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\begin{document}

\title{Partial Differential Equations}
\author{Final Exam}
\date{May 9, 2007}
\maketitle

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\begin{enumerate}
\item Consider the homogeneous Helmholtz equation on $\R^3$,
\[
  (1-\Delta) u = 0 \,.
\]
\begin{enumerate}
\item Show that if $u$ is a radial solution, i.e.\ $u(x)=v(r)$ with $r
= \lvert x \rvert$, then
\[
  v - \frac2r \, v' - v'' = 0 \,.
\]
\item Show that 
\[
  v_\pm(r) = \frac{\e^{\pm r}}r
\]
are two (independent) radial solutions.  Which one would you consider
to use as a fundamental solution for the Helmholtz equation?
\end{enumerate}
\hfill(10+10)
\newpage
\item Let $U \subset \R^n$ be open and assume that $u \colon U \to
\R$ is harmonic.  
\begin{enumerate}
\item Show that, for any $i=1,\dots,n$,
\[
  \lvert \partial_i u (x) \rvert 
  \leq \frac{n}r \, \lVert u \rVert_{\Space{L}{}{\infty}(\partial
  B(x,r))} 
\]
so long as $B(x,r) \subset U$.

\emph{Hint:} Mean value formula.
\item Then conclude that
\[
  \lvert \partial_i u (x) \rvert 
  \leq \frac{n}{\alpha(n)\,r^{n+1}} \, 
       \lVert u \rVert_{\Space{L}{}{1}(B(x,2r))} 
\]
provided $B(x,2r) \subset U$.

\emph{Hint:} Mean value formula.
\end{enumerate}
\hfill(10+5)
\newpage
\item Let $U\subset \R^n$ be open and bounded.  Consider the Poisson
equation with so-called Neumann boundary conditions,
\begin{gather*}
  -\Delta u = f \quad \text{in } U \,, \\
  \nu \cdot D u = 0 \quad \text{on } \partial U \,.
\end{gather*}
Show that this equation cannot have a solution unless
\[ 
  \int_U f \, dx = 0 \,.
\]
\hfill (10)
\newpage

\item A function $u \in L^1_{\operatorname{loc}}(\R \times \R)$ is
called a \emph{weak solution} of the wave equation on the line if
\[
  \int_\R \int_\R u(x,t) \,
    \bigl( v_{tt} (x,t) - v_{xx}(x,t) \bigr) \, dx \, dt = 0
\]
for every $v \in C_0^\infty(\R \times \R)$.
\begin{enumerate}
\item Show that if $u \in C^2 (\R \times \R)$ is a classical solution
of the wave equation, it is also a weak solution.

\item Verify, by explicit computation, that
\[
  u(x,t) =
  \begin{cases}
    1 & \text{for } x > t \\
    0 & \text{for } x \leq t 
  \end{cases}
\]
is a weak solution of the wave equation.
\end{enumerate}
\hfill (5+10)
\newpage

\item Consider the Korteweg--de Vries equation
\[
  u_t - 6 u u_x + u_{xxx} = 0 
\]
on $\T \times (0,\infty)$.  Show that
\begin{gather*}
  M = \int_\T u \, dx \,, \\
  E = \int_\T u^2 \, dx \,,
\intertext{and}
  H = \int_\T \bigl( \tfrac12 \, u_x^2 + u^3 \bigr) \, dx
\end{gather*}
are all constants of the motion.
\hfill(5+5+10)
\newpage
\item Let $H$ be a Hilbert space and $u_n$ a sequence in $H$.  Show
that the following are equivalent.
\begin{itemize}
\item[(i)] $u_n \to u$ strongly;
\item[(ii)] $u_n \rightharpoonup u$ weakly and $\lVert u_n \rVert \to
\lVert u \rVert$.
\end{itemize}
\hfill (10+10)
\newpage

\mbox{}
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\end{enumerate}

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