Some Open Problems in Elasticity — Lavrentiev phenomena

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. Let $\mathcal A_{\mathrm{Lip}}=\mathcal A\cap W^{1,\infty}(\Omega;\mathbb R^3)$, using the same prescribed trace, and require both compared classes to be nonempty with finite energy infima. A Lavrentiev gap between the full energy class and Lipschitz deformations is the strict inequality \[ \inf_{y\in\mathcal A} I(y)<\inf_{y\in\mathcal A_{\mathrm{Lip}}} I(y). \] More generally the same term denotes different infima after restricting to Sobolev classes $\mathcal A\cap W^{1,q}$ with larger integrability exponent $q$. Require growth conditions on $W$ under which every finite-energy deformation has a continuous representative in $\Omega$. For example, coercivity $W(A)\ge c|A|^p-C$ with $p>3$ ensures this by Sobolev embedding; this example does not restrict the question to that particular growth condition.

Statement recorded in the supplied source.

Can such a gap occur for elastostatics while every finite-energy deformation is continuous? Include the comparison between the unrestricted finite-energy class and more regular deformation classes, under the same displacement boundary conditions. [S2]

Short English statement

Can the least elastic energy fall strictly when less regular deformations are allowed, even though growth of the stored energy makes every finite-energy deformation continuous?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2000004 (AMR-019-0004). 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. Section 2.3, discussion of Lavrentiev phenomena and Problem 4, printed pp. 8--9. https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf.