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\title{Partial Differential Equations}
\author{Final Exam}
\date{May 23, 2014, 12:30--14:30}
\maketitle

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\begin{enumerate}

\item Show that $u \in W^{1,\infty}(\R^n)$ if and only if $u \colon
\R^n \to \R$ is bounded and Lipshitz continuous.
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\item Let $U \subset \R^n$ be open and bounded with $C^2$ boundary.
Consider the bi-harmonic equation
\begin{gather*}
  \Delta^2 u = f \qquad \text{in } U \,, \\
  u = 0 \qquad \text{on } \partial U \,, \\
  \nu \cdot D u = 0 \qquad \text{on } \partial U \,.
\end{gather*}
\begin{enumerate}
\item Define a notion of weak solution for the bi-harmonic equation.
Your answer should clearly state and define all the function spaces
involved.
\item Prove the existence of a unique weak solution according to your
definition. 

\emph{Hint:} It may be useful to refer to the regularity theorem for
uniformly elliptic second order problems, which says that a weak
solution $v \in H_0^1(U)$ to
\[
  L u = g \qquad \text{in } U
\]
satisfies
\[
  \lVert u \rVert_{H^2(U)}
  \leq c \, \bigl(
              \lVert u \rVert_{L^2(U)} +
              \lVert g \rVert_{L^2(U)}
            \bigr) 
\]
for some constant $c$ independent of $g$.
\end{enumerate}
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\item Let
\[
  L u = - \sum_{i,j=1}^n 
  D_i (a^{ij}(x) \, D_ju)
\]
be a uniformly elliptic symmetric second order operator with bounded
coefficients.  Suppose $u \in H_{\text{loc}}^1(\R^n)$ satisfies 
\[
  Lu=0
\]
in the sense of weak derivatives.  Show that $u \in L^2(\R^n)$ implies
that $u$ is a constant.
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\item Let $U \in \R^n$ be open and bounded with smooth boundary, and $T>0$.
Prove that there is at most one smooth solution of the
initial-boundary value problem
\begin{gather*}
  u_t - \Delta u = u^2 \qquad \text{in } U_T \,, \\
  \nu \cdot D u = 0 \qquad \text{on } \partial U \times [0,T] \,, \\
  u = g \qquad \text{on } U \times \{ t =0 \} \,.
\end{gather*}
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\item Let $U \in \R^n$ be open and bounded with smooth boundary.
Suppose $u \in W^{1,\infty}(U,\R^n)$ with $\nu \cdot u = 0$ on
$\partial U$ and assume that $\theta=\theta(x,t)$ is a smooth solution
to
\[
  \partial_t \theta + u \cdot D \theta = 0 \,. \tag{*}
\]

\begin{enumerate}
\item Show that there exists a constant $c$ such that 
\[
  \lVert \theta(t) \rVert_{L^p}^p 
  \leq \e^{ct} \, \lVert \theta(0) \rVert_{L^p}^p 
\]
for every $2 \leq p < \infty$ and $0 \leq t < \infty$.

\item Conclude that 
\[
  \lVert \theta(t) \rVert_{L^\infty}
  \leq \lVert \theta(0) \rVert_{L^\infty} \,.
\]

\emph{Hint:} You may use that
\[
  \lVert \theta \rVert_{L^\infty}
  = \lim_{p\to\infty} \lVert \theta \rVert_{L^p} \,.
\]

\item For every fixed $a \in U$ consider the autonomous ordinary
differential equation
\[
  \frac{\d \phi(a,t)}{\d t}
  = u(\phi(a,t)) \,.
\]
Show that $\theta(x,t)$, implicitly defined via
\[
  \theta(\phi(a,t),t) = \theta(a,0)
\]
solves (*).  You may assume sufficient smoothness of all objects
involved, and that $\phi(a,t) \in U$.

\item Do you see a connection between the above and the result from
Question~1? Explain.


\end{enumerate}
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\end{enumerate}

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