Result 368, Partial differential equations

The three-dimensional Ball–Evans approximation problem

Resolves the three-dimensional Ball–Evans approximation problem: every W1,pW^{1,p} homeomorphism between arbitrary bounded domains in ℝ3, for 1≤p<∞1\le p\lt \infty, is a strong W1,pW^{1,p} limit of smooth diffeomorphisms onto the same target.

Proof

The bigger picture

Why it matters

Can a rough but reversible transformation of a three-dimensional region be replaced by smooth, reversible transformations without losing its derivative information? Two unreviewed manuscripts report that this is possible across the full range of finite integrability exponents.

What changes?

Together, the manuscripts claim that every W^{1,p} homeomorphism between arbitrary bounded domains in three-dimensional space admits smooth diffeomorphisms onto the same target that converge strongly in W^{1,p}, for every finite p at least 1. A homeomorphism is a continuous bijection with continuous inverse; a diffeomorphism and its inverse are smooth. W^{1,p} requires the map and its weak first derivatives to have integrable pth powers. Strong convergence controls both the maps and those derivatives in this integral sense.

What does that help mathematicians do?

This would let researchers approximate an entire class of rough invertible maps without sacrificing invertibility or changing the target region. In particular, strong convergence implies convergence of the integrals of the pth power of derivative size. Thus, for this basic measure of deformation, smooth approximations would preserve the limiting value, not merely resemble the original map point by point.

Are there practical applications?

The immediate value is foundational for studying partial differential equations and variational problems involving invertible maps. The claimed result would bridge rough admissible transformations and smooth ones while controlling their derivatives. The abstracts establish no numerical procedure or computational guarantee; practical approximation schemes would require additional construction and analysis.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

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

September 24, 2026 62 pages

We resolve the three-dimensional Ball–Evans approximation problem for 1≤p≤21\le p\le2. Every W1,pW^{1,p} homeomorphism between arbitrary bounded domains in ℝ3 is a strong W1,pW^{1,p} limit of smooth diffeomorphisms onto the same target.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2-September-24-2026,
  author = {{OpenAI}},
  title = {{Strong diffeomorphic approximation in three dimensions for $1\le p\le2$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2-September-24-2026/main.pdf}{OAI:Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2-September-24-2026}},
  year = {2026}
}

Strong diffeomorphic approximation in three dimensions for p>2

September 24, 2026 130 pages

We resolve the three-dimensional Ball–Evans approximation problem for every finite p > 2. Every W1,pW^{1,p} homeomorphism between arbitrary bounded domains in ℝ3 can be approximated strongly in W1,pW^{1,p} by smooth diffeomorphisms onto the same target.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2-September-24-2026,
  author = {{OpenAI}},
  title = {{Strong diffeomorphic approximation in three dimensions for $p>2$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2-September-24-2026/main.pdf}{OAI:Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2-September-24-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.