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\title{Partial Differential Equations}
\author{Homework 2}
\date{due October 1, 2009}
\maketitle

\begin{enumerate}

\item Let $U \subset \R^n$ be open and bounded.  Let $\{u_m\}$ be a
sequence in $C^\infty(\bar U)$.  Assume that there are functions $u
\in W^{1,p}(U)$ and $v \in C^{0,\gamma}(U)$ such that
\begin{gather*}
  u_m \to u \quad \text{in } W^{1,p} \,,\\
  u_m \to v \quad \text{in } C^{0,\gamma} \,.
\end{gather*}
Show that $u=v$ almost everywhere.

\item Evans, p.\ 290, Problem 6

\item Evans, p.\ 290, Problem 8

\item Evans, p.\ 290, Problem 9

\item Evans, p.\ 291, Problem 10

\emph{Hint:} Look at the proof of Morrey's inequality.

\item Evans, p.\ 291, Problem 13

\end{enumerate}


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