Result 377, Partial differential equations

Interior C1,αC^{1,\alpha} regularity for infinity-harmonic functions

Proves uniform interior C1,αdC^{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.

New or sharp bound

The bigger picture

Why it matters

How erratic can the slope of an infinity-harmonic function be? The manuscript claims a dimension-dependent limit on how quickly that slope changes in the interior, giving quantitative control over these nonlinear equations' solutions.

What changes?

The manuscript reports uniform interior estimates for bounded infinity-harmonic functions, solutions of the nonlinear infinity-Laplace equation, in every dimension at least three. On the half unit ball, the gradient's maximum size and Hölder seminorm are bounded by a dimension-dependent constant times the oscillation (supremum minus infimum) on the unit ball. The positive Hölder exponent depends only on dimension, is at most one-third, and is not explicit. Neither endpoint nor boundary regularity is claimed.

What does that help mathematicians do?

The Hölder bound means that gradients at two nearby points differ by at most a controlled constant times their distance raised to the stated exponent. This rules out abrupt changes of slope inside the domain and supplies a uniform measure of smoothness. In particular, the reported result implies continuous first derivatives locally on every open set in these dimensions, not merely differentiability at individual points.

Are there practical applications?

The immediate value is foundational: researchers studying the infinity-Laplace equation gain a quantitative way to control interior slopes using only dimension and the function's variation on a larger ball. This supports analysis of local solution behavior. The unspecified exponent limits how numerically concrete the estimate is; the source reports no practical implementation.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Uniform Interior C1,αC^{1,\alpha} Estimates for Infinity-Harmonic Functions

October 4, 2026 46 pages

We prove a uniform interior C1,αdC^{1,\alpha_d} estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some αd∈(0,1/3]\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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Cite (BibTeX) 
@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}
}

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