Past exam of the mathematics course of the University of Cambridge / 2012 / ia / Paper 3 / 12C / ii / a / Solution

The Dirichlet boundary condition gives $B=-a^2/6$, so the unique radial solution isIts radial Laplacian is one, it is smooth at the origin, and its boundary value is zero. The uniqueness result in part (i) shows that this is also the only solution among all classical functions on the ball, not merely among spherically symmetric ones.