377. Interior C^{1,\alpha} regularity for infinity-harmonic functions

Interior $C^{1,\alpha}$ regularity for infinity-harmonic functions

1 paper

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.

Papers 1

$First page of Uniform Interior $`C^{1,\alpha}`$ Estimates for Infinity-Harmonic Functions$

Uniform Interior $C^{1,\alpha}$ Estimates for Infinity-Harmonic Functions

OpenAI · October 4, 2026

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 α_d-Hölder seminorm on B_1/2 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 ℝ^d, for d ≥ 3.

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@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}
}

Interior \(C^{1,\alpha}\) regularity for infinity-harmonic functions

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Uniform Interior \(C^{1,\alpha}\) Estimates for Infinity-Harmonic Functions

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