Let $M$ be an n dimensional Riemannian manifold without boundary. Let $\Omega \subset M$ be a bounded domain with smooth boundary. Let $f \in C^{\alpha}(\bar{\Omega})$, Consider the Dirichlet problem.
$$
\Delta u=f \ \text{ in }\Omega,
u|_{\partial \Omega}=0.
$$
**Do we have a solution $u\in C^{2,\alpha}(\bar{\Omega})$?**

The following argument is inspired by How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?.

Since $f$ is bounded, $f\in L^2(\Omega)$, by the existence of minizer of the energy functional (just like the Euclidean case), there exists a weak solution $u\in W_0^{1,2}(U)$. For $x\in \Omega$, let $U$ be a small neighborhood contained in a coordinate chart, consider $$ \Delta v=f \text{ in }U, v|_{\partial U}=0. $$ By the theory of Dirichlet problem for Euclidean space, we have a solution $v\in C^{2,\alpha}(\bar{U})$. Since $$ \Delta (u-v)=0 \text{ in } U, $$ then $u-v\in C^{\infty}(U)$, so $u\in C^{2,\alpha}(U)$. By the arbitrariness of x, we know $u\in C^{2,\alpha}(\Omega)$.

**But how to prove that $u\in C^{2,\alpha}(\bar{\Omega})$?**