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\title{Partial Differential Equations}
\author{Midterm Exam}
\date{October 27, 2009}
\maketitle

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\begin{enumerate}

\item Let $X$ be a Banach space such that for some $U \subset \R^n$
  open and $1 \leq p < p \leq \infty$, $X$ is compactly embedded in
  $L^p(U)$ and $X$ is continuously embedded in $L^q(U)$.

Prove that $X$ is compactly embedded in any $L^r(U)$ with $r\in[p,q)$.
\hfill (5)
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\item Let 
\[
  U = \{ x \in \R^n \colon 0 < x_n < L \}
\]
denote the region between two parallel hyperplanes.  Show that, for
all $u \in W_0^{1,p}(U)$,
\[
  \lVert u \rVert_{L^p(U)}
  \leq \frac{L}{p^{1/p}} \, \lVert Du \rVert_{L^p(U)} \,.
\]
\hfill (5)
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\item Let $U \subset \R^n$ be open and bounded.  Show that $\lambda$
  is the smallest eigenvalue of the Dirichlet Laplacian, i.e., is the
  smallest real number such that
\begin{gather*}
  -\Delta u - \lambda u = 0 \qquad \text{in } U \,, \\
  u = 0 \qquad \text{on } \partial U 
\end{gather*}
has weak solutions $u \in H_0^1(U)$ if and only if $1/\lambda$ is the
smallest constant $c$ such that
\[
  \lVert u \rVert_{L^2(U)}^2
  \leq c \, \lVert Du \rVert_{L^2(U)}^2 
\]
for all $u \in H_0^1(U)$.
\hfill (5)
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\item Let $U \subset \R^n$ be open and bounded with $C^1$ boundary.
  We say that $u \in
  H^1(U)$ is a weak solution if
\[
  \int_U Du \cdot Dv \, d x 
  + \int_U u \, v \, dx
  + \int_{\partial U} u \, v \, dS
  = \int_U f \, v \, dx
\]
for all $v \in H^1(U)$.

\begin{enumerate}
\item Assume, moreover, that $u \in C^2(U) \cap C^1(\bar U)$.  Which
  second order boundary value problem corresponds to the weak
  formulation above?
%\hfill (5)\newpage
\item Prove that the weak formulation has a unique solution $u$ for
  every $f \in L^2(U)$.
\mbox{~~~~}\hfill (5)
\end{enumerate}
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\item Let 
\[
  U = \{ x \in \R^3 \colon \lvert x \rvert \leq \pi \} \,.
\]
Show that the problem
\begin{gather*}
  - \Delta u - u = f \qquad \text{in }U \,, \\
  u = 0 \qquad \text{on } \partial U
\end{gather*}
has a weak solution in $H_0^1(U)$ only if
\[
  \int_U f(x) \, \frac{\sin(\lvert x \rvert)}{\lvert x \rvert} \, dx
  = 0 \,.
\]
\hfill (5)
\end{enumerate}
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