Comparison for viscosity solutions of infinity-fractional diffusion
Fix $n \ge 2$, $1/2 < s < 1$ and $T > 0$. For a unit vector $e$ set $$D_e^s \phi(x) = C_s \int_0^\infty \frac{\phi(x + re) + \phi(x - re) - 2\phi(x)}{r^{1+2s}}\,dr,$$ where $C_s > 0$ is fixed, and consider $$u_t = \Delta_\infty^s u \quad \text{on } \mathbb{R}^n \times (0, T)$$ with bounded uniformly continuous initial data, where $\Delta_\infty^s u = D_e^s u$ with $e = \nabla u / |\nabla u|$ wherever $\nabla u \ne 0$. Viscosity sub- and supersolutions are taken in the following sense. At a test contact with nonzero spatial gradient the right side is $D_e^s \phi$ with $e = \nabla\phi / |\nabla\phi|$; at a contact with zero gradient, subsolutions satisfy $\phi_t \le \sup_{|e| = 1} D_e^s \phi$ and supersolutions satisfy $\phi_t \ge \inf_{|e| = 1} D_e^s \phi$. In every test, the test function is replaced by the tested function outside a spatial neighbourhood of the contact point.
The infinity fractional Laplacian was introduced by Bjorland, Caffarelli and Figalli as the generator of a nonlocal tug-of-war game. Del Teso, Endal, Jakobsen and Vázquez constructed viscosity solutions of the Cauchy problem for bounded uniformly continuous data, with the same zero-gradient envelopes, and proved that these solutions lie between classical sub- and supersolutions with the same initial data. On smooth radial functions that are nonincreasing in the radius, the operator reduces to the one-dimensional fractional Laplacian.
Problem. Let $u$ be a bounded continuous viscosity subsolution and $v$ a bounded continuous viscosity supersolution, both attaining their initial data uniformly. Does $$u(\cdot, 0) \le v(\cdot, 0) \quad \Longrightarrow \quad u(x,t) \le v(x,t) \quad \text{for all } (x,t) \in \mathbb{R}^n \times [0, T]?$$ No radial symmetry or monotonicity is assumed.
Source: del Teso, Endal, Jakobsen and Vázquez (2023), §9, who ask for uniqueness and a general comparison principle for viscosity solutions. Stated by AIM — Open Applied Problems as comparison of sub- and supersolutions in their zero-gradient convention, with locally patched tests.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, comparison principles, infinity fractional Laplacian, viscosity solutions
References
@article{delteso2023,
author = {del Teso, F{\'e}lix and Endal, J{\o}rgen and Jakobsen, Espen R. and V{\'a}zquez, Juan Luis},
title = {Evolution driven by the infinity fractional {L}aplacian},
journal = {Calc. Var. Partial Differential Equations},
volume = {62},
number = {4},
year = {2023},
pages = {Paper No. 136},
doi = {10.1007/s00526-023-02475-w},
eprint = {2210.06414},
archivePrefix = {arXiv},
}
@article{bjorland2012,
author = {Bjorland, Clayton and Caffarelli, Luis and Figalli, Alessio},
title = {Nonlocal tug-of-war and the infinity fractional {L}aplacian},
journal = {Comm. Pure Appl. Math.},
volume = {65},
number = {3},
year = {2012},
pages = {337--380},
doi = {10.1002/cpa.21379},
eprint = {1011.1966},
archivePrefix = {arXiv},
}
@misc{aim-infinity-fractional-evolution-comparison,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Comparison for viscosity solutions of infinity-fractional diffusion}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/461-infinity-fractional-evolution-comparison.md},
note = {Entry 461 at commit 24edaad, 2026-10-15},
}
Resolution claims
We consider the infinity fractional evolution in dimension $n\ge2$, with $1/2<s<1$. At a zero spatial test gradient, the viscosity subsolution condition uses the supremum of symmetric directional integrals, and the supersolution condition uses their infimum. For this convention, comparison fails even for radial subsolutions and supersolutions with the same compactly supported initial data, attained uniformly. The construction uses two thin …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, 461-relaxed-infinity-fractional-comparison/paper.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00800, "Comparison for viscosity solutions of infinity-fractional diffusion", clopen.solutions.