Strict monotonicity without coercivity on H1 ($⟨A(v),h⟩=\int (\nabla v\cdot \nabla h+\tanh (v)h)$)

The log-cosh derivative operator is strictly monotone on $H^1$ but does not have $⟨A(v),v⟩/\Vert v{\Vert }_{H^1}\to \infty$. Constant functions $v=t$ make the ratio approach $|\Omega {|}^{1/2}$. Restricting to $H_0^1$ removes the uncontrolled constants: Poincaré gives $⟨A(v),v⟩\ge \Vert v{\Vert }_{H^1}^2/(1+C_P^2)$, which proves coercivity. Strict monotonicity and large-norm coercivity are distinct properties.