Let $u,v:[0,1]\to\mathbb{R}$ functions not absolutely continuous in $[0,1]$ but integrable in $(0,1)$. Let $w:(0,1)\times(0,1)\to\mathbb{R}$ such that
$$ (x,y)\mapsto u(x)+v(y) $$ then $w$ is locally integrable and don't have first order weak derivates. Prove that $w$ has second mixed weak derivates.
I'm trying by definition but stuck in
$$ \int w \partial^{(1,1)} \phi = \int u \partial^{(1,1)} \phi + \int v \partial^{(1,1)} \phi $$ Any suggestion is welcome