Let $D \subset \mathbb{R}^{2}$ be a two-dimensional domain with boundary $S=\partial D$, and let

$$G_{2}=G_{2}\left(\mathbf{r}, \mathbf{r}_{0}\right)=\frac{1}{2 \pi} \log \left|\mathbf{r}-\mathbf{r}_{0}\right|$$

where $\mathbf{r}_{0}$ is a point in the interior of $D$. From Green's second identity,

$$\int_{S}\left(\phi \frac{\partial \psi}{\partial n}-\psi \frac{\partial \phi}{\partial n}\right) d \ell=\int_{D}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d a$$

derive Green's third identity

$$u\left(\mathbf{r}_{0}\right)=\int_{D} G_{2} \nabla^{2} u d a+\int_{S}\left(u \frac{\partial G_{2}}{\partial n}-G_{2} \frac{\partial u}{\partial n}\right) d \ell$$

[Here $\frac{\partial}{\partial n}$ denotes the normal derivative on $S$.]

Consider the Dirichlet problem on the unit $\operatorname{disc} D_{1}=\left\{\mathbf{r} \in \mathbb{R}^{2}:|\mathbf{r}| \leqslant 1\right\}$ :

$$\begin{aligned} \nabla^{2} u=0, & \mathbf{r} \in D_{1} \\ u(\mathbf{r})=f(\mathbf{r}), & \mathbf{r} \in S_{1}=\partial D_{1} \end{aligned}$$

Show that, with an appropriate function $G\left(\mathbf{r}, \mathbf{r}_{0}\right)$, the solution can be obtained by the formula

$$u\left(\mathbf{r}_{0}\right)=\int_{S_{1}} f(\mathbf{r}) \frac{\partial}{\partial n} G\left(\mathbf{r}, \mathbf{r}_{0}\right) d \ell$$

State the boundary conditions on $G$ and explain how $G$ is related to $G_{2}$.

For $\mathbf{r}, \mathbf{r}_{0} \in \mathbb{R}^{2}$, prove the identity

$$\left|\frac{\mathbf{r}}{|\mathbf{r}|}-\mathbf{r}_{0}\right| \mathbf{r}||=\left|\frac{\mathbf{r}_{0}}{\left|\mathbf{r}_{0}\right|}-\mathbf{r}\right| \mathbf{r}_{0}|| \text {, }$$

and deduce that if the point $\mathbf{r}$ lies on the unit circle, then

$$\left|\mathbf{r}-\mathbf{r}_{0}\right|=\left|\mathbf{r}_{0}\right|\left|\mathbf{r}-\mathbf{r}_{0}^{*}\right|, \text { where } \mathbf{r}_{0}^{*}=\frac{\mathbf{r}_{0}}{\left|\mathbf{r}_{0}\right|^{2}}$$

Hence, using the method of images, or otherwise, find an expression for the function $G\left(\mathbf{r}, \mathbf{r}_{0}\right)$. [An expression for $\frac{\partial}{\partial n} G$ is not required.]