Approximation preserving a compressible elastic energy
Let $\Omega, \Delta \subset \mathbb{R}^2$ be bounded domains, $1 \le p < \infty$ and $a > 0$, and let $f\colon \Omega \to \Delta$ be an orientation-preserving Sobolev homeomorphism with $J_f = \det Df > 0$ almost everywhere and $$E(f) = \int_\Omega \bigl(|Df|^p + J_f^{-a}\bigr)\,dx < \infty.$$ The term $J_f^{-a}$ penalises compression. The Ball–Evans approximation problem asks for approximation of Sobolev homeomorphisms by diffeomorphisms in the Sobolev norm. In the plane, Iwaniec, Kovalev and Onninen ($1 < p < \infty$, 2011) and Hencl and Pratelli ($p = 1$, 2018) proved that every homeomorphism $f \in W^{1,p}(\Omega, \mathbb{R}^2)$ is a limit in $W^{1,p}$, and uniformly, of smooth diffeomorphisms of $\Omega$ onto $f(\Omega)$.
Problem. Must there exist orientation-preserving smooth diffeomorphisms $f_j\colon \Omega \to \Delta$ with $f_j \to f$ strongly in $W^{1,p}(\Omega)$ and $E(f_j) \to E(f)$? The question is for all $\Omega$, $\Delta$, $p$, $a$ and $f$ as above.
Source: Hencl (2025), §2, Open problem 4, for the functional E_1, as an extension of the Ball–Evans approximation problem. Stated by AIM — Open Applied Problems for bounded domains and a fixed target.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- FA
- Keywords
- AIM Open Applied Problems, diffeomorphic approximation, nonlinear elasticity, Sobolev homeomorphisms
References
@article{hencl2025,
author = {Hencl, Stanislav},
title = {Ball--{E}vans approximation problem: recent progress and open problems},
journal = {J. Math. Sci. (N.Y.)},
volume = {298},
number = {4},
year = {2025},
pages = {524--531},
doi = {10.1007/s10958-025-07739-8},
eprint = {2502.01336},
archivePrefix = {arXiv},
}
@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},
eprint = {1009.0286},
archivePrefix = {arXiv},
}
@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},
}
@incollection{ball2002,
author = {Ball, John M.},
title = {Some open problems in elasticity},
booktitle = {Geometry, Mechanics, and Dynamics},
publisher = {Springer},
address = {New York},
year = {2002},
pages = {3--59},
doi = {10.1007/0-387-21791-6_1},
}
@misc{aim-elastic-energy-approximation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Approximation preserving a compressible elastic energy}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/154-elastic-energy-approximation.md},
note = {Entry 154 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00857, "Approximation preserving a compressible elastic energy", clopen.solutions.