I am looking at the computation of weak derivative in the blog https://sunlimingbit.wordpress.com/2012/11/25/one-example-related-to-weak-derivative-2/
For equation (3), I have some confusions.
Here 1) I do not understand how he get $\Phi(x_2,x_2)$ rather than $\Phi(x_2,x_2)$ when evaluating the term $(1-x_1)\Phi|_{|x_2|}^1$ 2) I now understand my original second mud: the term $\int_{D_1}\Phi\,dx_1$ is the same as $-\int_{D_1} \partial_{x_1}u\Phi\,dx$.
Please, anyone help me with 1)
In (3) and (4), the absolute value sign is disregarded. Since the function $(1-|x_2|)\Phi(\cdot,x_2)$ is an even function w.r.t. $x_2$, hence we have $$-\int_{-1}^1(1-|x_2|)\Phi(x_2,x_2)dx_2$$ and $$\int_{-1}^1(1-|x_2|)\Phi(-x_2,x_2)dx_2.$$ rather than $$-\int_{-1}^1(1-|x_2|)\Phi(|x_2|,x_2)dx_2$$ and $$\int_{-1}^1(1-|x_2|)\Phi(-|x_2|,x_2)dx_2.$$