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Uniform Interior $C^{1,\alpha}$ Estimates for Infinity-Harmonic Functions
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Abstract
We prove a uniform interior $C^{1,\alpha_d}$ estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some $\alpha_d\in(0,1/3]$ depending only on the dimension, the gradient's supremum norm and αd-Hölder seminorm on B1/2 are bounded by a dimension-dependent constant times the oscillation on B1. The exponent is not explicit, and no endpoint or boundary regularity claim is made. In particular, infinity-harmonic functions are locally C1 on every open subset of ℝd, for d ≥ 3.
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Possibly relevant literature (found by the agent, not checked)
- Aelson Sobral, Eduardo V. Teixeira and José Miguel Urbano, Qualitative-to-quantitative regularity for degenerate equations (arXiv:2610.13137, 9 October 2026). Theorem 1.3 derives uniform interior $C^{1,\beta}$ estimates for planar infinity-harmonic functions for every $\beta<1/3$ from Xu's qualitative planar $C^{1,1/3}$ theorem, which can be compared with this manuscript's uniform estimate for $d\ge3$; it does not cite the OpenAI manuscript.
Digests and expositions
- Latent Seminar, Uniform gradient regularity for infinity-harmonic functions (6 October 2026). Explains the manuscript's three-point flatness estimate, anisotropic limit and Liouville argument, while explicitly limiting its review to statements, introductions and selected proof mechanisms rather than complete proof verification. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 377
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-October-4-2026,
author = {{OpenAI}},
title = {{Uniform Interior $C^{1,\alpha}$ Estimates for Infinity-Harmonic Functions}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-October-4-2026/interior-c1-infinity-harmonic.pdf}{OAI:Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-October-4-2026}},
year = {2026}
}Family 377: Interior $C^{1,\alpha}$ regularity for infinity-harmonic functions
Proves uniform interior $C^{1,\alpha_d}$ regularity for bounded infinity-harmonic functions in every dimension d ≥ 3, for a positive exponent depending only on dimension. The gradient's supremum norm and Hölder seminorm on the half unit ball are bounded by a dimension-dependent constant times the oscillation on the unit ball. The exponent is not explicit.
This is the only paper in this family.