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\begin{document}

\title{Partial Differential Equations}
\author{Homework 4}
\date{due October 8, 2008}
\maketitle

\begin{enumerate}

\item Let $U$ be open and bounded with a $C^1$ boundary.  For every $v
  \in \Space{C}{}{2}{(\bar U)}$, set
\[
  J[v] = \int_U \bigl(
                  \tfrac12 \, \lvert Dv \rvert^2 - f \, v
                \bigr) \, dx
         - \int_{\partial U} g \, v \, dS \,.
\]
Assume thoughout that $u \in \Space{C}{}{2}{(\bar U)}$.  Prove that
the following two statements are equivalent.

\begin{itemize}
\item[(i)] $u$ solves the so-called \emph{Neumann problem}
\begin{gather*}
  - \Delta u = f \qquad \text{in } U \,, \\
  \nu \cdot D u = g \qquad \text{on } \partial U \,.
\end{gather*}

\item[(ii)] $u$ minimizes $J$, i.e.
\[
  J[u] \leq J[w]
\]
for every $w \in \Space{C}{}{2}{(\bar U)}$.

\end{itemize}

\item Evans, p.~87 problem 10
\item Evans, p.~87 problem 11
\item Evans, p.~87 problem 12

\end{enumerate}

\end{document}
