Define the capacity of set $E\subset\subset\Omega$, $\Omega$ bounded, as $$ \text{cap}_p(E, \Omega)=\inf_{u\in \mathcal{F}_p(E, \Omega)}\int_{\Omega}|\nabla u|^p, $$ where the family of functions is $$\mathcal{F}_p(E, \Omega)=\{u\in W^{1, p}_0(\Omega):u\geq 1 \ \text{a.e.} \ \text{on some open} \ U\supset E\}. $$
I was told that, when $E$ is compact, we can replace the family of functions $\mathcal{F}_p(E, \Omega)$ in the definition of capacity by a family $$\widetilde{\mathcal{F}}_p(E, \Omega)=\{\phi\in C^{\infty}_c(\Omega): \phi\geq 1 \ \text{in} \ E\} $$ and still get the same capacity as before. I have no idea how to prove this. Could someone help me?