 Relaxed graph-area functional

For a BV space function with $Du=\nabla udx+D^{s}u$, relaxed graph area is $A(u)=\int \sqrt{1+|\nabla u{|}^2}dx+|D^{s}u|(\Omega )$. Its dual supremum uses the coupled constraint ${\phi }_0^2+|\phi {|}^2\le 1$ in the pairing $\int ({\phi }_0+u\mathrm{div}\phi )$. It is convex and lower semicontinuous in $L^1$. Independent bounds on the two test fields define a different functional.