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?

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?

Maybe Theorem 4.8 in Thierry Aubin - Some Nonlinear Problems in Riemannian Geometry will help. It said that

let $\bar{W}_n$ be a compact Riemannian manifold with boundary of then there exists a solution $\varphi \in C^{\infty}(\bar{W}_n)$ of $$\Delta \varphi =f$$ here $f$ is smooth and $\varphi$ vanishes on the boundary.

But Thierry Aubin used variational method, is there any Green function related lectures?