Graph area functional ($F(u)={\int }_{\Omega }\sqrt{1+|\nabla u{|}^2}dx$)

This functional measures the surface area of the graph of a function. On a bounded domain it is finite for $u\in H^1$. Its directional derivative is ${\partial }_{v}F(u)=\int \nabla u\cdot \nabla v/\sqrt{1+|\nabla u{|}^2}$, by the dominated convergence theorem, since the pointwise difference quotients are bounded by $|\nabla v|$. The resulting dual-valued operator is monotone: the scalar integrand $p\mapsto \sqrt{1+|p{|}^2}$ is convex and its gradient is $p/\sqrt{1+|p{|}^2}$. It is not a nonlinear coercive operator on the full $H^1$ space, since it vanishes on constant functions whose $H^1$ norms can tend to infinity.