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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

Tomás
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    See e.g. Theorem 2.1.11 in Weakly Differentiable Functions by W. Ziemer. – Umberto P. May 07 '13 at 14:19
  • have you checked the book of stampacchia with the same title I think? You can find the proof in the appendix. But I'm not sure whether he is also just considering the case with a finite number of discontinuities. Nevertheless Giovanni Leoni published a paper and a book that deals with general chain rules. – Quickbeam2k1 Oct 08 '13 at 18:34

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