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\centerline{{\sc Problem set 7}}
\centerline{Math 207B, Winter 2016}
\centerline{Due: Mon, Mar. 14}


\bigskip\noindent
{\bf 1.} (a) Use separation of variables to find the eigenvalues $\lambda$ and eigenfunctions $u(x,y)$ of the
Dirichlet Laplacian on the unit square that satisfy
\begin{align*}
&-\left(u_{xx} + u_{yy}\right) = \lambda u\qquad \text{$0<x<1$, $0<y<1$}
\\
&u(x,0) = 0, \quad u(x,1)=0\qquad \text{$0\le x \le 1$}
\\
&u(0,y) = 0, \quad u(1,y)=0\qquad \text{$0\le y \le 1$}.
\end{align*}
(b) What is the smallest eigenvalue that is not a simple eigenvalue?

\bigskip\noindent
{\bf 2.} (a) Let $\vec{x} = (x,y)$,  $\vec{\xi} = (\xi,\eta)$, and $\vec{\xi}^* = (\xi,-\eta)$ where $\eta > 0$. Show that
\[
G(\vec{x},\vec{\xi})  = -\frac{1}{2\pi} \log\left(\frac{|\vec{x}-\vec{\xi}|}{|\vec{x}-\vec{\xi}^*|}\right)
\]
is the solution of
\begin{align*}
-\left(G_{xx} + G_{yy}\right) &= \delta(\vec{x}-\vec{\xi}) \qquad \text{in $-\infty<x<\infty$, $y > 0$}
\\
G(\vec{x},\vec{\xi})&=0\qquad\qquad\quad \text{on $y=0$}.
\end{align*}
(b) Write down the Green's function representation for the solution $u(x,y)$  of the Dirichlet problem for the Laplacian in the upper half plane
\begin{align*}
&u_{xx} + u_{yy} = 0 \qquad \text{in $-\infty<x<\infty$, $y > 0$}
\\
& u(x,0)=f(x).
\end{align*}
You can assume that $u(x,y)\to 0$ sufficiently rapidly as $|(x,y)| \to \infty$. 

\smallskip\noindent
(c) Use the Green's function representation to show that
\[
u(x,y) = \frac{y}{\pi} \int_{-\infty}^\infty  \frac{f(t)}{(x-t)^2 + y^2}\, dt.
\]

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