we-solve (beta)

Math-2847 · mathematics · Pde

Uniform interior gradient Hölder estimates for infinity-harmonic functions

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every $d\geq 3$, are there $\alpha _d\in (0,1/3]$ and $C_d<infinity$ such that every bounded continuous viscosity solution of the infinity-Laplace equation on the unit ball belongs to $C^{1,\alpha _d}$ on the half ball and has ||grad $u||_infinity+[grad$ $u]_{C^\alpha _d}\leq C_d$ $osc_{B1}u$?

Short English statement

For every $d\geq 3$, are there $\alpha _d\in (0,1/3]$ and $C_d<infinity$ such that every bounded continuous viscosity solution of the infinity-Laplace equation on the unit ball belongs to $C^{1,\alpha _d}$ on the half ball and has ||grad $u||_infinity+[grad$ $u]_{C^\alpha _d}\leq C_d$ $osc_{B1}u$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2847-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2847

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.

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

Go to contribution →