Continuous gradients for three-dimensional infinity-harmonic functions
Let $\Omega \subset \mathbb{R}^3$ be open and let $u \in C(\Omega)$ be a viscosity solution of $$-\Delta_\infty u = -\sum_{i,j=1}^{3} (\partial_i u)(\partial_j u)\, \partial_{ij} u = 0,$$ the viscosity inequalities being tested against smooth functions touching $u$ from above or from below. Savin proved that infinity-harmonic functions in two dimensions are continuously differentiable, and Evans and Savin that they are $C^{1,\alpha}$ there for some $\alpha > 0$. Evans and Smart proved that in every dimension infinity-harmonic functions are differentiable at every point.
Problem. Must $u \in C^1_{\mathrm{loc}}(\Omega)$? This asks for continuity of the gradient, beyond its existence at every point, and no hypothesis that the gradient is nonzero is made.
Source: Savin (2005), Introduction, in every dimension; stated for three dimensions by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, gradient regularity, infinity Laplacian, viscosity solutions
References
@article{savin2005,
author = {Savin, Ovidiu},
title = {{$C^1$} regularity for infinity harmonic functions in two dimensions},
journal = {Arch. Ration. Mech. Anal.},
volume = {176},
number = {3},
year = {2005},
pages = {351--361},
doi = {10.1007/s00205-005-0355-8},
}
@article{evans2008,
author = {Evans, Lawrence C. and Savin, Ovidiu},
title = {{$C^{1,\alpha}$} regularity for infinity harmonic functions in two dimensions},
journal = {Calc. Var. Partial Differential Equations},
volume = {32},
number = {3},
year = {2008},
pages = {325--347},
doi = {10.1007/s00526-007-0143-4},
}
@article{evans2011,
author = {Evans, Lawrence C. and Smart, Charles K.},
title = {Everywhere differentiability of infinity harmonic functions},
journal = {Calc. Var. Partial Differential Equations},
volume = {42},
number = {1-2},
year = {2011},
pages = {289--299},
doi = {10.1007/s00526-010-0388-1},
}
@misc{aim-infinity-harmonic-c1-three-dimensions,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Continuous gradients for three-dimensional infinity-harmonic functions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/419-infinity-harmonic-c1-three-dimensions.md},
note = {Entry 419 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove a uniform interior $C^{1,\alpha_d}$ estimate for bounded infinity-harmonic functions in every dimension $d\ge3$. For some $\alpha_d\in(0,1/3]$ depending only on the dimension, the gradient's supremum norm and $\alpha_d$-Hölder seminorm on $B_{1/2}$ are bounded by a dimension-dependent constant times the oscillation on $B_1$. The exponent is not explicit, and no endpoint or boundary regularity claim is made. In particular, infinity-harmonic …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, manuscript/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00690, "Continuous gradients for three-dimensional infinity-harmonic functions", clopen.solutions.