If $\partial\Omega\in C^{2+\alpha}$ and $-\Delta\Theta=f\text{ in }\Omega$ with $f\in C_0^\infty(\Omega)$, then $\Theta\in C^{2+\alpha}$

Question

Let

• $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with $\partial\Omega\in C^{2+\alpha}$ for some $\alpha>0$
• $f\in C_0^\infty(\Omega)$
• $\Theta\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be the solution of $$-\Delta\left.\Theta\right|_{\Omega}\equiv f\;\;\;\text{and}\;\;\;\left.\Theta\right|_{\partial\Omega}\equiv 0$$

Why can we assume $\Theta\in C^{2+\alpha}(\overline{\Omega})$? I've read that this is due to $\partial\Omega\in C^{2+\alpha}$, but I don't understand why.

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Please note:$\;\;\;$ I'm unsure if I even really understood what is meant by $C^{2+\alpha}$. I assume that $C^{2+\alpha}$ is the space of all $C^2$-functions that are $\alpha$-Hölder continuous, but I may be wrong and someone who is more familiar with this context might be able to correct me.