The weak formulation of the Poisson equation of Dirichlet type in Euclidean space reads
For given source function $f \in H^{-1}(\Omega)$ find $u \in H_0^1(\Omega)$ such that \begin{equation} \int_{\Omega}\nabla u \cdot \nabla v \, dx = \int_{\Omega}fv \,dx \end{equation} for all $v \in H_0^1(\Omega)$ .
The existence and uniqueness of such $u$ is proved by the Lemma of Lax-Milgram. Do I have to take anything else into considerations if we now replace $\Omega$ by a compact manifold $M$ - say a two-dimensional surface? I am having a hard time to find literature on the Poisson equation on manifolds/surfaces.