Some Open Problems in Elasticity — Uniqueness of equilibrium

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\Omega\subset\mathbb R^3$ be a bounded Lipschitz domain. Write $\partial\Omega=\Gamma_D\cup\Gamma_N\cup N$, where $\Gamma_D,\Gamma_N$ are disjoint relatively open subsets, $\mathcal H^2(N)=0$, and $\mathcal H^2(\Gamma_D)>0$. Here $\mathcal H^2$ is two-dimensional Hausdorff measure. A deformation $y:\Omega\to\mathbb R^3$ belongs to the Sobolev space $W^{1,1}(\Omega;\mathbb R^3)$, so its weak derivative $Dy$ is defined almost everywhere. Prescribe $y=\bar y$ on $\Gamma_D$ in the trace sense and zero applied traction on $\Gamma_N$. Let $\mathbb M_+^{3\times3}=\{A\in\mathbb R^{3\times3}:\det A>0\}$. The homogeneous stored-energy density $W:\mathbb M_+^{3\times3}\to[0,\infty)$ is $C^1$, satisfies frame indifference $W(RA)=W(A)$ for $R\in\operatorname{SO}(3)$, and satisfies \[ W(A)\longrightarrow+\infty\quad\text{as}\quad\det A\longrightarrow0^+. \] Extend $W$ by $+\infty$ when $\det A\le0$. Put \[ I(y)=\int_\Omega W(Dy(x))\,dx,\qquad \mathcal A=\{y\in W^{1,1}(\Omega;\mathbb R^3):I(y)<\infty,\ y|_{\Gamma_D}=\bar y\}. \] Assume $\mathcal A\ne\varnothing$. For a matrix $A$, $\operatorname{cof}A$ denotes its cofactor matrix and $|A|$ its Frobenius norm. For the pure-displacement problem let $\Gamma_D=\partial\Omega$ and $\Gamma_N=\varnothing$, and suppose that $\overline\Omega$ is homeomorphic to a closed three-dimensional ball. A strictly polyconvex density has a representation \[W(A)=g(A,\operatorname{cof}A,\det A)\] with $g$ strictly convex. A sufficiently smooth, orientation-preserving equilibrium deformation solves \[ \operatorname{div}D_AW(Dy)=0\quad\text{in }\Omega, \qquad y=\bar y\quad\text{on }\partial\Omega, \qquad\det Dy>0. \] Equivalently its stress satisfies $\int_\Omega D_AW(Dy):D\varphi\,dx=0$ for test fields $\varphi$ vanishing on the boundary; $A:B=\operatorname{tr}(A^TB)$. Smoothness here is sufficient to interpret these equilibrium equations classically; the question concerns smooth solutions, rather than singular cavitating deformations.

Statement recorded in the supplied source.

For homogeneous bodies homeomorphic to a ball and strictly polyconvex $W$, are sufficiently smooth solutions of the pure-displacement equilibrium problem unique for the given boundary deformation, or can one construct two distinct such solutions? [S2]

Short English statement

Does strict polyconvexity force uniqueness of smooth equilibria with all boundary positions prescribed when the body has the topology of a ball?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2000008 (AMR-019-0008). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 John M. Ball, Some Open Problems in Elasticity, in Geometry, Mechanics, and Dynamics, Springer, 2002, pp. 3--59. Sections 2.1 and 2.6, Problem 8 and complete discussion, printed pp. 2--3 and 16--17. https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf.