Green's function for the Laplacian

Remarks

If such a function exists, it is unique, by the maximum principle applied to the difference of two of them. It exists whenever the Dirichlet problem on $\Omega$ is solvable for continuous boundary data: take $g(z, w) = h(z) - \log\abs{z - w}$, where $h$ is harmonic with boundary values $\log\abs{\zeta - w}$. Example: for the unit disc, $g(z, w) = \log\abs{\dfrac{1 - \bar w z}{z - w}}$. Here $\log\abs{1 - \bar w z}$ is harmonic on the disc because $1 - \bar w z \ne 0$ there, and on $\abs{z} = 1$ one has $\abs{1 - \bar w z} = \abs{\bar z - \bar w} = \abs{z - w}$, so $g = 0$ on the boundary. Non-example: for $w \ne 0$, the function $z \mapsto -\log\abs{z - w}$ on the unit disc satisfies clauses 1 and 2, but at the boundary point $\zeta = -w/\abs{w}$ its limit is $-\log(1 + \abs{w}) \ne 0$, so clause 3 fails. In the sense of distributions, $\Delta_z g(z, w) = -2\pi\delta_w$, where $\Delta$ is the Laplacian. The Green's function is positive on $\Omega \setminus \set{w}$ and symmetric: $g(z, w) = g(w, z)$. In $\R^n$ with $n \ge 3$, the analogous definition uses the singularity $\abs{x - y}^{2 - n}$ in place of $-\log\abs{z - w}$, up to normalizing constants.