Log-cosh energy on a Sobolev space ($I(v)={\int }_{\Omega }[|\nabla v{|}^2/2+\log \cosh v]$)

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 ${\mathrm{sech}}^{2}t\in (0,1]$, 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.