= Log-cosh energy on a Sobolev space
{title2=$I(v)=\int_\Omega[|\nabla v|^2/2+\log\cosh v]$}
The inequalities $0\le\log\cosh t\le t^2/2$ make this energy finite on $H^1$. Since the second <derivative> of its <scalar> potential is $\operatorname{sech}^2t\in(0,1]$, its <Fréchet derivative> is $DI(v)[h]=\int(\nabla v\cdot\nabla h+\tanh(v)h)$, with <Taylor remainder> between zero and half the squared $H^1$ <norm> of $h$. The <derivative> operator is strictly monotone because the <hyperbolic tangent> is strictly increasing.
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