Some Open Problems in Elasticity — Regularity of minimizers

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. Assume the hypotheses of the source's Theorem 2.1: $W(A)=g(A,\operatorname{cof}A,\det A)$ for a convex extended-real-valued function $g$, and \[ W(A)\ge c_0\left(|A|^2+|\operatorname{cof}A|^{3/2}\right)-c_1 \quad(c_0>0). \] Let $y^*\in\mathcal A$ be a global minimizer supplied by that theorem. The source uses smooth to mean $C^\infty$ in $\Omega$ and $C^\infty$ up to the boundary away from the displacement--traction interface $\overline{\Gamma_D}\cap\overline{\Gamma_N}$. Regularity there means local smooth extension in boundary coordinates at each boundary point outside that interface.

Statement recorded in the supplied source.

Determine the additional hypotheses on $W$, the domain, and the prescribed boundary deformation under which every minimizer $y^*$ from Theorem 2.1 has this smoothness, including interior and boundary regularity away from the mixed-boundary interface. In particular, determine what is required beyond smooth data and strict polyconvexity, meaning that the representing function $g$ is strictly convex. [S2]

Short English statement

Find conditions that make the minimizers in Ball’s precise existence theorem smooth, both inside the body and at smooth boundary portions away from the displacement--traction interface.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2000003 (AMR-019-0003). 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. Theorem 2.1, printed p. 4; Section 2.3, Problem 3 and its complete following paragraph, pp. 6--7. https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf.