Equivalent norms in the subspace of Sobolev space containing only functions with zero integral

Question

In a problem I ended up with the vector space $V=\{f\in H^1(\Omega):\int_\Omega f=0\}$. I think it can be proven (and it would be really helpful) that $||f||_{L^2(\Omega)}\le c ||\nabla f||_{L^2(\Omega)}$ because this would show norm equivalence. But how do I show that?

When I try to bound $f$ by $\nabla f$ I end up with the integral of $f$ over $\partial\Omega$ and I cannot see how this bounds the integral of $f$ in $\Omega$. Is there something I don't see or is there an other way to do it?

EDIT: $\Omega$ is a bounded subset of $\mathbb{R}^n$.

Answer 1

If you know about Sobolev embedding theorems (and specifically the Rellich compactness theorem) then a simple proof can be found on page 1 of this note.

Answer 2

That is the Poincaré inequality if $\Omega$ is bounded. The result is false in general when $\Omega$ is unbounded.

To see the latter, consider $\Omega=\mathbb{R}$ and let $$f(x)=\begin{cases}x&|x|\le n\\2n-x&n<x<2n\\x-2n&-2n<x<n\\0&|x|\ge 2n\end{cases}$$