Hölder regularity under directional nonlocal ellipticity alone
Fix $n \ge 2$, $0 < s < 1$, $\lambda > 0$ and $\Lambda > 1$. For each $x \in B_1$ let $K_x$ be an even nonnegative measure on $\mathbb{R}^n \setminus \{0\}$, depending measurably on $x$, such that for all $r > 0$ and all unit vectors $e$ $$r^{2s} K_x(B_{2r} \setminus B_r) \le \Lambda, \qquad r^{2s-2} \int_{B_{\Lambda r} \setminus B_r} |e \cdot y|^2 \, K_x(dy) \ge \lambda,$$ and for smooth bounded $u$ let $$L_x u(x) = \frac{1}{2} \int_{\mathbb{R}^n} \bigl[2u(x) - u(x+y) - u(x-y)\bigr] \, K_x(dy).$$ No continuity or small oscillation in $x$ and no pointwise lower bound on a density of $K_x$ are assumed. Viscosity inequalities are defined with test functions touching $u$ at a point and patched with $u$ outside a neighbourhood of it. For operators that do not depend on $x$, Fernández-Real and Ros-Oton develop the interior regularity theory under these two conditions alone. Schwab and Silvestre proved the Hölder estimate below when, in addition, for some $\mu > 0$ and every $r \in (0,1)$ the kernel satisfies $K(x,y) \ge \lambda |y|^{-n-2s}$ on a subset of $B_{2r} \setminus B_r$ of measure at least $\mu |B_{2r} \setminus B_r|$; Fernández-Real and Ros-Oton proved it when the kernels have small oscillation in $x$.
Problem. Are there $\gamma > 0$ and $C < \infty$, depending only on $n$, $s$, $\lambda$ and $\Lambda$, such that every bounded continuous viscosity solution of $L_x u(x) = 0$ in $B_1$ satisfies $$\|u\|_{C^\gamma(B_{1/2})} \le C \|u\|_{L^\infty(\mathbb{R}^n)}?$$
Source: Fernández-Real and Ros-Oton (2024), Open question 3.4, with conditions (3.6.3)–(3.6.5). Stated by AIM — Open Applied Problems for bounded continuous viscosity solutions.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- PR
- Keywords
- AIM Open Applied Problems, Hölder estimates, integro-differential equations, jump processes
References
@book{fernandezreal2024,
author = {Fern{\'a}ndez-Real, Xavier and Ros-Oton, Xavier},
title = {Integro-Differential Elliptic Equations},
series = {Progress in Mathematics},
volume = {350},
publisher = {Birkh{\"a}user},
address = {Cham},
year = {2024},
doi = {10.1007/978-3-031-54242-8},
eprint = {2411.12455},
archivePrefix = {arXiv},
}
@article{schwab2016,
author = {Schwab, Russell W. and Silvestre, Luis},
title = {Regularity for parabolic integro-differential equations with very irregular kernels},
journal = {Anal. PDE},
volume = {9},
number = {3},
year = {2016},
pages = {727--772},
doi = {10.2140/apde.2016.9.727},
eprint = {1412.3790},
archivePrefix = {arXiv},
}
@article{fernandezreal2024plms,
author = {Fern{\'a}ndez-Real, Xavier and Ros-Oton, Xavier},
title = {Schauder and {C}ordes--{N}irenberg estimates for nonlocal elliptic equations with singular kernels},
journal = {Proc. Lond. Math. Soc. (3)},
volume = {129},
number = {3},
year = {2024},
pages = {e12629},
doi = {10.1112/plms.12629},
eprint = {2308.11383},
archivePrefix = {arXiv},
}
@misc{aim-rough-jump-kernel-holder-regularity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Hölder regularity under directional nonlocal ellipticity alone}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/464-rough-jump-kernel-holder-regularity.md},
note = {Entry 464 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00639, "Hölder regularity under directional nonlocal ellipticity alone", clopen.solutions.