\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts}

\begin{document}

\title{Partial Differential Equations}
\author{Homework 3}
\date{due March 18, 2014}
\maketitle

\begin{enumerate}

\item Prove the following version of the Poincar\'e inequality: Let $U
\subset \mathbb R^n$ 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 Let $H$ be a Hilbert space and $A \colon H \to H$ a bounded
linear operator.  Show that
\[
  \overline{\operatorname{Range} A} 
  = (\operatorname{Ker} A^*)^\bot \,.
\]

\item Let $H$ be a Hilbert space, and suppose $x_k \rightharpoonup x$
weakly in $H$.  Show that
\[
  \lVert x \rVert \leq \liminf_{k\to\infty} \lVert x_k \rVert \,.
\]

\item Study the proof of Theorem~6 on pp.\ 306--307 of Evans.  Show
that the Theorem actually holds true in the following stronger
version.

\emph{If $\lambda \notin \Sigma$, there exists a constant $C$ such that 
\[
  \lVert u \rVert_{L^2(U)} \leq C \, \lVert f \rVert_{H^{-1}(U)} 
\]
whenever $f \in H^{-1}(U)$ and $u \in H_0^1(U)$ is the unique weak
solution of
\begin{gather*}
  Lu = \lambda h + f \qquad \text{in }U \,, \\
  u = 0 \qquad \text{on } \partial U \,.
\end{gather*}
The constant $C$ depends only on $\lambda$, $U$, and on the
coefficients of $L$.}

Explain, in particular, what is meant by ``according to the usual
energy estimates\ldots''  

\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}


\end{document}

