Let $\mathcal{D}$ be a 2-dimensional region in $\mathbb{R}^{2}$ with boundary $\partial \mathcal{D}$. In this question you may assume Green's second identity:

$$\int_{\mathcal{D}}\left(f \nabla^{2} g-g \nabla^{2} f\right) d A=\int_{\partial \mathcal{D}}\left(f \frac{\partial g}{\partial n}-g \frac{\partial f}{\partial n}\right) d l$$

where $\frac{\partial}{\partial n}$ denotes the outward normal derivative on $\partial \mathcal{D}$, and $f$ and $g$ are suitably regular functions that include the free space Green's function in two dimensions. You may also assume that the free space Green's function for the Laplace equation in two dimensions is given by

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

(a) State the conditions required on a function $G\left(\boldsymbol{r}, \boldsymbol{r}_{\mathbf{0}}\right)$ for it to be a Dirichlet Green's function for the Laplace operator on $\mathcal{D}$. Suppose that $\nabla^{2} \psi=0$ on $\mathcal{D}$. Show that if $G$ is a Dirichlet Green's function for $\mathcal{D}$ then

$$\psi\left(\boldsymbol{r}_{\mathbf{0}}\right)=\int_{\partial \mathcal{D}} \psi(\boldsymbol{r}) \frac{\partial}{\partial n} G\left(\boldsymbol{r}, \boldsymbol{r}_{\mathbf{0}}\right) d l \quad \text { for } \boldsymbol{r}_{\mathbf{0}} \in \mathcal{D}$$

(b) Consider the Laplace equation $\nabla^{2} \phi=0$ in the quarter space

$$\mathcal{D}=\{(x, y): x \geqslant 0 \text { and } y \geqslant 0\}$$

with boundary conditions

$$\phi(x, 0)=e^{-x^{2}}, \phi(0, y)=e^{-y^{2}} \text { and } \phi(x, y) \rightarrow 0 \text { as } \sqrt{x^{2}+y^{2}} \rightarrow \infty$$

Using the method of images, show that the solution is given by

$$\phi\left(x_{0}, y_{0}\right)=F\left(x_{0}, y_{0}\right)+F\left(y_{0}, x_{0}\right),$$

where

$$F\left(x_{0}, y_{0}\right)=\frac{4 x_{0} y_{0}}{\pi} \int_{0}^{\infty} \frac{t e^{-t^{2}}}{\left[\left(t-x_{0}\right)^{2}+y_{0}^{2}\right]\left[\left(t+x_{0}\right)^{2}+y_{0}^{2}\right]} d t$$