course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Methods |
IB |
53 |
|
Paper 4, Section II, A |
2011 |
Let
where
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
Suppose 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,
with boundary conditions
[ 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 $\nabla^{2} \log \left|\mathbf{r}-\mathbf{r}{0}\right|=2 \pi \delta\left(\mathbf{r}-\mathbf{r}_{0}\right)$. ]