Let $\Omega\subset\mathbb{R}^n$ be an open bounded set with continuous boundary. Prove that for each $\epsilon>0$ there is a constant $C(\epsilon)>0$ s.t. $$ \int_{\Omega}|f(x)|^pdx\leq C(\epsilon)\int_{\Omega\backslash\Omega_\epsilon}|f(x)|^pdx+C(\epsilon)\int_{\Omega}\|\nabla f(x)\|^pdx $$ for all $f\in W^{1,p}(\Omega)$. where $\Omega_{\epsilon}=\{x\in\Omega: d(x,\partial\Omega)>\epsilon\}$.
I managed to prove the inequality when $n=1$ and $\Omega$ is an open interval. But I do not know how to prove it when $n\geq2$. How can I do that?