TITLE: Invertibility of laplacian operator
QUESTION [6 upvotes]: Let $\Omega\in\mathbb{R}^n$ be a bounded open set with smooth boundary. How to prove the invertibility of $$- \triangle:H^2_0(\Omega) \to L²(\Omega) $$
The injectivity is easy. But how to prove surjectivity without the use of weak notion of solution (when the domain becomes $H^1_0(\Omega)$ and this can be easily found in books)?

REPLY [6 votes]: The operator $$- \triangle:H^1_0(\Omega) \to L²(\Omega) $$
is "surjective" in the weak sense (as a direct use of Riesz representation theorem).
Then you can use regularity theorems to prove that, in fact, this weak solution is in $H^2(\Omega)$. A good reference to a  result like this is the Brezis book  "functional analysis, Sobolev spaces and PDE". See theorem 9.25 and note that a $C^2$ domain $\Omega$ is required. So your result is valid from $H^1_0(\Omega)\cap H^2(\Omega) $ to $L²(\Omega) $.