Solution
= Solution
The zero-Dirichlet <Green function of the Laplacian> $G_D(x,y)$ is symmetric, vanishes at the boundary in the appropriate sense, is harmonic in $x$ away from $y$, and satisfies
$$
-\Delta_xG_D(x,y)=2\pi\delta_y
$$
as a <distribution>. Equivalently, for suitable $\phi$,
$$
(-2\pi\Delta^{-1}\phi)(x)
=\int_DG_D(x,y)\phi(y)\,dy.
$$