Partial differential equations·OpenAI Math·Result 377
Interior C1,α regularity for infinity-harmonic functions
One of the 372 results in OpenAI's public mathematics catalogue, filed under partial differential equations: 1 preprint, the latest dated 4 October 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a result titled ‘Interior C^(1,α) regularity for infinity-harmonic functions’ (partial differential equations) in a preprint dated 4 October 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More, is on YouTube.
Chapter 58 of 84 in Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · starts at 19:52 · watch from there on YouTube →
What OpenAI claims
Proves uniform interior C1,αd 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.
Paper
Uniform Interior C1,α Estimates for Infinity-Harmonic Functions
OpenAI's abstract We prove a uniform interior C1,αd estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some αd ∈ (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.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 4 October 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Proves uniform interior C1,αd 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.
- Where is the proof?
- In the paper, Uniform Interior C1,α Estimates for Infinity-Harmonic Functions (PDF).
- Is there a video?
- Yes, on YouTube (29:20, chapter 58).
Also in
- Chapter 9 of 9 in Boltzmann Equation, Hot Spots Conjecture and 7 More in Partial Differential Equations · starts at 2:47 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · Compilation · 29:20 · chapter 58 from 19:52 · description, sources and chapters →
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