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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.

Using the dual pairing, a monotone operator satisfies $⟨A(u)-A(v),u-v⟩\ge 0$ for all $u,v\in M$. It is a strongly monotone operator if some $m>0$ givesA nonlinear coercive operator with respect to $u_0\in M$ satisfies $⟨A(u),u-u_0⟩/\Vert u-u_0\Vert \to +\infty$ as $\Vert u-u_0\Vert \to \infty$ in $M$.

Strong monotonicity, with $v=u_0$, givesAfter division, the lower bound is $m\Vert u-u_0\Vert -\Vert A(u_0){\Vert }_{X^{\ast }}\to +\infty$, proving coercivity. Reflexivity and separability are not needed for this particular estimate.

Using the dual pairing between $X^{\ast }$ and $X$, a monotone operator satisfiesA nonlinear coercive operator with respect to $u_0$ satisfiesThe directional derivative is ${\partial }_{v}F(u)={\lim }_{t\to 0}[F(u+tv)-F(u)]/t$, when this finite limit exists through admissible points. At a boundary point of $M$ only an inward direction may be admissible; then one uses the corresponding one-sided limit.

If $A:X\to X^{\ast }$ is monotone and $s<t$, apply its defining inequality at $u+tv$ and $u+sv$. It becomes $(t-s)[\phi (t)-\phi (s)]\ge 0$, so $\phi$ is nondecreasing. Conversely choose $u=y$, $v=x-y$. Nondecrease from zero to one gives $⟨A(x)-A(y),x-y⟩\ge 0$. This proves the line criterion for a monotone operator.

A map from a subset of a real Banach space to its continuous dual space is strongly monotone if the displayed inequality holds for every pair in its domain, with one constant $m>0$. The brackets denote the dual pairing. This implies strict monotonicity for distinct arguments. Fixing $v=u_0$ and applying the dual norm inequality gives $⟨A(u),u-u_0⟩\ge m\Vert u-u_0{\Vert }^2-\Vert A(u_0)\Vert \Vert u-u_0\Vert$, so it is a nonlinear coercive operator with respect to $u_0$.