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\begin{document}

\title{Introduction to Partial Differential Equations}
\author{Homework 5}
\date{due March 21, 2013}
\maketitle

\begin{enumerate}
\item Finish the proof of the mean value formula for the heat equation
  by showing that
\[
  \iint_{E(0,0;1)} \frac{|y|^2}{s^2} \, dy \, ds = 4 \,,
\]
where
\[
  E(x,t;r) = \{ (y,s) \colon \Phi(x-y,t-s) \geq r^{-n} \}
\]
denotes the heat ball ``centered'' at $(x,t)$.

\emph{Hint:} Use polar coordinates in space, and an appropriate change
of variables in time.  The remaining one-dimensional integral is
\textsc{Mathematica}-integrable.  You can also use that
\begin{gather*}
  \int_0^\infty t^{\lambda+1} \, e^{-\lambda t} \, dt
  = \frac{\Gamma (\lambda+2)}{\lambda^{2+\lambda}} \,, \\
  \Gamma(x+1) = x \, \Gamma(x) \,, \\
  \alpha(n) = \frac{\pi^{n/2}}{\Gamma(\tfrac{n}2+1)} \,.
\end{gather*}

\item Evans, p.~87 problem 12
\item Evans, p.~87 problem 13

\end{enumerate}


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