Let $\phi \in C_0^{\infty}(\Omega)$ with $\Omega = (0,1)\times(0,1)$. Let $u\in L_2(\Omega)$ defined by $u(x,y) = 1$ for $x>y, u(x,y) = 0$ for $x\leq y$
Is there a way we can conclude $$\int_{\Omega}(\partial_x^2-\partial_y^2)u\phi d\omega =0 $$
using that $\phi$ has compact support?
Notice that $u$ is differentiable almost everywhere, which also follows straightforward from definition.
You have $$\int_{\Omega}\partial_{x}^{2}u\phi=-\int_{\Omega}\partial_{x}u\partial_{x}\phi+\int_{\partial \Omega}\partial_{x}u\phi$$
where the second term vanishes since $\phi$ has compact support. Similarly for $\partial_{y}$. But we know $\partial_{x}u=0$ almost everywhere in $\Omega$ except a measure zero set $x=y$. This showed the summation must be zero.