Some Open Problems in Elasticity — Existence 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. The extended density is quasiconvex when, for every matrix $A$, bounded open set $U\subset\mathbb R^3$ and $\varphi\in C_c^\infty(U;\mathbb R^3)$, \[ |U|W(A)\le\int_U W(A+D\varphi)\,dx. \] The inequality is interpreted for extended real values. Assume also the source's coercivity hypothesis: for constants $c_0>0$ and $c_1\ge0$, \[ W(A)\ge c_0\left(|A|^2+|\operatorname{cof}A|^{3/2}\right)-c_1. \]

Statement recorded in the supplied source.

Does $I$ attain its infimum on $\mathcal A$ for every quasiconvex density and admissible boundary data satisfying these hypotheses? More generally, establish quasiconvex existence theorems compatible with the determinant blow-up condition, rather than upper polynomial growth on all matrices. [S2]

Short English statement

Prove existence of an energy-minimizing deformation when quasiconvexity replaces polyconvexity, while keeping the physically necessary infinite energy as the determinant tends to zero.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2000001 (AMR-019-0001). 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--2.2, equations (2.1)--(2.10), Theorem 2.1 (H2), and Problem 1, printed pp. 2--6. https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf.