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\title{Partial Differential Equations}
\author{Homework 3}
\date{due October 15, 2009}
\maketitle

\begin{enumerate}

\item Prove the following version of the Poincar\'e inequality: Let $U
\subset \R^n$ be open, bounded, with smooth boundary.  Then there
exists a constant $c$ such that for every $u \in H_0^1(U)$,
\[
  \lVert u \rVert_{L^2(U)} \leq c \, \lVert Du \rVert_{L^2(U)} \,.
\]

\item Evans, p.\ 345, Problem 1

\item Evans, p.\ 345, Problem 2

\item Solve Evans, p.\ 345, Problem 3, by direct computation.

\item Solve Evans, p.\ 345, Problem 3, by referring to the solvability
condition arising through the Fredholm alternative.


\end{enumerate}


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