Independent box constraints in a graph-area supremum ($A_{\mathrm{box}}(u)=|\Omega |+|Du|(\Omega )$)

The independent constraints $|{\phi }_0|\le 1$ and $|\phi |\le 1$ separate the dual supremum into $|\Omega |+|Du|(\Omega )$. They do not define square-root graph area. For $|\nabla u|=1$ the box-constrained value is $2|\Omega |$, while the relaxed graph-area functional is $\sqrt{2}|\Omega |$.