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\begin{document}

\title{Partial Differential Equations}
\author{Homework 3}
\date{due September 26, 2008}
\maketitle

\begin{enumerate}
\item Let
\[
  K(x,y) = \frac2{n \alpha(n)} \, \frac{x_n}{\lvert x-y \rvert^n}
\]
be the Poisson kernel for the half-plane.  Show that
\[
  \int_{\partial \mathbb{R}_+^n} K(x,y) \, dy = 1
\]
for any $x \in \mathbb{R}_+^n$.

\item In class we have used Liouville's theorem to show that any
bounded solution of the Poisson equation
\[
  -\Delta u = f
\]
for $f \in \Space{C}{\mathrm{c}}{2}(\mathbb{R}^n)$, $n \geq 3$, is
given by the solution formula
\[
  u(x) = \int_{\mathbb{R}^n} \Phi(x-y) \, f(y) \, dy 
\]
up to an arbitrary additive constant.  (Evans, p.~30, Theorem~8.)

This statement does not hold in dimension $n=2$ since solutions are
generically unbounded.  Use Liouville's theorem to conjecture and
prove the corresponding theorem for $n=2$.

\item Evans, p.~86 problem 5
\item Evans, p.~86 problem 6

\end{enumerate}


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