= Graph area functional
{title2=$F(u)=\int_\Omega\sqrt{1+|\nabla u|^2}\,dx$}
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 $\partial_vF(u)=\int\nabla u\cdot\nabla v/\sqrt{1+|\nabla u|^2}$, 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.
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