I need to show that the function \begin{equation} u(x_1,x_2) = 1-x_1^2 \quad x_1>0 \\ u(x_1,x_2)=1+x_1^2 \quad x_1 \leq 0 \end{equation} is weakly differentiable on the unit ball. It is clear what the weak derivative will be, but I am having a hard time showing that, when tested with a smooth function of compact support, the integration by parts does indeed work out as it should.
I have tried moving to polar coordinates to make the integration easier but I end up not being able to switch back into the regular coordinate system. This happens because when you do IBP you do not get $rdrd\theta$ just $drd\theta$.
Rewrite your function as $u(x_1, x_2)=1-x_1\lvert x_1\rvert$. This makes evident that $u\in C^1$.