For poisson equation $\Delta u = f$ in bounded domain in $\mathbb{R}^n$, we can directly get the solution by Green function. For poisson equation $\Delta u = f$ on closed Riemannian manifold, the necessary and sufficient condition for the existence of the solution is $\int_M f dV =0$.

What I want to ask is that for the Riemannian manifold with boundary, do we still have the Green function to directly have a solution for $\Delta u = f$. Can you share some lectures on this topic with me?