Uniqueness of positive harmonic profiles for singular stable operators in cones
Let $n \ge 2$ and $0 < s < 1$, and let $\mu$ be a finite even nonnegative measure on $S^{n-1}$ with $$\mu(S^{n-1}) \le \Lambda, \qquad \inf_{|e| = 1} \int_{S^{n-1}} |e \cdot \theta|^{2s}\,\mu(d\theta) \ge \lambda > 0.$$ The associated stable operator is $$Lu(x) = \frac{1}{2} \int_{S^{n-1}} \int_0^\infty \bigl[2u(x) - u(x + r\theta) - u(x - r\theta)\bigr] \frac{dr}{r^{1+2s}}\,\mu(d\theta).$$ When the kernel of $L$ is comparable to that of the fractional Laplacian, two nonnegative solutions of $Lw = 0$ with integrable tails outside a closed convex cone that vanish on the cone are proportional, a consequence of the boundary Harnack principle (Fernández-Real and Ros-Oton). For general stable operators even the interior Harnack inequality may fail. Ros-Oton and Weidner proved that two such positive solutions of growth below $2s$, each monotone in a direction outside the cone, are comparable up to constant factors when $\mu$ has a density in $L^p(S^{n-1})$ for some $p > n/(2s)$.
Problem. Let $\Sigma \subsetneq \mathbb{R}^n$ be a closed convex cone with nonempty interior, and let $w_1, w_2$ be continuous on $\mathbb{R}^n$, zero on $\Sigma$, strictly positive on $\mathbb{R}^n \setminus \Sigma$, with $Lw_i = 0$ in $\mathbb{R}^n \setminus \Sigma$ in the viscosity sense and $|w_i(x)| \le C_i (1 + |x|)^{\beta_i}$ for some $\beta_i < 2s$, so that the tails in the equation are integrable. Must $w_1 = c\,w_2$ for a constant $c > 0$? The measure $\mu$ may be singular, including measures supported on finitely many directions.
Source: Fernández-Real and Ros-Oton (2024), §4.6.2, Open question 4.4, for any closed convex cone and any positive solutions. Stated by AIM — Open Applied Problems for cones with nonempty interior and profiles of growth below 2s.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, nonlocal elliptic equations, positive solutions in cones, 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},
}
@misc{aim-singular-stable-cone-positive-solution-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of positive harmonic profiles for singular stable operators in cones}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/460-singular-stable-cone-positive-solution-uniqueness.md},
note = {Entry 460 at commit 24edaad, 2026-10-15},
}
Resolution claims
We prove uniqueness, up to a positive factor, of positive harmonic functions outside a proper closed convex cone with nonempty interior for every nondegenerate symmetric stable operator. The spectral measure may be singular or supported on finitely many directions. Each profile is assumed continuous, zero on the cone, and of polynomial growth strictly below the operator's order. The proof controls …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-20 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00833, "Uniqueness of positive harmonic profiles for singular stable operators in cones", clopen.solutions.