Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 3 / a / Solution

Use the convention $\Delta G=+{\delta }^{(3)}$. The Dirichlet Green function for this Poisson equation satisfies, for each source point ${\mathbf{r}}^{\prime }\in V$,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).