= Solution
Impose a homogeneous <Dirichlet boundary condition> on the Green function in its first variable, $G(r,r')=0$ for $r\in S$. The second boundary term then vanishes, giving
$$
\boxed{\phi(r')=\int_S\phi(r)\partial_{n_r}G(r,r')\,dS.}
$$
In a bounded domain, a Dirichlet Green function requires $k^2$ not to be a Dirichlet eigenvalue of $-\Delta$; otherwise a resonance can obstruct its existence. The representation assumes such a Green function exists.
In all space, put $\rho=|r-r'|$ and try $G=Ae^{ik\rho}/\rho$. For $\rho>0$, direct differentiation gives $(\rho^2G')'+k^2\rho^2G=0$, so the <Helmholtz equation> holds away from the source. Integrate its inhomogeneous version over a small source-centred ball. The $k^2G$ volume integral tends to zero, while the flux is
$$
\lim_{\varepsilon\downarrow0}4\pi\varepsilon^2G'(\varepsilon)=-4\pi A.
$$
It must equal the integrated delta, one. Therefore
$$
\boxed{A=-\frac1{4\pi},\qquad G(r,r')=-\frac{e^{ik|r-r'|}}{4\pi|r-r'|}.}
$$
This is the negative of the convention used in the <outgoing Green function for the three-dimensional Helmholtz equation>, where the source is $-\delta$. The sign here follows the present $+\delta$ convention. The factor $e^{ik\rho}$ chooses the outgoing solution, with time convention $e^{-i\omega t}$ and the <Sommerfeld radiation condition>; one could add homogeneous solutions without a radiation prescription.
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