Let $D$ be a two dimensional domain with boundary $\partial D$. Establish Green's second identity

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

where $\frac{\partial}{\partial n}$ denotes the outward normal derivative on $\partial D$.

State the differential equation and boundary conditions which are satisfied by a Dirichlet Green's function $G\left(\mathbf{r}, \mathbf{r}_{0}\right)$ for the Laplace operator on the domain $D$, where $\mathbf{r}_{0}$ is a fixed point in the interior of $D$.

Suppose that $\nabla^{2} \psi=0$ on $D$. Show that

$$\psi\left(\mathbf{r}_{0}\right)=\int_{\partial D} \psi(\mathbf{r}) \frac{\partial}{\partial n} G\left(\mathbf{r}, \mathbf{r}_{\mathbf{0}}\right) d s$$

Consider Laplace's equation in the upper half plane,

$$\nabla^{2} \psi(x, y)=0, \quad-\infty<x<\infty \quad \text { and } \quad y>0$$

with boundary conditions $\psi(x, 0)=f(x)$ where $f(x) \rightarrow 0$ as $|x| \rightarrow \infty$, and $\psi(x, y) \rightarrow 0$ as $\sqrt{x^{2}+y^{2}} \rightarrow \infty$. Show that the solution is given by the integral formula

$$\psi\left(x_{0}, y_{0}\right)=\frac{y_{0}}{\pi} \int_{-\infty}^{\infty} \frac{f(x)}{\left(x-x_{0}\right)^{2}+y_{0}^{2}} d x$$

[ Hint: It might be useful to consider

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

for suitable $\tilde{\mathbf{r}}_{\mathbf{0}}$. You may assume $\nabla^{2} \log \left|\mathbf{r}-\mathbf{r}_{0}\right|=2 \pi \delta\left(\mathbf{r}-\mathbf{r}_{0}\right)$. ]