The three-dimensional Ball–Evans approximation problem

Claims to solve the Ball–Evans problem in 3D: every Sobolev homeomorphism is a strong limit of smooth diffeomorphisms.

2 papers September 24, 2026 Explained by AI

Engineers model a squashed solid as a map that sends each point to its new place. The map must be one-to-one, so no two bits of material overlap. Mathematicians asked whether such a rough map can be approached by smooth one-to-one maps, with the derivatives also close. This was known in the plane and false in dimension four and up for some exponents. Two papers claim a yes in three dimensions, for every finite exponent.

First step

2001

John Ball, who attributed the question to Lawrence Evans

Significance

Major

It claims to settle a well-known problem from nonlinear elasticity in the physical dimension three, for every finite exponent.

Trust

Claimed, not checked

Field

Partial differential equations

2 papers

The explanation

Start with a ball of rubber of radius 1. Move each point x to a point in the same direction, at distance √r, where r is the old distance from the centre. A point at r = 0.25 moves out to 0.5. A point at r = 0.01 moves out to 0.1. This map is one-to-one and fills the same ball. Near the centre it stretches hard. At r = 0.01 it stretches lengths around the centre by 10 and lengths away from it by 5. At r = 0.25 these numbers are 2 and 1. So the map has no derivative at the centre itself. Still, the total of the squared stretch over the ball is finite: it is 4.5π, about 14.14.

Maps like this are Sobolev homeomorphisms. A homeomorphism is a one-to-one, onto map that is continuous both ways. It is in W^{1,p} if its values and its weak derivative, a derivative defined by averages, have finite p-th power totals. Engineers use such maps for elastic solids. One-to-one means no two pieces of material overlap.

The question is whether smooth maps can copy this one. A diffeomorphism is a smooth one-to-one map with a smooth inverse. We want diffeomorphisms onto the same target whose values and derivatives get close in the p-th power sense. Ball posed this and credited the question to EvansSource 1. Smoothing alone is easy, but the usual averaging, called convolution, can break one-to-oneSource 2. For example, average the identity map with the half-turn x ↦ −x. Both are one-to-one, but the average sends every point to 0.

In the plane, Iwaniec, Kovalev and Onninen solved the problem for p > 1Source 3. Hencl and Pratelli handled p = 1Source 4. In dimension four and up, Campbell, Hencl and Tengvall built counterexamples for some small pSource 5. Dimension three stayed open.

The first paper claims the answer is yes in three dimensions for 1 ≤ p ≤ 2Source 6. The second paper claims yes for every finite p > 2Source 7. Neither paper assumes anything about the boundary of either setSource 8. The main obstacle is that three-dimensional topology gives one-to-one replacements but no bound on their derivativesSource 9. So each paper must measure energy directly. The first paper builds a one-to-one map that stretches only by bounded amounts and is close in energy. It then smooths that map with fine controlSource 10. The papers do not claim the inverse maps converge in the Sobolev senseSource 11.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. Both proofs are long and technical. No specialist review of either paper has been published.

What it doesn't show. This does not claim the inverse maps converge in a Sobolev sense, and it says nothing about dimensions four and up. It has no Lean check.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

A major claim if the proofs hold: it would settle a well-known problem in three dimensions. The proofs are long, there is no Lean check, and no specialist review is published.

Words used

Homeomorphism: A one-to-one, onto map that is continuous, with a continuous inverse.
Diffeomorphism: A smooth one-to-one, onto map whose inverse is also smooth.
Weak derivative: A derivative defined through averages, so it can exist for maps with corners or spikes.
Sobolev space W^{1,p}: The maps whose values and weak derivatives have finite p-th power totals over the domain.
Strong approximation: Getting close in both values and derivatives, measured by p-th power totals.
Convolution: Smoothing a map by replacing each value with a weighted average of nearby values.
Domain: A nonempty, connected open set.

The original papers (2)

Strong diffeomorphic approximation in three dimensions for 1≤p≤2 (PDF, opens in a new tab)
September 24, 2026
Strong diffeomorphic approximation in three dimensions for p>2 (PDF, opens in a new tab)
September 24, 2026

All quoted lines (11)

Ball's formulation explicitly attributes the question to Evans; his later discussion and Hencl's survey explain the surrounding variational questions.
Strong diffeomorphic approximation in three dimensions for 1≤p≤2 Back to text
Smooth approximation is elementary if injectivity is discarded, but convolution need not preserve it.
Strong diffeomorphic approximation in three dimensions for 1≤p≤2 Back to text
In the plane, Iwaniec, Kovalev, and Onninen proved strong diffeomorphic approximation for $1<p<\infty$ .
Strong diffeomorphic approximation in three dimensions for 1≤p≤2 Back to text
Hencl and Pratelli established the endpoint $p=1$ , including approximation by locally finite piecewise affine homeomorphisms.
Strong diffeomorphic approximation in three dimensions for 1≤p≤2 Back to text
Campbell, Hencl, and Tengvall corrected a gap in that construction and extended the range to $1\le p<\lfloor n/2\rfloor$ in dimensions $n\ge4$ , with Jacobian determinants of both signs on sets of positive measure.
Strong diffeomorphic approximation in three dimensions for 1≤p≤2 Back to text
Every $W^{1,p}$ homeomorphism between arbitrary bounded domains in $\R^3$ is a strong $W^{1,p}$ limit of smooth diffeomorphisms onto the same target.
Strong diffeomorphic approximation in three dimensions for 1≤p≤2 Back to text
We resolve the three-dimensional Ball–Evans approximation problem for every finite $p>2$ .
Strong diffeomorphic approximation in three dimensions for p>2 Back to text
No smoothness at the boundary, boundary extension of $f$ , or regularity of either boundary is assumed.
Strong diffeomorphic approximation in three dimensions for 1≤p≤2 Back to text
It supplies no strong derivative estimate.
Strong diffeomorphic approximation in three dimensions for p>2 Back to text
It first constructs an onto locally bi-Lipschitz map with small strong Sobolev error, and then proves a smoothing theorem with arbitrarily fine pointwise control.
Strong diffeomorphic approximation in three dimensions for 1≤p≤2 Back to text
Smooth invertibility of each approximant in Theorem … is not a claim of Sobolev convergence of the inverses.
Strong diffeomorphic approximation in three dimensions for 1≤p≤2 Back to text

Other sources

openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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