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Let

  • $\Omega\subseteq\mathbb{R}^n$ be bounded
  • $u\in \mathcal{L}^2(\Omega)$ be weakly differentiable, i.e. $$\int_\Omega u\nabla\psi\;d\lambda^n=-\int\psi\nabla u\;d\lambda^n\;\;\;\text{for all }\psi\in C_0^\infty(\Omega)$$ (where we carefully denote the weak derivative of $u$ by $\nabla u$)
  • $x^+:=\max(x,0)$ and $x^-:=\max(-x,0)$ for $x\in\mathbb{R}$

How can we show, that $$\int_\Omega\left|\nabla u\right|^2\;d\lambda^n=\int_\Omega\left|\nabla u^+\right|^2\;d\lambda^n+\int_\Omega\left|\nabla u^-\right|^2\;d\lambda^n\tag{1}$$ and what can we derive for $\nabla u^\pm$? (e.g. $\nabla u^\pm=\nabla u$ almost everywhere in $\left\{u^\pm >0\right\}$ or even $\nabla u=\nabla u^+-\nabla u^-$ almost everywhere).

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I think theorem 4 (iii) on page 131 in this book is good enough to answer your question.

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