Do finite-energy elastic minimizers satisfy force balance?
Let $\Omega \subset \mathbb{R}^3$ be smooth and bounded, and let $y_0$ be a smooth orientation-preserving diffeomorphism on a neighbourhood of $\overline{\Omega}$. Let $W \in C^1(GL^+(3))$ be a frame-indifferent, polyconvex stored energy, extended by $+\infty$ when $\det F \le 0$, such that for some $p > 3$ and positive constants $c, C$ $$W(F) \ge c|F|^p - C, \qquad W(F) \to \infty \ \text{as } \det F \downarrow 0,$$ $$|DW(F)F^T| + |F^T DW(F)| \le C\bigl(1 + W(F)\bigr),$$ after adding a constant so that $W \ge 0$. Polyconvex means that $W(F)$ is a convex function of $(F, \operatorname{cof} F, \det F)$; the class includes energies with inverse powers of the determinant. The difficulty is that $\det(Dy + \tau D\varphi)$ need not be positive, so the difference quotients of the energy along $y + \tau\varphi$ cannot simply be passed to the limit; if $y \in W^{1,\infty}$ and $\det Dy \ge \varepsilon > 0$ almost everywhere, a local minimizer does satisfy the identity below (Ball). Under the bound $|DW(F)F^T| + |F^T DW(F)| \le C(1 + W(F))$, Ball proved that every $W^{1,p}$ local minimizer satisfies $$\int_\Omega \bigl[DW(Dy)\,Dy^T\bigr] : D\phi(y)\, dx = 0 \quad\text{and}\quad \int_\Omega \bigl[W(Dy)\,\mathbf{1} - Dy^T DW(Dy)\bigr] : D\varphi\, dx = 0$$ for all $\varphi \in C_c^1(\Omega; \mathbb{R}^3)$ and all $\phi \in C^1(\mathbb{R}^3; \mathbb{R}^3)$ with $\phi$ and $D\phi$ bounded and $\phi(y) = 0$ on $\partial\Omega$.
Problem. If $y$ globally minimizes $\int_\Omega W(Dy)\, dx$ over $W^{1,p}$ maps with trace $y_0$, must $DW(Dy) \in L^1_{\mathrm{loc}}$ and $$\int_\Omega DW(Dy) : D\varphi\, dx = 0 \qquad \bigl(\varphi \in C_c^\infty(\Omega; \mathbb{R}^3)\bigr)?$$
Source: Ball (2002), Problem 5, after Ball (1984). Stated by AIM — Open Applied Problems for global minimizers under specific growth hypotheses.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, Euler–Lagrange equations, nonlinear elasticity, polyconvexity
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},
}
@incollection{ball1984,
author = {Ball, J. M.},
title = {Minimizers and the {E}uler--{L}agrange equations},
booktitle = {Trends and Applications of Pure Mathematics to Mechanics ({P}alaiseau, 1983)},
series = {Lecture Notes in Physics},
volume = {195},
publisher = {Springer},
address = {Berlin},
year = {1984},
pages = {1--4},
doi = {10.1007/3-540-12916-2_47},
}
@misc{aim-elastic-minimizer-equilibrium,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Do finite-energy elastic minimizers satisfy force balance?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/222-elastic-minimizer-equilibrium.md},
note = {Entry 222 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-00587, "Do finite-energy elastic minimizers satisfy force balance?", clopen.solutions.