Dirichlet problem and schauder estimate for manifolds

Let $$M$$ be an n dimensional Riemannian manifold without boundary. Let $$U\subset M$$ be a bounded domain with smooth boundary. Let $$f \in C^{\alpha}(U)$$, $$g\in C^{2,\alpha}(\partial U)$$. Consider the Dirichlet problem. $$\Delta u=f \text{ in }U,\\ u|_{\partial U}=g.$$ Do we have the existence, uniqueness and schauder estimate?

Can we extend $$g$$ to a function $$\tilde{g} \in C^{2,\alpha}(U) \cap C(\bar{U})$$?

If so, let $$v=u-\tilde{g}$$, then $$\Delta v=f-\Delta \tilde{g}:=\tilde{f}\in C^{\alpha},\\ v|_{\partial U}=0.$$ by the existence of minizer of the energy functional (just like the Euclidean case), there exists a weak solution $$v\in W_0^{1,2}(U)$$.

For any $$x\in U$$, consider a small nbh $$U_x$$ (with smooth boundary) contained in a coordinate chart, and consider the equation $$\Delta w=\tilde{f} \text{ in } {U_x},\\ w|_{\partial U_x}=0.$$ Then $$w\in C^{2,\alpha}(U_x)$$. Since $$\Delta(v-w)=0$$ in $$U_x$$, we know $$v-w\in C^{\infty}(U_x)$$, then $$v\in C^{2,\alpha}$$, hence $$u\in C^{2,\alpha}$$.

But I don't know whether we have $$u|_{\partial U}=g$$.

Do we have a schauder estimate? i.e. Is there a C, s.t. $$\|u\|_{C^{2,\alpha}(U)}\leqslant C(\|f\|_{C^{\alpha}(U)}+\|g\|_{C^{2,\alpha}(U)})?$$ I don't know how to use Euclidean schauder estimate and partion of unity to get the estimate on manifolds.