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.

Manuscript

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

October 4, 2026 · source · cite as \cite{OAI:Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-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.

Result 377 · Partial differential equations. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.