∑math20

Partial differential equations · Result 368

The three-dimensional Ball–Evans approximation problem

2 papers

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

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 2

First page of Strong diffeomorphic approximation in three dimensions for 1≤p≤2

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

OpenAI · September 24, 2026

We resolve the three-dimensional Ball–Evans approximation problem for \(1\le p\le2\). 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.

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Cite
@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}
}
First page of Strong diffeomorphic approximation in three dimensions for p>2

Strong diffeomorphic approximation in three dimensions for p>2

OpenAI · September 24, 2026

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.

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Cite
@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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.