Higher regularity for concave stable nonlocal equations
Fix $n \ge 2$, $0 < s < 1$ and $0 < \lambda \le \Lambda$. For an arbitrary index set $A$ and $a \in A$ let $$L_a u(x) = \operatorname{P.V.}\int_{\mathbb{R}^n} [u(x) - u(x+y)] \frac{k_a(y/|y|)}{|y|^{n+2s}}\,dy,$$ where each $k_a$ is even on $S^{n-1}$, $\lambda \le k_a \le \Lambda$, and $\sup_{a \in A} \|k_a\|_{C^j(S^{n-1})} \le M_j$ for every integer $j \ge 1$. By the nonlocal Krylov–Safonov and Evans–Krylov theorems of Caffarelli and Silvestre, solutions of concave equations such as the one below are $C^{\max\{1, 2s\} + \alpha}$ in the interior for some $\alpha > 0$; by analogy with $s = 1$, Fernández-Real and Ros-Oton do not expect regularity beyond $C^{1+2s}$. Ros-Oton, Torres-Latorre and Weidner proved a one-sided $C^{1,1}$ estimate: solutions are semiconcave in $B_{1/2}$, with constant $C\|u\|_{L^\infty(\mathbb{R}^n)}$.
Problem. Is there $\alpha \in (0, \min\{s, 1 - s\})$ such that every bounded continuous viscosity solution $u$ of $$\inf_{a \in A} \bigl(-L_a u\bigr) = 0 \quad \text{in } B_1$$ satisfies $u \in C^{1+s+\alpha}(B_{1/2})$ and $$\|u\|_{C^{1+s+\alpha}(B_{1/2})} \le C \|u\|_{L^\infty(\mathbb{R}^n)},$$ with $\alpha$ and $C$ depending only on $n$, $s$, $\lambda$, $\Lambda$ and the bounds $(M_j)_{j \ge 1}$, and not on $u$ or on the family $A$? No differentiability in $x$ of the index attaining the infimum is assumed.
Source: Fernández-Real and Ros-Oton (2024), Open question 3.1, for kernels that are regular enough; see also Ros-Oton, Torres-Latorre and Weidner (2025), Introduction. Stated by AIM — Open Applied Problems for the equation without source terms, with stable kernels whose angular parts are uniformly smooth.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, Bellman equations, fully nonlinear integro-differential equations
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{caffarelli2009,
author = {Caffarelli, Luis and Silvestre, Luis},
title = {Regularity theory for fully nonlinear integro-differential equations},
journal = {Comm. Pure Appl. Math.},
volume = {62},
number = {5},
year = {2009},
pages = {597--638},
doi = {10.1002/cpa.20274},
eprint = {0709.4681},
archivePrefix = {arXiv},
}
@article{caffarelli2011,
author = {Caffarelli, Luis and Silvestre, Luis},
title = {The {E}vans--{K}rylov theorem for nonlocal fully nonlinear equations},
journal = {Ann. of Math. (2)},
volume = {174},
number = {2},
year = {2011},
pages = {1163--1187},
doi = {10.4007/annals.2011.174.2.9},
eprint = {0905.1339},
archivePrefix = {arXiv},
}
@misc{aim-concave-nonlocal-higher-regularity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Higher regularity for concave stable nonlocal equations}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/459-concave-nonlocal-higher-regularity.md},
note = {Entry 459 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-00583, "Higher regularity for concave stable nonlocal equations", clopen.solutions.