 Solution

ID: 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

Δ G = + δ^{(3)}

. The Dirichlet Green function for this Poisson equation satisfies, for each source point

r^{'} \in V

Δ_{r} G (r, r^{'}) = δ^{(3)} (r - r^{'}) in V, G (r, r^{'}) = 0 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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