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\begin{document}

\title{Partial Differential Equations}
\author{Homework 2}
\date{due September 25, 2002}
\maketitle

\begin{enumerate}
\item Verify the prefactors in the definition of the averages
\[
  \PVint_{B(x,r)} u(y) \, dy
  \equiv
  \frac1{\alpha(n) \, r^n}
  \int_{B(x,r)} u(y) \, dy 
\]
and
\[
 \PVint_{\partial B(x,r)} u(y) \, dy
  \equiv
  \frac1{n \, \alpha(n) \, r^{n-1}}
  \int_{\partial B(x,r)} u(y) \, dy   \,.
\]

\item Let $X \subset \mathbb{R}^n$.  Show that
\begin{enumerate}
\item $X$ is connected iff $\emptyset$ and $X$ are the only subsets of
$X$ that are both relatively open and relatively closed in $X$.
\item If $\{ W_\alpha \}_{\alpha \in A}$ is a collection of connected
subsets of $X$ such that
\[
  \bigcap_{\alpha \in A} W_\alpha \neq \emptyset\,,
\]
then
$\cup_{\alpha \in A} W_\alpha$ is connected.

\item If $X$ is connected, then $\overline X$ is connected.
\item Every point $x\in X$ is contained in a unique maximal connected
subset of $X$, and this subset is relatively closed in $X$.

\end{enumerate}

The relevant definitions from point-set topology in $\mathbb{R}^n$:

\begin{itemize}
\item $A \subset X$ is called \emph{relatively open in} $X$ if for
every $x \in A$ there exists an $\varepsilon > 0$ such that $X \cap
B(x,\varepsilon) \subset A$.

\item $B \subset X$ is called \emph{relatively closed in} X if
$A=X\setminus B$ is relatively open in $X$.

\item $X$ is called \emph{disconnected} if there exist disjoint,
nonempty subsets $A_1, A_2 \subset X$ that are relatively open in $X$
and $X = A_1 \cup A_2$.

\item $X$ is called \emph{connected} if it is not disconnected.

\end{itemize}

\item Evans, p. 85/86 problems 3 and 4.

\end{enumerate}


\end{document}
