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\title{Partial Differential Equations}
\author{Homework 4}
\date{due November 12, 2009}
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\begin{enumerate}

\item We have shown in class that if $u \in L^2((0,T);H_0^1(U))$ and
  $u' \in L^2((0,T);H^{-1}(U))$, then 
\[
  u \in C([0,T];L^2(U))
\]
and the map $t \mapsto \lVert u(t) \rVert_{L^2(U)}^2$ is absolutely
continuous with
\[
  \frac{d}{dt} \lVert u(t) \rVert_{L^2(U)}^2
  = 2 \, \langle u'(t),u(t) \rangle_{H^{-1},H_0^1}
\]
for a.e.\ $t \in [0,T]$.

Show that, moreover, there exists a constant $c=c(T)$ such that
\[
  \lVert u \rVert_{C([0,T];L^2(U))}
  \leq c \, \bigl(
              \lVert u \rVert_{L^2((0,T);H_0^1(U))}
              + \lVert u' \rVert_{L^2((0,T);H^{-1}(U))}
            \bigr) \,.
\]

\item Let $\{w_k\} \subset H_0^1(U)$ be an orthonormal basis of
  $L^2(U)$.  Show that the set of functions of the form
\[
  v = \sum_{k=1}^N d_k(t) \, w_k \,,
\]
where the $d_k$ are smooth functions of $t$, is dense in
$L^2((0,T);H_0^1(U))$.

\item Evans, p.\ 425, Problem 1
\item Evans, p.\ 425, Problem 2
\item Evans, p.\ 425, Problem 3
\item Evans, p.\ 425, Problem 4


\end{enumerate}


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