Optimal nonlocal obstacle regularity with a minimally smooth obstacle
Let $n \ge 2$ and $0 < s < 1$, let $\beta > 1 + s$ be non-integer, and let $a \in \operatorname{Lip}(S^{n-1})$ be even with $0 < \lambda \le a \le \Lambda$. Let $$Lu(x) = \operatorname{P.V.}\int_{\mathbb{R}^n} [u(x) - u(x+y)] \frac{a(y/|y|)}{|y|^{n+2s}}\,dy,$$ and consider bounded continuous viscosity solutions of the obstacle problem $$\min\{Lu, \, u - \varphi\} = 0 \quad \text{in } B_1$$ with $\varphi \in C^\beta(B_1)$. The values of $u$ outside $B_1$ are arbitrary bounded data, and the contact set $\{u = \varphi\}$ may reach $\partial B_1$. For kernels $K$ of this kind with $|\nabla K(y)| \le \Lambda |y|^{-1} K(y)$, Ros-Oton, Torres-Latorre and Weidner proved the estimate below for $\beta > 1 + 2s$, and $\|u\|_{C^\beta(B_{1/2})} \le C(\|\varphi\|_{C^\beta(B_1)} + \|u\|_{L^\infty(\mathbb{R}^n)})$ for $\beta < 1 + s$, with $C$ depending only on $n$, $s$, $\beta$, $\lambda$ and $\Lambda$. For the fractional Laplacian, Caffarelli, De Silva and Savin proved a local $C^{1+s}$ estimate for every $\beta > 1 + s$.
Problem. Does every such $u$ satisfy $$\|u\|_{C^{1+s}(B_{1/2})} \le C \bigl(\|\varphi\|_{C^\beta(B_1)} + \|u\|_{L^\infty(\mathbb{R}^n)}\bigr),$$ with $C$ depending only on $n$, $s$, $\beta$, $\lambda$, $\Lambda$ and $\|a\|_{\operatorname{Lip}}$? The range of interest is $1 + s < \beta \le 1 + 2s$.
Source: Fernández-Real and Ros-Oton (2024), Open question 4.1, after the estimate of Ros-Oton, Torres-Latorre and Weidner (2025) for β > 1 + 2s, who expect the same for β > 1 + s (footnote 5; arXiv 2023). Stated by AIM — Open Applied Problems for non-integer β and bounded continuous viscosity solutions.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, integro-differential operators, obstacle problems, optimal stopping
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{rosoton2025,
author = {Ros-Oton, Xavier and Torres-Latorre, Clara and Weidner, Marvin},
title = {Semiconvexity estimates for nonlinear integro-differential equations},
journal = {Comm. Pure Appl. Math.},
volume = {78},
number = {3},
year = {2025},
pages = {592--647},
doi = {10.1002/cpa.22237},
eprint = {2306.16751},
archivePrefix = {arXiv},
}
@article{caffarelli2017,
author = {Caffarelli, Luis and De Silva, Daniela and Savin, Ovidiu},
title = {The two membranes problem for different operators},
journal = {Ann. Inst. H. Poincar{\'e} C Anal. Non Lin{\'e}aire},
volume = {34},
number = {4},
year = {2017},
pages = {899--932},
doi = {10.1016/j.anihpc.2016.05.006},
eprint = {1601.02548},
archivePrefix = {arXiv},
}
@misc{aim-nonlocal-obstacle-minimal-smoothness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Optimal nonlocal obstacle regularity with a minimally smooth obstacle}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/465-nonlocal-obstacle-minimal-smoothness.md},
note = {Entry 465 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-00624, "Optimal nonlocal obstacle regularity with a minimally smooth obstacle", clopen.solutions.