Let $H^1(\mathbb{R}^d)$ be the usual Sobolev space and let $\phi: \mathbb{R} \to \mathbb{R}$ be a non decreasing Lipschitz function with $\phi(0)=0$. Is the operator $\Delta \phi $ on $H^1(\mathbb{R}^d)$ monotone? i.e. Do we have $\langle \Delta \phi(u) -\Delta \phi(v) , u -v \rangle \ge 0$( or $\le$) for $u,v \in H^1(\mathbb{R}^d)$?
This is easily verified for $\phi(x)=x$, but I'm not able to conclude anything for general $\phi$. $$\langle \Delta \phi(u) -\Delta \phi(v) , u -v \rangle = -\langle \nabla \phi(u)-\nabla \phi(v), \nabla u -\nabla v \rangle = -\langle \phi'(u)\nabla u-\phi'(v)\nabla v, \nabla u -\nabla v \rangle$$ If $\phi'$ is constant then we are done, otherwise I'm not sure where the above equality leads us.
