First I apologise for not writing properly, but I'm using a cell phone.
We know that having a Sobolev function $u$, if we have a good enough function $f$, then $f\circ u$ is Sobolev and the chain rule holds. It is easy to prove this if for example $f$ is pointwise $\mathcal{C}^1$ with bounded derivative and the set is bounded. I'm reading for elliptic PDE Analysis and there is a lemma where we know that $f\circ u$ is Sobolev, $u$ is an $H^1$ solution to an elliptic PDE, but for $f$ we only know that it is locally Lipschitz and convex, however the writer claims that the chain rule holds and this is what I don't see. I have used mollifiers and several techniques but I cannot prove this without having a bound on the derivative of $f$. Maybe I should use convexity somehow, but I don't know how. Can anyone help?