Optimal obstacle regularity for stable jump kernels with only angular integrability
Let $n \ge 2$, $0 < s < 1$ and $0 < \lambda \le \Lambda$. Let $a \in L^1(\mathbb{S}^{n-1})$ be nonnegative and even, with $$\int_{\mathbb{S}^{n-1}} a(\theta)\,d\theta \le \Lambda, \qquad \inf_{e \in \mathbb{S}^{n-1}} \int_{\mathbb{S}^{n-1}} |e \cdot \theta|^2 a(\theta)\,d\theta \ge \lambda,$$ put $K(y) = a(y/|y|)\,|y|^{-n-2s}$ and $$\mathcal{E}_a(w) = \frac14 \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} |w(x) - w(x+y)|^2 K(y)\,dy\,dx.$$ For a nonnegative obstacle $\phi \in C_c^\infty(\mathbb{R}^n)$, let $u$ be the unique minimizer of $\mathcal{E}_a$ over $\{w \in L^{2n/(n-2s)}(\mathbb{R}^n) : \mathcal{E}_a(w) < \infty,\ w \ge \phi \text{ a.e.}\}$, the finite-energy solution of the obstacle problem for $Lu = \operatorname{p.v.} \int (u(x) - u(x+y)) K(y)\,dy$. Write $\|u\|_{C^{1,s}(\mathbb{R}^n)} = \|u\|_\infty + \|\nabla u\|_\infty + \sup_{x \ne y} |\nabla u(x) - \nabla u(y)|/|x - y|^s$. For kernels with $\lambda \le a \le \Lambda$ on the sphere, the optimal $C^{1+s}$ regularity of solutions was proved by Caffarelli, Ros-Oton and Serra. Ros-Oton and Weidner proved $\|u\|_{C^{1+s}(\mathbb{R}^n)} \le C \|\phi\|_{C^{2,\varepsilon}(\mathbb{R}^n)}$, for $\varepsilon > \max\{2s - 1, 0\}$ and with $C$ depending only on $n, s, \lambda, \Lambda$, when in addition $\|a\|_{L^p(\mathbb{S}^{n-1})} \le \Lambda$ for some $p > n/(2s)$.
Problem. Does a constant $C = C(n, s, \lambda, \Lambda)$ always exist such that $$\|u\|_{C^{1,s}(\mathbb{R}^n)} \le C \|\phi\|_{C^3(\mathbb{R}^n)}$$ for every such $a$ and $\phi$? It is not assumed that $a$ belongs to any $L^p(\mathbb{S}^{n-1})$ with $p > 1$, or that the kernel is positive in every direction.
Source: Fernández-Real and Ros-Oton (2024), Open Question 4.3, for every stable operator; Ros-Oton and Weidner (2026), after Theorem 1.3, for kernels satisfying only their ellipticity conditions. Stated by AIM — Open Applied Problems as a global C^{1,s} estimate for integrable angular densities.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, nonlocal obstacle problems, stable operators
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{rosoton2026,
author = {Ros-Oton, Xavier and Weidner, Marvin},
title = {Obstacle problems for nonlocal operators with singular kernels},
journal = {Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)},
volume = {27},
number = {1},
year = {2026},
pages = {535--588},
doi = {10.2422/2036-2145.202309_010},
eprint = {2308.01695},
archivePrefix = {arXiv},
}
@article{caffarelli2017,
author = {Caffarelli, Luis and Ros-Oton, Xavier and Serra, Joaquim},
title = {Obstacle problems for integro-differential operators: regularity of solutions and free boundaries},
journal = {Invent. Math.},
volume = {208},
number = {3},
year = {2017},
pages = {1155--1211},
doi = {10.1007/s00222-016-0703-3},
}
@misc{aim-general-stable-obstacle-optimal-regularity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Optimal obstacle regularity for stable jump kernels with only angular integrability}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/485-general-stable-obstacle-optimal-regularity.md},
note = {Entry 485 at commit 24edaad, 2026-10-15},
}
Resolution claims
We prove a global $C^{1,s}$ estimate for the finite-energy obstacle problem of a symmetric stable operator whose angular density is only integrable. The constant depends on the order, dimension, angular mass bound, and quadratic nondegeneracy bound, without stronger angular integrability or positivity in every direction. The classification step uses the convex primitive: a derivative associated with a boundary ray of …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00796, "Optimal obstacle regularity for stable jump kernels with only angular integrability", clopen.solutions.