Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 3 / 12B (Vector Calculus)

Green's first identity and the boundary condition giveThus $\nabla u=0$, so $u$ is constant on each connected component of $V$; its zero boundary value forces $\boxed{u=0}$.

Since $w-v=0$ on $S$ and ${\nabla }^{2}v=0$, another application of Green's first identity givesConsequentlyExpanding $\nabla w=\nabla v+\nabla (w-v)$ and using the vanishing cross term yields the Dirichlet principle

For a spherically symmetric function $\Phi (r)$, the Laplacian in spherical coordinates isIntegrating from zero to $r$ and using regularity at the origin gives

For the stated density, integration and the boundary condition $\Phi (a)=0$ giveThe two pieces and their first derivatives agree at $r=b$. If another solution existed, its difference from $\Phi$ would be harmonic with zero boundary data, so part (a) proves Uniqueness of the Dirichlet problem.

On the shell $U(b,a)$, normalize the outer radial part to obtain the harmonic functionBy the Dirichlet principle, this function minimizes the energy among all $w$ with the same boundary values. Sinceits energy isTherefore