\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts}

\newcommand*{\Space}[3]{{#1}_{{\vphantom 1}#2}^{{\vphantom +}#3}}
\newcommand\PVint{\mathop{\setbox0\hbox{$\displaystyle\intop$}%
    \hskip0.2\wd0%
    \vcenter{\hrule width0.6\wd0height0.5pt depth0.5pt}%
    \hskip-0.8\wd0%
    }\mskip-\thinmuskip\intop\nolimits}
    
\begin{document}

\title{Partial Differential Equations}
\author{Homework 2}
\date{due September 21, 2004}
\maketitle

\begin{enumerate}
\item
\begin{enumerate}
\item The \emph{standard mollifier} is defined by
\[
  \eta(x) \equiv
  \begin{cases}
    c(n) \, \exp \biggl( \dfrac1{\lvert x \rvert^2 -1} \biggr)
    & \text{if } \lvert x \rvert < 1 \\
    0 & \text{otherwise}\,,
  \end{cases}
\]
where $c(n)$ is chosen such that
\[
  \int_{\mathbb{R}^n} \eta(x) \, d x = 1 \,.
\]
Show that $\eta \in C^\infty(\mathbb{R}^n)$.

\item Show that if $\eta_\varepsilon$ is a radial mollifier, and $u$
is a radial, locally integrable function, then its mollification
\[
  u_\varepsilon (x) = (\eta_\varepsilon * u ) (x)
  = \int_{\mathbb{R}^n} \eta_\varepsilon(y) \, u(x-y) \, dy
\]
is also radial.
\end{enumerate}


\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 problem 3.
\item Evans, p. 86 problem 4.

\end{enumerate}


\end{document}
