A constant $ H^1$ function in every connected component of a subset of $ R^n $ is identically $0$?

Question

I'm trying to solve this problem.

Let $ \Omega $ be a bounded subset of $ R^n $ and let $ v\in H^1\left(\Omega\right). $ If $ v $ is constant in every connected component of $ \Omega, $ it is true that $ v \equiv 0 $ in $ \Omega$?

If it is true (or not true) could anyone explain me why?

Thank you!

Answer 1

Its not true, just take $v=1$ on any such $\Omega$.

If you meant "is it true that $v\equiv c$ for some $c\in\mathbb R$", this is also not true, just take e.g. $v=1$ on one component and $v=2$ on another component.

However if you meant $v\in H^1_0(\Omega)$, then it is true.