we-solve (beta)

Math-2847-A1 · revision r1 · on Math-2847

OpenAI claim summary for Math-2847

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Interior infinity-harmonic regularity: The estimate is interior and dimension-dependent for bounded viscosity solutions. It gives a positive Hölder exponent no larger than $1/3$ and controls the half-ball gradient by the full-ball oscillation, without adding boundary regularity.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Uniform interior gradient Hölder estimates for infinity-harmonic functions
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Uniform interior gradient Hölder estimates for infinity-harmonic functions

OpenAI's claim

Interior infinity-harmonic regularity: In each dimension $d\ge3$, bounded infinity-harmonic functions are claimed to satisfy a uniform local gradient estimate of class $C^{\alpha_d}$. The exponent is positive and dimension-dependent, with no explicit endpoint or boundary claim.

Mathematical statement of the claimed result

Mathematical claim 1.
For every integer $d\ge3$ there are $0<\alpha_d\le1/3$ and $C_d<\infty$ such that each bounded continuous viscosity solution of \[\Delta_\infty u=\sum_{i,j=1}^d u_iu_ju_{ij}=0\quad\text{in }B_1\subset\mathbb R^d\] is claimed to lie in $C^{1,\alpha_d}(B_{1/2})$ and satisfy \[\|\nabla u\|_{L^\infty(B_{1/2})}+[\nabla u]_{C^{0,\alpha_d}(B_{1/2})}\le C_d(\sup_{B_1}u-\inf_{B_1}u).\] The equation is the unnormalized infinity Laplacian, with upper and lower $C^2$ tests including those of zero gradient.

Outline of the argument

Interior infinity-harmonic regularity: If uniform affine approximation were too slow, a rescaling would produce a global limiting solution that remains nonaffine. Intersecting-cylinder estimates retain affine fits at every center and scale. A Liouville argument uses a variance-penalized horizontal envelope and localized contacts to force that limit to be affine, contradicting the failure and recovering a Hölder gradient estimate.

Formal verification

Interior infinity-harmonic regularity: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources