Green function with boundary

Question

Let $(M,g)$ be a riemannian manifold with boundary $\partial M$, the Green operator is the inverse of the Laplacian operator $\Delta \phi = f$ on $M$ and $\phi =0$ on $\partial M$. Then have we the decomposition:

$$G(\phi)(x)=\int_M g(x,y) \phi (y) dy +\int_{\partial M} <\hat{g}(x,y), d\phi (y)> dy$$

$g$ is the Green function, and $\hat{g}$ is the boundary part.

Comments

Is this a question or a statement?