Gradient Hölder regularity for singular fractional p-Laplace equations
Abstract
We prove interior C1,α regularity for globally bounded weak solutions to the fractional p-Laplace equation (−Δp)s u=0 under the assumptions n≥2, 1<p<2, 0<s<1, and sp>p−1, where α=α(n,p,s)>0. This resolves the open problem of gradient Hölder continuity for globally bounded solutions in the singular range sp>p−1. The proof is based on affine approximation and estimates independent of the normalized affine slope. When these slopes tend to infinity, we prove convergence to a solution of a linear nonlocal equation of order sp−p+2, together with convergence of the normalized energies to the corresponding quadratic energy. The regularity of the limiting equation yields an improved affine approximation. For bounded slopes, we prove that every globally Lipschitz weak solution of (−Δp)s u=0 in ℝn is affine and use this result in a compactness argument. The two estimates are combined in an iteration that preserves the required energy and tail bounds and gives Hölder continuity of the gradient.
Related
- Asymptotically Hölder gradient regularity for degenerate or singular parabolic normalized $ p$-Laplace equations
- Gradient rearrangement estimates for singular and degenerate anisotropic p-Laplace equations
- Asymptotic gradient regularity for degenerate or singular fully nonlinear parabolic equations
- Local and global C1,β-regularity for uniformly elliptic quasilinear equations of p-Laplace and Orlicz-Laplace type
- Local behavior of solutions to inhomogeneous singular parabolic p-Laplace equations
BibTeX
@misc{zhang2026gradient,
title = {{Gradient Hölder regularity for singular fractional $p$-Laplace equations}},
author = {Chao Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.16243},
eprint = {2608.16243},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16243},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gradient Hölder regularity for singular fractional p-Laplace equations
AU - Chao Zhang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2608.16243
UR - https://arxiv.org/abs/2608.16243
AB - We prove interior C^1,α regularity for globally bounded weak solutions to the fractional p-Laplace equation (−Δ_p)^s u=0 under the assumptions n≥2, 1<p<2, 0<s<1, and sp>p−1, where α=α(n,p,s)>0. This resolves the open problem of gradient Hölder continuity for globally bounded solutions in the singular range sp>p−1. The proof is based on affine approximation and estimates independent of the normalized affine slope. When these slopes tend to infinity, we prove convergence to a solution of a linear nonlocal equation of order sp−p+2, together with convergence of the normalized energies to the corresponding quadratic energy. The regularity of the limiting equation yields an improved affine approximation. For bounded slopes, we prove that every globally Lipschitz weak solution of (−Δ_p)^s u=0 in ℝ^n is affine and use this result in a compactness argument. The two estimates are combined in an iteration that preserves the required energy and tail bounds and gives Hölder continuity of the gradient.
ER -