Let $U$ a bounded and open subset of $R^n$. Let $u \in H^{1}(U)$ a bounded function , $v \in H^{1}_{0}(U)$ a non negative function. Consider $\varphi : R \rightarrow R$ a convex and smooth function.Is true that $\varphi^{'}(u)v \in H^{1}_{0}(U)$?
I am trying , but nothing works... Someone can give me a hint ?