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\title{Partial Differential Equations}
\author{Midterm Exam}
\date{March 27, 2014}
\maketitle

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\begin{enumerate}

\item Let $U \subset \R^n$ be open and bounded.  Show that $H_0^1(U)$
is a strict subspace of $H^1(U)$.
\mbox{~~~}\hfill(5)

\item Let $U \subset \R^n$ be open and bounded with $C^1$ boundary.
Let $1<p<\infty$ and define the H\"older conjugate $p'$ by
\[
  \frac1p + \frac1{p'} = 1 \,.
\]

Show that $u \in W^{1,p}_0(U)$ if and only if $u \in L^p(U)$ and there
exists a constant $C=C(u)$ such that
\[
  \biggl| \int_U u \, D\phi \, \d x \biggr|
  \leq C \, \lVert \phi \rVert_{L^{p'}(U)}
\]
for all $\phi \in C^\infty(\R^n)$.
\hfill (``only if'' 5 + ``if'' 10)

\emph{Hints:} Note the class of test functions is arbitrary smooth
functions on $\R^n$ restricted to $U$!  Further, recall that $u \in
W^{1,p}_0(U)$ if and only if $u \in W^{1,p}(U)$ and its trace is zero.
\hfill(10)

\item Let $U\subset \R^n$ be open and bounded with $C^1$ boundary.
Let $h \in C^2(U)\cap C^1(\bar U)$ and suppose further that there
exists a constant $\theta>0$ such that $h(x) \geq \theta$ and $\Delta
h(x) \geq \theta$ for all $x \in U$.

Show that the equation
\begin{gather*}
  - h \, \Delta u - 3 \, Dh \cdot Du = f \qquad \text{in } U \,, \\
  u = 0 \qquad \text{on } \partial U
\end{gather*}
has a unique weak solution $u \in H_0^1(U)$ for every $f \in
H^{-1}(U)$.
\hfill(10)


\item State a condition on $f \in L^2(0,\pi)$ such that the equation
\begin{gather*}
  - \frac{\d^2 u}{\d x^2} - u = f(x) 
  \qquad \text{on } (0,\pi) \,, \\
  u(0) = u(\pi) = 0 
\end{gather*}
has a weak solution $u \in H_0^1(0,\pi)$.  If it does, will the
solution be unique?
\hfill(10)


\item Let $c \in C(\R^n)$ with $c(x) \geq 1$ for all $x \in \R^n$.
Let $u \in C^2(\R^n)$ satisfy
\[
  - \Delta u + c \, u = 1
\]
and assume further that $\operatorname{supp}(Du)$ is compact.

Show that $u \leq 1$.

\emph{Hint:} Consider the function $v=u-1$.
\hfill(10)


\end{enumerate}

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