Exercise 8.2: Polygons.
Consider the Dirichlet problem
$\Delta u = f$ in $\Omega \subset \mathbb{R}^2$, $u = g$ on $\partial \Omega$
with $f \in C(\overline{\Omega})$ and $g \in C(\partial \Omega)$.
a) Let $\Omega \subset \mathbb{R}^2$ be a polygon with the $k$ corner points $y_1, \dots, y_k$. Let $\alpha_j$ denote the interior
angle enclosed by the two edges meeting in $y_j$, $j \in \{1, \dots, k\}$. For $u \in C^2(\overline{\Omega})$ verify the
following representation formula
$\sigma(x)u(x) = \int_{\Omega} K_2(y-x)\Delta u(y) dy - \int_{\partial \Omega} \left(K_2(y-x)\frac{\partial u}{\partial \nu} - u(y)\frac{\partial K_2(y-x)}{\partial \nu}\right) da_y$
with
$\sigma(x) = \begin{cases} 1 & \text{if } x \in \Omega, \
1/2 & \text{if } x \in \partial \Omega \setminus \{y_1, \dots, y_k\} \
\alpha_j/(2\pi) & \text{if } x \in \{y_1, \dots, y_k\} \end{cases}$
and $K_2(x) = \frac{1}{2\pi} \ln|x|$. Hint: To check for $y_j$, start from the 2nd Green's formula on
$\Omega_\varepsilon = \Omega \setminus S_\varepsilon(y_j)$ with $S_\varepsilon(y_j)$ being the segment of the ball of radius $\varepsilon$ around $y_j$.
b) Show that, for a polygon with a reentrant corner, there does not always exist a solution
$u \in C^2(\overline{\Omega})$ of ().
Hint: You may consider the open polygon $\Omega \subset \mathbb{R}^2$ with the corner points
$\{(-1, -1), (0, -1), (0, 0), (1, 0), (1, 1), (-1, 1)\}$ and $u$ of the form $u(r, \varphi) = r^\alpha \sin(\alpha \varphi)$; then
$f$ and $g$ have to be determined suitably.
