Does directional ellipticity force a fractional energy bound?
Fix $n \ge 2$, $0 < s < 1$, $\lambda > 0$ and $\Lambda > 1$. Let $K(x,z) = K(z,x) \ge 0$ be measurable on $\mathbb{R}^n \times \mathbb{R}^n$ and suppose that for almost every $x$, every $r > 0$ and every unit vector $e$, $$r^{2s} \int_{B_{2r}(x) \setminus B_r(x)} K(x,z)\,dz \le \Lambda, \qquad r^{2s-2} \int_{B_{\Lambda r}(x) \setminus B_r(x)} |e \cdot (x - z)|^2 K(x,z)\,dz \ge \lambda.$$ These are the ellipticity conditions under which Fernández-Real and Ros-Oton consider divergence-form integro-differential equations $\operatorname{P.V.}\int_{\mathbb{R}^n} (u(x) - u(z)) K(x,z)\,dz = 0$ with bounded measurable coefficients. As they record, solutions are Hölder continuous when in addition, for some $\theta \in (0,1)$ and $\mu > 0$, $|\{z \in B : K(x,z) \ge \mu |x - z|^{-n-2s}\}| \ge \theta |B|$ for every ball $B$ and every $x \in B$ (Chaker and Silvestre, Dyda and Kassmann, Imbert and Silvestre), and the question of Hölder continuity under the conditions above essentially reduces to the energy inequality below.
Problem. Must there be a constant $c = c(n, s, \lambda, \Lambda) > 0$ such that every $v \in C_c^\infty(\mathbb{R}^n)$ satisfies $$\iint_{\mathbb{R}^n \times \mathbb{R}^n} |v(x) - v(z)|^2 K(x,z)\,dx\,dz \ge c \iint_{\mathbb{R}^n \times \mathbb{R}^n} \frac{|v(x) - v(z)|^2}{|x - z|^{n+2s}}\,dx\,dz?$$ No regularity of $K$ in the base point and no lower bound on the measure of directions carrying a pointwise kernel lower bound are assumed.
Source: Fernández-Real and Ros-Oton (2024), §2.8.1, the coercivity question following Open Question 2.1. Stated by AIM — Open Applied Problems for smooth compactly supported functions.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- FA
- Keywords
- AIM Open Applied Problems, coercivity estimates, fractional Sobolev spaces, integro-differential 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{chaker2020,
author = {Chaker, Jamil and Silvestre, Luis},
title = {Coercivity estimates for integro-differential operators},
journal = {Calc. Var. Partial Differential Equations},
volume = {59},
number = {4},
year = {2020},
pages = {106},
doi = {10.1007/s00526-020-01764-y},
eprint = {1904.13014},
archivePrefix = {arXiv},
}
@misc{aim-nonlocal-directional-ellipticity-coercivity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Does directional ellipticity force a fractional energy bound?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/658-nonlocal-directional-ellipticity-coercivity.md},
note = {Entry 658 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
No: for n = 2, s = 1/2, λ = 1, Λ = 4 the half-plane kernel 1{tr > 0}/|x − z|^3 satisfies both conditions, while smooth v_ε have bounded K-energy and H^{1/2} energy growing like log(1/ε)..
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, 471-nonlocal-coercivity/PROOF.md. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00701, "Does directional ellipticity force a fractional energy bound?", clopen.solutions.