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Strong diffeomorphic approximation in three dimensions for 1≤p≤2
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Abstract
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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Possibly relevant literature (found by the agent, not checked)
- Jani Onninen and Pekka Pankka, Bing meets Sobolev (arXiv:1908.09193, 2019). Constructs wild involutions of the 3-sphere in $W^{1,p}$ for every $1\le p<2$, i.e. three-dimensional Sobolev homeomorphisms with wild topological behaviour in the exponent range of the paper's main theorem, which assumes no boundary or inverse regularity.
- Stanislav Hencl, Aleksis Koski and Jani Onninen, Sobolev homeomorphic extensions from two to three dimensions (arXiv:2112.14767, 27 December 2021). Its study of homeomorphic extensions from sphere boundaries into three-dimensional interiors may provide context for the manuscript's cell-completion construction, which fills volumes while preserving boundary values and controlling Sobolev energy.
Digests and expositions
- Gradient Damage, Boltzmann Equation, Hot Spots Conjecture and 7 More in Partial Differential Equations (video compilation, 15 October 2026). The chapter starting at 1:04 introduces the three-dimensional Ball–Evans problem through nonlinear elasticity and presents the family's claimed approximation result as an illustration rather than a proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 368
Independent sites; their content is not reviewed here.
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}
}Family 368: The three-dimensional Ball–Evans approximation problem
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.
Other papers in this family:
- Strong diffeomorphic approximation in three dimensions for p>2 · September 24, 2026