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C1 regularity of infinity-harmonic functions in dimension three and higher

A continuous function uu is infinity-harmonic if it is a viscosity solution of Δ∞u=∑i,juiujuij=0\Delta_\infty u=\sum_{i,j}u_iu_ju_{ij}=0; by Jensen (1993) and Crandall-Evans-Gariepy (2001) these are the absolutely minimizing Lipschitz extensions studied by Aronsson (1967). Solutions are locally Lipschitz, but the equation controls second derivatives only in the gradient direction, and Aronsson's example ∣x1∣4/3−∣x2∣4/3|x_1|^{4/3}-|x_2|^{4/3} shows the gradient is at best C0,1/3C^{0,1/3}. In the plane, Savin (2005) proved u∈C1u\in C^1 and Evans and Savin (2008) proved C1,αC^{1,\alpha}; Evans and Smart (2011) proved differentiability at every point in every dimension, which does not give continuity of the gradient. In dimensions d≥3d\ge3, is every infinity-harmonic function continuously differentiable, with a Holder modulus for the gradient?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Nonlinear elliptic PDE: the infinity Laplacian
Posed by
Open problem of the infinity-Laplacian regularity theory after Savin (2005), Evans and Savin (2008) and Evans and Smart (2011); the manuscript cites no explicit statement of the question
Year posed
—
Years open
—
Solved
2026-10-04
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
42 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for each d≥3d\ge3 there are αd∈(0,1/3]\alpha_d\in(0,1/3] and CdC_d such that every bounded infinity-harmonic uu on B1⊂RdB_1\subset\mathbb R^d satisfies ∥∇u∥L∞(B1/2)+[∇u]C0,αd(B1/2)≤Cd oscB1u\|\nabla u\|_{L^\infty(B_{1/2})}+[\nabla u]_{C^{0,\alpha_d}(B_{1/2})}\le C_d\,\mathrm{osc}_{B_1}u; hence infinity-harmonic functions are Cloc1C^1_{loc} on every open set in d≥3d\ge3. The proof is by compactness, using the Evans-Savin anisotropic blow-up and a new Liouville theorem for the limiting equation. The exponent αd\alpha_d is not explicit, and no claim is made about the optimal exponent 1/31/3, boundary regularity or the planar endpoint.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source; the proof was not refereed. No Lean formalization exists for this family at the pinned commit (lean/docs/377.md does not exist). The manuscript cites no source that poses the question, so its standing as a posed problem rests on the introduction's account of prior work.

Source

Changelog1 change

Discussion