized Harnack inequality for bounded quotients, retaining the full exterior interactions. Ordered variational deformations and a common fractional p-harmonic majorant yield comparison for solutions with unrelated exterior data. Truncation of distant data and fractional p-harmonic replacement extend the estimates from the variational energy class to the natural tail class.
ER -