\documentclass[12pt]{article}
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\newcommand*  {\R} {{\mathbb R}}

\newcommand\PVint{\mathop{\setbox0\hbox{$\displaystyle\intop$}%
    \hskip0.2\wd0%
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    }\mskip-\thinmuskip\intop\nolimits}

\begin{document}

\title{Introduction to Partial Differential Equations}
\author{Homework 2}
\date{due February 18, 2015}
\maketitle

\begin{enumerate}

\item Show that 
\begin{equation*}
  \Delta u = 2n \lim_{\varepsilon\to 0} 
  \frac{1}{\varepsilon^2} \, 
  \PVint_{\partial B(x,\varepsilon)} (u(y)-u(x)) \, dS(y) \,.
\end{equation*}

\item Evans, p. 85 problem 3.

\emph{Note:} This problem can be hard when you try it.  Please ask in
class in case you get stuck.

\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 In the proof of Evans, p.\ 23, Theorem 1, we have used that
\[
  D \int_{\mathbb R^n} \Phi(y) \, f(x-y) \, dy  
  = \int_{\mathbb R^n} \Phi(y) \, Df(x-y) \, dy \,.
\]
State sufficient assumptions under which this manipulation is
permitted, and why.

\end{enumerate}

\end{document}

