I need to prove the following inequality:
Let $\Omega\subset \mathbb{R}^n$ be a bounded open set. Let $\overline{u}=\frac{1}{|\Omega|}\int_{\Omega}u(x)dx$ be the average of a function $u$ defined in $\Omega$.
If $u:\Omega\to\mathbb{R}$, $u\in C^1(\Omega)$ and $1<p\leq \infty$ then $$\int_{\Omega}|u(x)-\overline{u}|^pdx\leq C\int_{\Omega}|\nabla u(x)|^pdx$$
I know this inequality is really well-known and its proof is probably in every book about Sobolev Spaces. The problem is that every statement I've seen has additional hypothesis like $\Omega$ has Lipschitz boundary and is connected (for example in Evans' book by using Rellich-Kondrachov Theorem) or $\Omega$ has $C^1$ boundary, also in some texts it is proved for $u\in W_0^{1,p}$. I'm not sure if the hypothesis that $u\in C^1(\Omega)$ makes up for the "missing hypothesis".
The exercise gives the hint to extend $u$ by $0$ outside $\Omega$, but AFAIK the extension operator is somewhat troubling and requires smooth properties of $\Omega$'s boundary.
Any help is highly appreciated. Regards!