What does this ensemble of symbols mean: $C^{2}(\bar{\Omega})$? For Context, I'll provid the sentence in which the symbols lie; it sets up the conditions for a proof of Green's Identity.

Let $\Omega \subset \Re^n$ be domain with a smooth boundary $\partial \Omega$. Let u,v $\in $ $C^{2}(\bar{\Omega})$, where $\bar{\Omega}$ denotes the closure of $\Omega$. 

What does this ensemble of symbols mean: $C^{2}(\bar{\Omega})$ ?
From [Adams-Fournier, p. 80]:

$C^j(\bar{\Omega})$ denotes the closed subspace of $C^j_b(\Omega)$ consisting of functions having bounded, uniformly continuous derivatives up to order $j$ on $\Omega$ normed by 
$$
\|\phi\|_{C^j(\bar{\Omega})} = \max_{0 \leq \alpha \leq j} \sup_{x\in \Omega} |D^{\alpha}\phi(x)|.
$$

Here $C^j_b(\Omega)$ is the space of functions having bounded, continuous derivatives up to order $j$ on $\Omega$.
$C^2 (\bar{\Omega})$ represents the $C^2$ functions on this domain i.e. functions whose second derivatives (both pure and mixed) exist and are continuous.

Edit: As Shifrin points out below, this is incorrect.