Conjecture on the three-dimensional vectorial Gamma-limit for thin films with delamination

Let $\mathcal{A}$ be the space of kinematically admissible three-dimensional vectorial displacements, and let $\mathcal{E}_\varepsilon$ and $\mathcal{E}_0$ be the corresponding thin-film energy functionals, with $\mathcal{E}_0$ defined using membrane and bending energies, fracture and delamination costs, and delamination set

$$\Delta:=\left\{x' \in \omega: |\bar{\boldsymbol u}(x')| > \sqrt{\frac{2\kappa_b}{\mu_b}}\right\}\cup\left\{x' \in \omega: u_3\neq 0\right\}.$$

Three-dimensional vectorial Gamma-limit conjecture. If $\boldsymbol u\in L^2(\mathcal{O};\mathbb{R}^3)$ and $(\boldsymbol u_\varepsilon)_{\varepsilon>0}\subset L^2(\mathcal{O};\mathbb{R}^3)$ converges strongly to $\boldsymbol u$ in $L^2(\mathcal{O}_f;\mathbb{R}^3)$, then

$$\mathcal{E}_0(\boldsymbol u)\leq\liminf_{\varepsilon\to 0}\mathcal{E}_\varepsilon(\boldsymbol u_\varepsilon).$$

This is the lower-bound inequality needed to establish the conjectured $\Gamma$-convergence of the reduced energies in the general three-dimensional vectorial setting; the authors state that they have been unable to prove it.