I have this question:
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 $H_0^1(\Omega)$.
It is a classical result that we have $$\nabla (u^+)(x) = \begin{cases} \nabla u(x) & \text{if } u(x) > 0 \\ 0 & \text{if } u(x) \le 0 \end{cases}$$ and $$\nabla (u^-)(x) = \begin{cases} -\nabla u(x) & \text{if } u(x) < 0 \\ 0 & \text{if } u(x) \ge 0 \end{cases}$$ for almost all $x \in \Omega$. Your result follows.