Why is this Linear map closable?

Question

Let $\Delta$ be the laplacian, $\Omega \subset \mathbb{R}^n$ be open and bounded. Then define $$\Delta: C^{\infty}_c(\Omega) \subset L^2(\Omega) \rightarrow L^2(\Omega), f \mapsto \Delta f.$$

The question is now why do we need $\Omega$ to be open and bounded for this map to be closable?

Thanks in advance!

Answer 1

Boundedness is not needed. The closure is given by $$ \Delta : \{v\in L^2(\Omega): \Delta v\in L^2(\Omega)\} \to L^2(\Omega). $$