Existence of elastic equilibria under quasiconvexity and infinite compression energy
Let $\Omega \subset \mathbb{R}^3$ be a bounded domain with Lipschitz boundary and let $p > 3$. Let $W : \mathbb{R}^{3 \times 3} \to [0, \infty]$ be finite and $C^1$ on $\mathrm{GL}^+(3) = \{F : \det F > 0\}$ and $+\infty$ elsewhere, with $W(RF) = W(F)$ for all $R \in SO(3)$, $W(F) \ge c|F|^p - C$ for some constants $c > 0$ and $C$, and $W(F_j) \to \infty$ whenever $\det F_j \downarrow 0$. Assume that $W$ is quasiconvex: $$W(F) \le \int_{(0,1)^3} W(F + D\varphi(x)) \, dx \quad \text{for every } F \in \mathbb{R}^{3 \times 3} \text{ and } \varphi \in C_c^\infty((0,1)^3; \mathbb{R}^3).$$ For a boundary datum $g$ let $$\mathcal{A}_g = \Bigl\{ y \in W^{1,p}(\Omega; \mathbb{R}^3) : \operatorname{Tr} y = g, \ I(y) := \int_\Omega W(Dy) \, dx < \infty \Bigr\}.$$ Ball proved that minimizers exist when $W$ is polyconvex and satisfies suitable growth conditions from below. By the theorems of Morrey and of Acerbi and Fusco, minimizers also exist for quasiconvex integrands $f$ with $C_1 |A|^p - C_0 \le f(A) \le C_2 (|A|^p + 1)$; such an upper bound is incompatible with $W \to \infty$ as $\det F \downarrow 0$.
Problem. For every such $\Omega$, $p$ and $W$, and every $g$ for which $\mathcal{A}_g$ is nonempty, does $I$ attain its infimum on $\mathcal{A}_g$? No polyconvexity, no polynomial upper growth and no global injectivity constraint is assumed.
Source: Ball (2002), §2.2, Problem 1; stated in this form, with Dirichlet data on the whole boundary and p-growth from below with p > 3, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, calculus of variations, nonlinear elasticity, quasiconvexity
References
@incollection{ball2002,
author = {Ball, John M.},
title = {Some open problems in elasticity},
booktitle = {Geometry, Mechanics, and Dynamics},
editor = {Newton, Paul and Holmes, Philip and Weinstein, Alan},
publisher = {Springer},
address = {New York},
year = {2002},
pages = {3--59},
doi = {10.1007/0-387-21791-6_1},
}
@article{ball1977,
author = {Ball, John M.},
title = {Convexity conditions and existence theorems in nonlinear elasticity},
journal = {Arch. Rational Mech. Anal.},
volume = {63},
number = {4},
year = {1977},
pages = {337--403},
doi = {10.1007/BF00279992},
}
@article{morrey1952,
author = {Morrey, Charles B., Jr.},
title = {Quasi-convexity and the lower semicontinuity of multiple integrals},
journal = {Pacific J. Math.},
volume = {2},
number = {1},
year = {1952},
pages = {25--53},
doi = {10.2140/pjm.1952.2.25},
}
@article{acerbi1984,
author = {Acerbi, Emilio and Fusco, Nicola},
title = {Semicontinuity problems in the calculus of variations},
journal = {Arch. Rational Mech. Anal.},
volume = {86},
number = {2},
year = {1984},
pages = {125--145},
doi = {10.1007/BF00275731},
}
@misc{aim-quasiconvex-physical-elastic-minimizers,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Existence of elastic equilibria under quasiconvexity and infinite compression energy}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/368-quasiconvex-physical-elastic-minimizers.md},
note = {Entry 368 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-00634, "Existence of elastic equilibria under quasiconvexity and infinite compression energy", clopen.solutions.