Hölder regularity of solutions to parabolic equations associated with fractional p-Laplacian type operators
Abstract
In this paper, we study Hölder regularity of local weak super-solutions to the parabolic equation associated with fractional p-Laplacian type operators ∂t u(t,x)−p.v.∫Rd|u(t,y)−u(t,x)|p−2(u(t,y)−u(t,x))J(t;x,y) dy=0, (t,x)∈ R× Rd, where p.v. denotes the principal value, p∈(1,∞) and the kernel J(t;x,y) is comparable (uniformly with respect to (t;x,y)∈R×Rd×Rd) to the fractional p-Laplacian kernel |x−y|−d−sp with s∈(0,1). Our contribution extends the fundamental work of Kassmann and Weidner to the nonlinear case p≠ 2. In particular, we provide an explicit nonlocal parabolic tail condition in terms of an L1+ε-norm in the time variable (for a suitable ε>0) for all p∈(1,∞), under which Hölder regularity holds. Our proof is based on the method of expansion of positivity, which is the key ingredient in the reduction-of-oscillation argument.
Related
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BibTeX
@misc{kumagai2026hölder,
title = {{Hölder regularity of solutions to parabolic equations associated with fractional $p$-Laplacian type operators}},
author = {Takashi Kumagai and Jian Wang and Meng-ge Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13539},
eprint = {2610.13539},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13539},
}
RIS
TY - UNPB
TI - Hölder regularity of solutions to parabolic equations associated with fractional p-Laplacian type operators
AU - Takashi Kumagai
AU - Jian Wang
AU - Meng-ge Zhang
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13539
UR - https://arxiv.org/abs/2610.13539
AB - In this paper, we study Hölder regularity of local weak super-solutions to the parabolic equation associated with fractional p-Laplacian type operators ∂_t u(t,x)−p.v.∫_R^d|u(t,y)−u(t,x)|^p−2(u(t,y)−u(t,x))J(t;x,y) dy=0, (t,x)∈ R× R^d, where p.v. denotes the principal value, p∈(1,∞) and the kernel J(t;x,y) is comparable (uniformly with respect to (t;x,y)∈R×R^d×R^d) to the fractional p-Laplacian kernel |x−y|^−d−sp with s∈(0,1). Our contribution extends the fundamental work of Kassmann and Weidner to the nonlinear case p≠ 2. In particular, we provide an explicit nonlocal parabolic tail condition in terms of an L^1+ε-norm in the time variable (for a suitable ε>0) for all p∈(1,∞), under which Hölder regularity holds. Our proof is based on the method of expansion of positivity, which is the key ingredient in the reduction-of-oscillation argument.
ER -