I am not able to solve the following question that I came across: Let $\Omega \subset R^n$ be a bounded set. A function $u$ is called a weak solution of the differential inequality $\begin{cases} -\Delta u \ge 0 &\text{in} \ \Omega \\ u = 0 &\text{in} \ \partial\Omega, \end{cases} $ if $u\in H_0^1(\Omega)$ and $\int_\Omega \nabla u \cdot \nabla \phi \ dx \ge 0$ for all $\phi \in H_0^1$ such that $\phi \ge 0$ a.e
What I want to show is that any such weak solution $u$ satisfies weak minimum principle that $u\ge 0$ a.e in $\Omega$.
Thank you for your help .