= Solution
Use the convention $\Delta G=+\delta^{(3)}$. The <Dirichlet Green function> for this <Poisson equation> satisfies, for each source point $\mathbf r'\in V$,
$$
\boxed{\Delta_{\mathbf r}G(\mathbf r,\mathbf r')=\delta^{(3)}(\mathbf r-\mathbf r')\quad\text{in }V,
\qquad G(\mathbf r,\mathbf r')=0\quad\text{on }S}.
$$
It is a <harmonic function> away from its source, with a unit delta singularity there. Its <boundary condition> is homogeneous even when the original potential has nonzero boundary values; those values enter through the boundary integral. This sign convention is the one giving the plus signs in the formula requested in part (b).
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