Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $G:\mathbb{R}\to\mathbb{R}$ a Lipschitz function with $G(0)=0$. Stampacchia's Theorem states that if $u\in W_0^{1,p}(\Omega)$, then $G(u)\in W_0^{1,p}(\Omega)$, where $W_0^{1,p}$ is na Sobolev space. Well, this part of the theorem I can understand, the problem is that he also says $$\nabla G(u)=G'(u)\nabla u$$
in the distributional sense. He does not prove this equality and I dont have idea how to proceed. In some particular cases, I can prove it, for example if $G'$ has only a finite number of discontinuities (or all discontinuities are isolated). Any help is appreciated.
Remark 1: This theorem appears in the paper of Stampacchia (Lemma 1.1 page 5): Equations elliptiques du second ordre a coecients discontinus,
Remark 2: If $G$ is Lipschitz, then $G$ is almost everywhere differentiable and $G'$ is bounded by the Lipschitz constant.
Thank you