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.


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