positive and negative parts of a function?

Question

I have this question that I found in a demonstration of a theorem:

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and $u^+,u^-$ the positive and negative parts of $u$ respectively. Why do we have this equality:

$$\int_\Omega \nabla u(x).\nabla u^-(x)dx=-\int_\Omega \nabla u^-(x).\nabla u^-(x)dx,$$

where $u$ is a function in the sobolev space $H_0^1(\Omega)$.

Answer 1

If you define $u^-=\max \{-u,0\}$ then by [Section 4.2.2, Th 4][L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions] you have $\nabla u^-=\chi _{\{u<0\}}\nabla (-u)=(\chi _{\{u<0\}})^2\nabla (-u)$ and the result follows...