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The Ball-Evans approximation problem in three dimensions: strong approximation of Sobolev homeomorphisms by diffeomorphisms

In nonlinear elasticity a deformation is a map between domains whose injectivity expresses non-interpenetration. Smooth approximation of a Sobolev map is elementary if injectivity is dropped, but convolution destroys it. Ball, attributing the question to Evans, asked whether a Sobolev homeomorphism f∈W1,p(Ω;Rn)f\in W^{1,p}(\Omega;\mathbb R^n) can be approximated in the W1,pW^{1,p} norm by diffeomorphisms (or piecewise affine homeomorphisms). In the plane this was proved for 1<p<∞1<p<\infty (Iwaniec-Kovalev-Onninen) and p=1p=1 (Hencl-Pratelli); in dimensions n≥4n\ge4 it fails for 1≤p<⌊n/2⌋1\le p<\lfloor n/2\rfloor (Hencl-Vejnar, Campbell-Hencl-Tengvall). In dimension three: for bounded domains Ω,Λ⊂R3\Omega,\Lambda\subset\mathbb R^3 and 1≤p<∞1\le p<\infty, is every W1,pW^{1,p} homeomorphism f:Ω→Λf:\Omega\to\Lambda a strong W1,pW^{1,p} limit of smooth diffeomorphisms onto Λ\Lambda?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Construction
Field
Geometric function theory; Sobolev homeomorphisms and nonlinear elasticity
Posed by
John M. Ball, attributing the question to Lawrence C. Evans
Year posed
2001
Years open
25y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
35 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorems: for nonempty bounded connected open Ω,Λ⊂R3\Omega,\Lambda\subset\mathbb R^3 and a homeomorphism f∈W1,p(Ω;R3)f\in W^{1,p}(\Omega;\mathbb R^3) onto Λ\Lambda, there are C∞C^\infty diffeomorphisms fj:Ω→Λf_j:\Omega\to\Lambda with ∥fj−f∥W1,p→0\|f_j-f\|_{W^{1,p}}\to0; proved for 1≤p≤21\le p\le2 in one manuscript and for all finite p>2p>2 in the other. The method first builds onto locally bi-Lipschitz approximants with small Sobolev error, then smooths them with fine pointwise control using three-dimensional PL topology (Moise, Bing, Hamilton). It does not give convergence of inverses, approximation by piecewise affine maps with inverse control, or anything in dimensions n≥4n\ge4 for p≥⌊n/2⌋p\ge\lfloor n/2\rfloor.

What the AI did

The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts dated September 24, 2026: one for 1≤p≤21\le p\le2 and one for every finite p>2p>2.

Verification

No independent mathematician has checked this yet. Checked here: the main theorems of both manuscripts were read against Ball's formulation. Together they state that for every 1≤p<∞1\le p<\infty and arbitrary nonempty bounded connected open Ω,Λ⊂R3\Omega,\Lambda\subset\mathbb R^3, each W1,pW^{1,p} homeomorphism f:Ω→Λf:\Omega\to\Lambda is a W1,pW^{1,p} limit of C∞C^\infty diffeomorphisms onto the same Λ\Lambda, with no boundary regularity, inverse Sobolev regularity or Jacobian hypothesis. Classified partial because the posed question is dimension-free: dimension two was already known, dimensions four and up are negative for small pp and otherwise untouched. The papers state that no convergence of inverses is claimed. Not refereed; no Lean formalization.

Sources

Changelog1 change

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