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\begin{document}

\title{Partial Differential Equations}
\author{Final Exam}
\date{December 11, 2008}
\maketitle

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\begin{enumerate}

\item 
\begin{enumerate}
\item Fix $x_0 \in \R^n$ with $n \geq 2$ and set $R = \lvert x_0
  \rvert$.  Show that
\[
  h(x) = \frac{\lvert x \rvert^2 - R^2}{\lvert x - x_0 \rvert^n} +
  R^{2-n}
\]
is a harmonic function on $\R^n \setminus \{x_0\}$ and that $h(0)=0$.

\item \emph{Kuran's theorem.} Let $U \subset \R^n$ be an open,
  bounded, and connected set which contains the origin.  Suppose that
\[
  \PVint_U u(x) \, dx = u(0)
\]
for every function $u$ which is harmonic on $U$.  Prove that $U$ must
be a ball centered at the origin.

\emph{Hint:} Use the function from (a) with a clever choice of $R$ and
$x_0$ to test the mean value property.
\end{enumerate}
\hfill(10+10)

\item \label{1} Let $U \subset \R^n$ be open and bounded and fix
$T>0$.  Recall
  the definition of the parabolic cylinder
\[
  U_T \equiv U \times (0,T] 
\]
and the parabolic boundary 
\[
  \Gamma_T \equiv \overline U_T \setminus U_T \,.
\]

Show that there exists at most one solution $u \in C_1^2(\overline
U_T)$ to the reaction diffusion equation
\begin{gather*}
  u_t = \Delta u - u^2 \qquad \text{in } U_T \,, \\
  u = g \qquad \text{on } \Gamma_T \,.
\end{gather*}
\hfill(10)

\item 
\begin{enumerate}
\item Show that the solution to the wave equation 
\begin{gather*}
  u_{tt} - u_{xx} = 0 \qquad \text{in } \R \times (0, \infty) \,, \\
  u = g \qquad \text{on } \R \times \{t =0 \} \,, \\
  u_t = h \qquad \text{on } \R \times \{t =0 \} 
\end{gather*}
for $g \in C^2$ and $h\in C^1$ is given by
\[
  u(x,t) = \frac{g(x+t) + g(x-t)}2 
  + \frac12 \int_{x-t}^{x+t} h(y) \, dy \,.
\]
What is the speed of propagation?

\item Let $U=[a,b]$ and define $U_T$ and $\Gamma_T$ as in
Question~\ref{1}. 

Show that $u \in C^2(\overline U_T)$ solves the wave
  equation in $U_T$ if and only if
\[
  u(x-\xi,t-\tau) + u(x+\xi,t+\tau) 
  = u(x + \tau,t+\xi) + u(x-\tau,t-\xi)
\]
for all $\xi,\tau\geq 0$ such that the function arguments are
contained in $\overline U_T$.

\emph{Hint:} The ``only if'' part is difficult.  One way of proving it
starts from the factorization of the wave equation $u_{tt}-u_{xx} =
(\partial_t-\partial_x)(\partial_t+\partial_x)u=0$ used in class for
the constructive derivation of the solution formula from (a).

\item Prove the existence of a solution to the initial-boundary value
problem 
\begin{gather*}
  u_{tt} - u_{xx} = 0 \qquad \text{in } U_T \,, \\
  u = g \qquad \text{on } \Gamma_T \,, \\
  u_t = h \qquad \text{on } U \times \{t =0 \} \,.
\end{gather*}

\emph{Hint:} No computations are necessary.  Think about how to
construct the solutions using (a) and (b) on different segments of the
$x$-$t$ plane:
\begin{center}
\includegraphics[width=7cm]{wave-bvp.eps}\\
\hfill(10+10+10)
\end{center}


\end{enumerate}

\item Let $u$ be a weak solution of Burgers' equation
\begin{gather*}
  u_t + u \, u_x = 0 \qquad \text{in } \R \times [0,\infty) \,,\\
  u=g \qquad \text{on } \R \times \{ t=0 \}\,,
\end{gather*}
where $g$ is a smooth bounded function all of whose derivatives go to
zero as $\lvert x \rvert \to \infty$.

Show that the characteristic curve which first runs into a shock is
one which originates from a point of inflection of $g$ at time $t=0$.

\emph{Hint:} Find an equation for $u_x$ along a characteristic line.
A shock is characterized by $\vert u_x \rvert \to \infty$.  \hfill(10)

\item Let $U \subset \R^n$ be an open set with sufficiently smooth
boundary, and $u \colon U \to \R$ a smooth function with $u=0$ on
$\partial U$.  

Show that
\[
  \int_U \lvert Du \rvert^2 \, dx
  \leq \biggl( \int_U \lvert u \rvert^2 \, dx \biggr)^{\tfrac12} 
       \biggl( \int_U \lvert \Delta u \rvert^2 \, dx
       \biggr)^{\tfrac12} \,.
\]
\end{enumerate}
\hfill(10)

\end{document}
