Attainment of the three-dimensional neo-Hookean energy in the regular admissible class
Let $\Omega \subset \mathbb{R}^3$ be a smooth bounded domain, $\widetilde\Omega \Subset \Omega$ a smooth subdomain, and $b \colon \overline\Omega \to b(\overline\Omega)$ an orientation-preserving bi-Lipschitz homeomorphism. Let $H \colon (0,\infty) \to [0,\infty)$ be convex with $H(t) \to \infty$ as $t \downarrow 0$ and $H(t)/t \to \infty$ as $t \to \infty$, and let $$E(u) = \int_\Omega \bigl(|Du|^2 + H(\det Du)\bigr)\,dx$$ be the neo-Hookean energy. Let $\mathcal{A}$ be the set of $u \in H^1(\Omega; \mathbb{R}^3)$ that equal $b$ on $\Omega \setminus \widetilde\Omega$, are one-to-one almost everywhere, satisfy $\det Du > 0$ almost everywhere and $E(u) < \infty$, and satisfy the divergence identities $$\operatorname{Div}\bigl((\operatorname{adj} Du)\, g(u)\bigr) = (\operatorname{div} g)(u) \det Du \quad \text{in } \mathcal{D}'(\Omega) \qquad \text{for every } g \in C_c^1(\mathbb{R}^3; \mathbb{R}^3),$$ where $Du = (\partial_j u_i)_{ij}$ and $\operatorname{adj} Du = (\operatorname{cof} Du)^T$. Conti and De Lellis showed that such classes are not closed under weak convergence in $H^1$; with $|Du|^2$ replaced by $|Du|^p$ for some $p > 2$, minimisers exist in classes of this kind (Müller and Spector; Henao and Mora-Corral). Barchiesi, Henao, Mora-Corral and Rodiac proved that $E$ plus twice the total variation of the singular part of the derivative of $u^{-1}$, which equals $E$ on $\mathcal{A}$, attains its minimum on a larger class (2024). For $\Omega$ a ball and $H$ of moderate growth at $0$ and at $\infty$, they constructed $\widetilde\Omega$ and $b$ for which $\inf_{\mathcal{A}} E$ is strictly larger than the infimum of $E$ over the weak closure of $\mathcal{A}$ in $H^1$ (2026).
Problem. Whenever $\mathcal{A} \ne \varnothing$, is $\inf_{u \in \mathcal{A}} E(u)$ attained by an element of $\mathcal{A}$?
Source: Barchiesi, Henao, Mora-Corral and Rodiac (2024), abstract and §1.2, for the admissible class defined there; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, existence of minimizers, neo-Hookean materials, nonlinear elasticity
References
@article{barchiesi2024,
author = {Barchiesi, Marco and Henao, Duvan and Mora-Corral, Carlos and Rodiac, R{\'e}my},
title = {A relaxation approach to the minimization of the neo-{H}ookean energy in {3D}},
journal = {SIAM J. Math. Anal.},
volume = {56},
number = {6},
year = {2024},
pages = {7830--7845},
doi = {10.1137/23M1614547},
eprint = {2311.02952},
archivePrefix = {arXiv},
}
@misc{barchiesi2026,
author = {Barchiesi, Marco and Henao, Duvan and Mora-Corral, Carlos and Rodiac, R{\'e}my},
title = {A {L}avrentiev phenomenon in the neo-{H}ookean model},
year = {2026},
eprint = {2603.22873},
archivePrefix = {arXiv},
}
@article{conti2003,
author = {Conti, Sergio and De Lellis, Camillo},
title = {Some remarks on the theory of elasticity for compressible {N}eohookean materials},
journal = {Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)},
volume = {2},
number = {3},
year = {2003},
pages = {521--549},
}
@article{muller1995,
author = {M{\"u}ller, Stefan and Spector, Scott J.},
title = {An existence theory for nonlinear elasticity that allows for cavitation},
journal = {Arch. Rational Mech. Anal.},
volume = {131},
number = {1},
year = {1995},
pages = {1--66},
doi = {10.1007/BF00386070},
}
@article{henao2010,
author = {Henao, Duvan and Mora-Corral, Carlos},
title = {Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity},
journal = {Arch. Ration. Mech. Anal.},
volume = {197},
number = {2},
year = {2010},
pages = {619--655},
doi = {10.1007/s00205-009-0271-4},
}
@misc{aim-neo-hookean-critical-attainment,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Attainment of the three-dimensional neo-Hookean energy in the regular admissible class}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/381-neo-hookean-critical-attainment.md},
note = {Entry 381 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-00619, "Attainment of the three-dimensional neo-Hookean energy in the regular admissible class", clopen.solutions.