Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 3 / 11B / b

Since $q=-\kappa \nabla T$ and $\nabla \cdot q=H$, the divergence theorem givesSpherical symmetry gives $(r^2\kappa T^{\prime }{)}^{\prime }=-r^{2}H$. Regularity at zero and decay at infinity yieldFor $\alpha \le -1$, the exterior solution cannot tend to zero: it is logarithmic at $-1$ and grows below it.

On the exterior domain, the energy-minimizing solution with boundary value one is $\varphi =r^{-\alpha -1}$. Applying the Dirichlet principle to any admissible $w$ gives