Smooth approximation of three-dimensional Sobolev homeomorphisms
Let $\Omega, \Delta \subset \mathbb{R}^3$ be bounded domains and $1 \le p < \infty$. Motivated by nonlinear elasticity, Ball, ascribing the question to Evans, asked whether a homeomorphism in $W^{1,p}$ can be approximated in the $W^{1,p}$ norm by diffeomorphisms or by piecewise affine homeomorphisms. In the plane, Iwaniec, Kovalev and Onninen proved this for $1 < p < \infty$, and Hencl and Pratelli for $p = 1$, with approximants that have the same image as the given map. In dimensions $n \ge 4$, Hencl and Vejnar for $p = 1$, and Campbell, Hencl and Tengvall for $1 \le p < \lfloor n/2 \rfloor$, constructed $W^{1,p}$ homeomorphisms of a cube that are not $W^{1,p}_{\mathrm{loc}}$ limits of diffeomorphisms. Campbell, D'Onofrio and Vítek approximated locally finitely piecewise affine homeomorphisms in dimensions three and four by diffeomorphisms.
Problem. Is it true that for every such pair $\Omega, \Delta$, every $1 \le p < \infty$ and every orientation-preserving homeomorphism $f \colon \Omega \to \Delta$ in $W^{1,p}(\Omega; \mathbb{R}^3)$, there are smooth diffeomorphisms $f_j \colon \Omega \to \Delta$ with $$\|f_j - f\|_{W^{1,p}(\Omega)} \to 0?$$ Equality of boundary values is not required.
Source: Ball (2001), who ascribes the question to Evans; Hencl (2025), Open problem 1, for dimensions n ≥ 3. Stated by AIM — Open Applied Problems in dimension three, for orientation-preserving homeomorphisms and diffeomorphisms onto the same target.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, diffeomorphic approximation, nonlinear elasticity, Sobolev homeomorphisms
References
@incollection{ball2001,
author = {Ball, John M.},
title = {Singularities and computation of minimizers for variational problems},
booktitle = {Foundations of Computational Mathematics ({O}xford, 1999)},
editor = {DeVore, Ronald A. and Iserles, Arieh and S{\"u}li, Endre},
series = {London Math. Soc. Lecture Note Ser.},
volume = {284},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2001},
pages = {1--20},
doi = {10.1017/CBO9781107360198.002},
}
@misc{hencl2025,
author = {Hencl, Stanislav},
title = {Ball--{E}vans approximation problem: recent progress and open problems},
year = {2025},
eprint = {2502.01336},
archivePrefix = {arXiv},
note = {Section 2, Open problem 1},
}
@article{iwaniec2011,
author = {Iwaniec, Tadeusz and Kovalev, Leonid V. and Onninen, Jani},
title = {Diffeomorphic approximation of {S}obolev homeomorphisms},
journal = {Arch. Ration. Mech. Anal.},
volume = {201},
number = {3},
year = {2011},
pages = {1047--1067},
doi = {10.1007/s00205-011-0404-4},
}
@article{hencl2018,
author = {Hencl, Stanislav and Pratelli, Aldo},
title = {Diffeomorphic approximation of {$W^{1,1}$} planar {S}obolev homeomorphisms},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {20},
number = {3},
year = {2018},
pages = {597--656},
doi = {10.4171/JEMS/774},
}
@article{hencl2016,
author = {Hencl, Stanislav and Vejnar, Benjamin},
title = {Sobolev homeomorphism that cannot be approximated by diffeomorphisms in {$W^{1,1}$}},
journal = {Arch. Ration. Mech. Anal.},
volume = {219},
number = {1},
year = {2016},
pages = {183--202},
doi = {10.1007/s00205-015-0895-5},
}
@article{campbell2018,
author = {Campbell, Daniel and Hencl, Stanislav and Tengvall, Ville},
title = {Approximation of {$W^{1,p}$} {S}obolev homeomorphism by diffeomorphisms and the signs of the {J}acobian},
journal = {Adv. Math.},
volume = {331},
year = {2018},
pages = {748--829},
doi = {10.1016/j.aim.2018.04.017},
}
@article{campbell2026,
author = {Campbell, Daniel and D'Onofrio, Luigi and V{\'i}tek, Tom{\'a}{\v s}},
title = {Diffeomorphic approximation of piecewise affine homeomorphisms},
journal = {J. Geom. Anal.},
volume = {36},
number = {5},
year = {2026},
pages = {157},
doi = {10.1007/s12220-026-02398-w},
eprint = {2507.02854},
archivePrefix = {arXiv},
}
@misc{aim-ball-evans-three-dimensions,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Smooth approximation of three-dimensional Sobolev homeomorphisms}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/153-ball-evans-three-dimensions.md},
note = {Entry 153 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We resolve the three-dimensional Ball--Evans approximation problem for $1\le p\le2$. 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.
We resolve the three-dimensional Ball--Evans approximation problem for every finite $p>2$. Every $W^{1,p}$ homeomorphism between arbitrary bounded domains in $\R^3$ can be approximated strongly in $W^{1,p}$ by smooth diffeomorphisms onto the same target.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00665, "Smooth approximation of three-dimensional Sobolev homeomorphisms", clopen.solutions.