Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 3 / a / Solution

Use the Dirichlet Green function with the sign conventionThe normal derivative is taken with respect to the integration variable and the outward normal of $V$. Green's second identity givesThe Dirac delta distribution evaluates the first volume term at ${\mathbf{r}}^{\prime }$. Substituting $\Delta u=\rho$, $u=f$ on $S$ and $G=0$ there yieldsHenceOne can justify applying the identity to the singular Green function by removing a small sphere around the source and then taking its radius to zero.