is there some sort of fundamental lemma of calculus of variations for non-negative functions? Let $u\geq 0$ and $u\in L^1(\Omega)$ on a domain $\Omega$ in $\mathbb{R}^N$. If $$\displaystyle\int_\Omega u\phi \leq 0$$
for any $\phi\in C^\infty_0(\Omega)$ with $\phi\geq 0$, can we conclude $u=0$ a.e. on the whole of $\Omega$?
