I'm looking for a reference for the following. Take the obstacle problem: $$\int_\Omega \nabla u \nabla (v-u) \leq \int_\Omega f(v-u)$$ for a function $u \in K:=\{v \in H^1_0(\Omega) : v \geq \varphi\}$ for a sufficiently nice given function $\varphi$, and holding for all test functions $v \in K$. Say $\Omega$ is a bounded smooth domain.
Denote $u=\sigma(f)$ as the solution map.
I read, some time ago, that depending on the nature of the coincidence set $\{u = \varphi\}$ and the obstacle $\varphi$, the solution map $\sigma$ can be Frechet or directionally differentiable or not differentiable at all (but I may be misremembering some aspects of this). Unfortunately I can't find a reference to these issues. I would appreciate any help. Thanks.
