As of writing I have not managed to found a single literature reference nor mathexchange post which proves explicitly why the Laplacian can be interpreted as bounded linera operator in some Sobolev space situation. In particular, suppose that we have restricted ourselves to some $X\subset\mathbb{R}^n$ with $X$ being compact and we take the domain of $\Delta$ to be $H^2(X)\cap H^1_0(X)$. How do we then show that $\Delta$ is bounded, that is
$$||\Delta u||_{L^2(X)}\leq C||u||_{???}$$
where I am not sure whether $???$ should be the norm of $H^1(X)$ or $H^2(X)$. But despite what the norm of $u$ should be in the upper bound, I don't know how to finish the proof nor where to find an example discussing this?