Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 2 / b

With $u=\cos \theta$, the boundary data have the finite Legendre polynomial expansionInside the sphere, every $r^{-\ell -1}$ term is singular at the origin and must be absent. Matching at $r=a$ therefore givesIn the exterior, decay at infinity excludes all $r^{\ell }$ terms, including the constant term at $\ell =0$. ThusBoth harmonic functions have the prescribed Dirichlet boundary condition, and each is finite throughout its own domain. The interior expression is regular even at the origin because $r\cos \theta =z$ and $r^2(3{\cos }^2\theta -1)=2z^2-x^2-y^2$. Uniqueness of the Dirichlet problem identifies these as the respective solutions. Their normal derivatives need not match: the two Dirichlet problems do not require a single harmonic function across the bounding sphere.