= Strict monotonicity without coercivity on H1
{title2=$\langle A(v),h\rangle=\int(\nabla v\cdot\nabla h+\tanh(v)h)$}
The log-cosh <derivative> operator is strictly monotone on $H^1$ but does not have $\langle A(v),v\rangle/\|v\|_{H^1}\to\infty$. <Constant functions> $v=t$ make the ratio approach $|\Omega|^{1/2}$. Restricting to $H_0^1$ removes the uncontrolled constants: <Poincaré inequality>[Poincaré] gives $\langle A(v),v\rangle\ge\|v\|_{H^1}^2/(1+C_P^2)$, which proves coercivity. <strictly monotone operator>[Strict monotonicity] and large-norm coercivity are distinct properties.
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