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\begin{document}

\title{Partial Differential Equations}
\author{Final Exam}
\date{December 9, 2004}
\maketitle

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\begin{enumerate}

\item Consider a modified Laplace equation,
\[
  D \cdot (r\,Du(x)) = 0 \,,
\]
where $u \colon \R^n \to \R$ for $n \geq 2$, and $r = \lvert x
\rvert$.
\begin{enumerate}
\item Show that if $u$ is a radial solution, i.e.\ $u(x)=v(r)$, then
\[
  r \, v'' + n \, v' = 0 \,.
\]
\item Conclude that any radial solution with $u(r)\to 0$ as
$r\to\infty$ is a multiple of
\[
  u(x) = \frac1{r^{n-1}} \,.
\]
\end{enumerate}
\hfill (5+5)


\item Let $u(x,t) \geq 0$ denote a smooth temperature distribution
within a homogeneous, heat-conducting medium $U$.  Hence, $u$
satisfies the heat equation.  It is known that the heat energy
contained in $U$ is proportional to
\begin{gather*}
  E = \int_U u(x,t) \, dx \,.
\end{gather*}
\begin{enumerate}
\item Show that the boundary condition $u(x,t)=0$ for $x \in \partial
U$ corresponds to cooling, i.e.\ that the heat energy is
non-increasing.

\item What boundary condition would you use to describe a perfectly
insulated medium?

\item \textbf{Extra credit:} Prove that in case (a), provided $U$ is
bounded, the heat energy decreases exponentially.
\end{enumerate}
\hfill (5+5\textit{+10})



\item   
\begin{enumerate}
\item Let $U \in \R^n$ be open and simply connected, with sufficiently
smooth boundary, and assume that the \emph{Poincar\'e inequality} 
\begin{equation*}
  \norm{u}{\l2} \leq C \, \norm{D u}{\l2}
\end{equation*}
holds for some constant $C$ and for every $u\in\h1(U)$ with $u=0$ on
$\partial U$.

Show that 
\begin{equation*}
  C \geq \frac1{\sqrt{\lvert\lambda\rvert}} \,,
\end{equation*}
where $\lambda$ is an eigenvalue of the Laplacian corresponding to an
eigenfunction that vanishes on $\partial U$.

\item Show that there is no Poincar\'e inequality for $U=\R^n$.

\end{enumerate}
\hfill (5+5)


\item Assume that $U \subset \R^n$ is open and connected.  Show that
if $u \in \Space{C}{}{2}(U)$ solves the \emph{Neumann problem}
\begin{gather*}
  -\Delta u = 0 \quad \text{in }U \,, \\
  \nu \cdot D u = 0 \quad \text{on } \partial U \,,
\end{gather*}
then $u = \operatorname{const}$.
\hfill (10)



\item \emph{Note:} The different parts of this question can be worked
on independently.  If you get stuck, move on.
\begin{enumerate}
\item Let $u \in \h1([0,1])$ with 
\begin{equation*}
  u(0) = u(1) = 0 \,.
\end{equation*}
Prove that
\begin{equation*}
  \lvert u^{m}(x) \rvert 
  \leq m \,
       \biggl(
         \int_0^1 \lvert u \rvert ^{m} \, d x
       \biggr)^{\tfrac12}
       \biggl(
         \int_0^1 \lvert u \rvert ^{m-2} \,
         \lvert u_x \rvert^2 \, d x 
       \biggr)^{\tfrac12}
\end{equation*}

\emph{Hint:} Apply the Fundamental Theorem of Calculus to $v=u^{m}$;
Cauchy--Schwarz inequality.

\item Deduce from (a) that
\begin{gather*}
  \int_0^1 \lvert u \rvert^{2m} \, dx
  \leq \frac{m}2 \,
       \biggl(
         \int_0^1 \lvert u \rvert^m \, d x
       \biggr)^3
       + \frac{m}2
         \int_0^1 \lvert u \rvert ^{m-2} \,
         \lvert u_x \rvert^2 \, d x \,.
\end{gather*}


\item Consider the partial differential equation 
\begin{gather*}
  u_t = u_{xx} + u^{m+1} \\
  u(0) = u(1) = 0 
\end{gather*}
on $U=(0,1) \times (0,T)$, where $m \geq 2$ is an even integer.

Show that
\begin{gather*}
  \frac1{m} \, \frac{d}{dt}
  \int_0^1 u^{m} \, dx
  = - (m-1) \int_0^1 u^{m-2} \, u_x^2 \, dx
    + \int_0^1 u^{2m} \, dx \,.
\end{gather*}

\item Combine (b) and (c) to find that
\begin{gather*}
  \frac{d}{dt} \int_0^1 u^{m} \, dx
  \leq \frac{m^2}2 \,
       \biggl(
         \int_0^1 \lvert u \rvert^m \, d x
       \biggr)^3 \,.
\end{gather*}
Finally, conclude that if $u(0) \in \l{m}([0,1])$, then $u(t) \in
\l{m}([0,1])$ for every $t \in [0,T)$.  Give a lower bound for the
(potential) blow-up time $T$.

\end{enumerate}
\hfill(5+5+5+5)

\end{enumerate}

\end{document}



\begin{enumerate}
\item Let $u \in \h1(\T)$ with 
\begin{equation*}
  \PVint_\T u \, dx = 0 \,.
\end{equation*}
Prove that
\begin{equation*}
  \lvert u(x) \rvert 
  \leq \norm[{\tfrac12}]{u}{\l2} \, \norm[{\tfrac12}]{u_x}{\l2} \,.
\end{equation*}

\emph{Hint:} Apply the Fundamental Theorem of Calculus to $v=u^2$. 

\item Show that, if additionally $u \in \h2(\T)$, then
\begin{equation*}
  \lvert u(x) \rvert 
  \leq \norm[{\tfrac34}]{u}{\l2} \, 
       \norm[{\tfrac14}]{u_{xx}}{\l2} \,.
\end{equation*}

\item Conclude that (b) implies that
\begin{equation*}
  \norm{u}{\l{p}} 
  \leq \norm[{\tfrac{2+3p}{4p}}]{u}{\l2} \, 
       \norm[{\tfrac{p-2}{4p}}]{u_{xx}}{\l2} \,.
\end{equation*}

\end{enumerate}
