The three-dimensional Ball–Evans approximation problem

1 · The essentials

The manuscripts consider one-to-one, continuous ways of mapping one bounded three-dimensional region onto another, with a continuous way back. These maps need not be smooth, but their rates of change must satisfy a specified mathematical condition. The question is whether they can be approached arbitrarily closely by smooth maps that also have smooth ways back. The approximation must capture both the map and its rates of change, while keeping exactly the same target region. The first manuscript claims this for the parameter range $1 \leq p \leq 2$ , and the companion claims it for every finite $p > 2$ . Together, their stated results cover every finite $p \geq 1$ . The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine replacing a rough reshaping of a piece of clay with increasingly smooth reshapings, without folding different points onto one another and while filling exactly the same final shape. The required closeness concerns not just where the clay ends up, but also how its reshaping changes from place to place.

Why it matters

The claimed results connect continuous, invertible maps with smooth, invertible maps under an approximation requirement that also controls derivatives. Their stated scope includes arbitrary bounded domains in three dimensions and every finite exponent $p \geq 1$ . The repository describes the combined claim as a resolution of the three-dimensional Ball–Evans approximation problem.

2 · Going further

The precise claim concerns a homeomorphism $f : Ω \to Ω^{'}$ belonging to $W^{1, p}$ , where $Ω$ and $Ω^{'}$ are arbitrary bounded domains in $R^{3}$ . The manuscripts claim that there are smooth diffeomorphisms $f_{k} : Ω \to Ω^{'}$ converging strongly to $f$ in $W^{1, p}$ . Strong convergence here requires convergence in the Sobolev norm, so both the maps and their weak first derivatives are approximated, rather than only their pointwise values. Each approximating map is onto the same target $Ω^{'}$ .

The source identifies this as the three-dimensional Ball–Evans approximation problem. Manuscript 1 states the result for $1 \leq p \leq 2$ , while Manuscript 2 states the corresponding result for every finite $p > 2$ . Thus the two stated ranges together cover $1 \leq p < \infty$ . The supplied abstracts state the conclusions but give no construction, intermediate steps, or main idea of the argument; a description of the method cannot be supplied from this source.

Useful notions

Continuous maps and continuous inverses
Bounded domains in three-dimensional space
Derivatives and weak derivatives
Sobolev spaces $W^{1, p}$
Convergence in a norm

What was known before / what is new

Before

The source names the question as the three-dimensional Ball–Evans approximation problem. It gives no account of earlier results, previously settled exponent ranges, or the history of the problem.

New

The main manuscript claims strong diffeomorphic approximation for every $W^{1, p}$ homeomorphism between arbitrary bounded three-dimensional domains when $1 \leq p \leq 2$ . The companion manuscript claims the same conclusion for every finite $p > 2$ . In both statements, the smooth approximations map onto exactly the original target.

Limits and assumptions

The statements require a $W^{1, p}$ homeomorphism between bounded domains in $R^{3}$ , with $1 \leq p < \infty$ . They do not state conclusions for $p = \infty$ , other dimensions, unbounded domains, or maps lacking the homeomorphism hypothesis. The supplied abstracts contain no argument to assess, and the main result is not formalized in Lean and has not been formally verified. The source does not identify what remains open beyond the stated scope.

Mini glossary

Homeomorphism: A continuous bijection with a continuous inverse.
Diffeomorphism: A smooth bijection with a smooth inverse.
Sobolev space $W^{1, p}$: A space of maps whose values and weak first derivatives are integrable to the power $p$ .
Strong $W^{1, p}$ convergence: Convergence in the Sobolev norm, controlling both the maps and their weak first derivatives.
Weak derivative: A derivative understood through integration against smooth test functions rather than requiring ordinary differentiation at every point.

Keywords : Ball–Evans approximation · Sobolev homeomorphisms · Diffeomorphic approximation

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Resolves the three-dimensional Ball–Evans approximation problem: every $W^{1, p}$ homeomorphism between arbitrary bounded domains in ℝ³, for $1 \leq p < \infty$ , is a strong $W^{1, p}$ limit of smooth diffeomorphisms onto the same target.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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Strong diffeomorphic approximation in three dimensions for 1≤p≤2

24/09/2026

To be verified

We resolve the three-dimensional Ball–Evans approximation problem for $1 \leq p \leq 2$ . Every $W^{1, p}$ homeomorphism between arbitrary bounded domains in ℝ³ is a strong $W^{1, p}$ limit of smooth diffeomorphisms onto the same target.

PDF GitHub folder

Strong diffeomorphic approximation in three dimensions for p>2

24/09/2026

To be verified

We resolve the three-dimensional Ball–Evans approximation problem for every finite p > 2. Every $W^{1, p}$ homeomorphism between arbitrary bounded domains in ℝ³ can be approximated strongly in $W^{1, p}$ by smooth diffeomorphisms onto the same target.

PDF GitHub folder