Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 3 / 11B (Vector Calculus)

The identity follows immediately from the product rule:If ${\varphi }_1,{\varphi }_2$ have the same boundary data, put $\psi ={\varphi }_1-{\varphi }_2$. Integrating the identity with $\varphi =\psi$ givesbecause the volume equation and boundary term vanish. Positivity of $\kappa$ makes $\psi$ constant, and its boundary value makes it zero, proving uniqueness.

For any $w=\varphi +\psi$ with $\psi =0$ on the boundary, expansion givesbecause the cross term vanishes by the same integration by parts. This proves the inequality and the Dirichlet principle.

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