Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 3 / c / Solution

Fix the observation point $\mathbf{x}$ and apply Green's second identity in the source variable $\mathbf{y}$. Since $\Delta u=0$, ${\Delta }_{\mathbf{y}}{G}_D={\delta }_{\mathbf{x}}$ and $G_D=0$ on the boundary, it givesThe identity can be justified by deleting a small circle about $\mathbf{x}$ and letting its radius tend to zero; the logarithmic flux contributes $u(\mathbf{x})$.

Set $\mathbf{x}=(r,\varphi )$ and $\mathbf{y}=(\rho ,{\varphi }^{\prime })$, and write $c=\cos (\varphi -{\varphi }^{\prime })$. The symmetric expression for the Green's function isAt the boundary, the outward normal derivative is ${\partial }_{\rho }$ at $\rho =1$, and henceSince $ds=d{\varphi }^{\prime }$ on the unit circle, the Poisson integral on the unit disk follows, with Poisson kernel