I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim brezis, and I am a bit confused about the proof. The theorem is stated as:
Let $\Omega \subset \mathbb{R}^n$ be an open set of class $C^2$ with $\partial \Omega$ bounded. Let $f \in L^2(\Omega)$ and $u \in H_0^1(\Omega)$ satisfies the weak formulation (Laplacian): $$\int_\Omega \nabla u \nabla \varphi + \int_{\Omega}u \varphi=\int_{\Omega}f\varphi \quad \forall \varphi \in H_0^1(\Omega)$$ Then $u \in H^2(\Omega)$ and $||u||_{H^2} \leq C ||f||_{L^2}$. Furthermore, if $\Omega$ is of class $C^{m+2}$ and $f \in H^m(\Omega)$, then: $$ u \in H^{m+2} \ and \ ||u||_{H^{m+2}} \leq C ||f||_{H^m}$$ In particular, if $f \in H^m(\Omega)$ with $m > n/2$, then: $$u \in C^2(\overline{\Omega}) $$
The statement is clear. The problem is that the proof only shows that if $f \in H^m(\Omega)$ then $u \in H^{m+2}(\Omega)$, and it doesn't even mention the last part (Does not show that $u \in C^2(\overline{\Omega}) $).
I suppose that this is because it is trivial? Because I haven't been able to show it. Can someone clarify this to me?
This is a consequence of the Sobolev Embedding Theorems, which is explained, e.g., here
(and it's not trivial, but well known. If this is a textbook what you are reading it should be mentioned somewhere).