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\begin{document}           


\centerline{{\sc Problem set 2}}
\centerline{Math 218B, Spring 2001}

\bigskip\noindent
{\bf 1.} Give a counterexample to show that
if $\Omega' \subset \subset \Omega$, then
$u\in L^1_{\mathrm{loc}}(\Omega)$, and
\[
\|D^h u\|_{L^1(\Omega')} \le C,
\]
for all sufficiently small $h > 0$, does not imply
that $D u \in L^1(\Omega')$. Where does the
proof of the analogous result for $1<p<\infty$ fail?


\bigskip\noindent
{\bf 2.} Consider the following problem for $u\in H^1(\Omega)$:
\[
\int_{\Omega} \left\{a^{ij} u_{x^i} v_{x^j} +  b^{i} u_{x^i} v + cu\right\} \, dx = 0
\qquad\mbox{for all $v\in H^1(\Omega)$}.
\]
If the function $u$ is smooth, and the coefficient functions
and the domain are also smooth, derive the PDE and boundary conditions
that correspond to this weak formulation.



\bigskip\noindent
{\bf 3.} Give a precise weak formulation of the \emph{biharmonic} equation
\beann
&\Delta^2 u = f,\qquad&\mbox{in $\Omega$}\\
&u = \frac{\partial u}{\partial n} = 0\qquad&\mbox{on $\partial\Omega$}.
\eeann
Prove that there exists a unique weak solution $u\in H^2_0(\Omega)$
for every $f\in L^2(\Omega)$.

\bigskip\noindent
{\bf 4.} Suppose that $L$ is an elliptic operator
\[
Lu = a^{ij}u_{x^i x^j} + b^i u_{x^i} + c u
\]
with constant coefficients $a^{ij}$, $b^i$, $c$, acting on functions
$u : \Rl^n \to \Rl$. Use the Fourier transform to show that if
$u\in L^2(\Rl^n)$ and $L u \in L^2(\Rl^n)$, then $u\in H^2(\Rl^n)$,
and there exists a constant $C$ such that
\[
\|u\|_{H^2(\Rl^n)} \le C \left(\|L u\|_{L^2(\Rl^n)} + \|u\|_{L^2(\Rl^n)}\right).
\]

\bigskip\noindent
{\bf 5.} Let $\Omega_\alpha\subset \Rl^2$ be the
wedge
\[
\Omega_\alpha = \left\{ (r,\theta) \mid 0 < r < 1, \quad 0 < \theta <
\alpha\right\},
\]
where $(r,\theta)$ are polar coordinates. Use separation of variables to solve
the following boundary value problem for Laplace's equation in the wedge:
\beann
&\Delta u = 0\qquad&\mbox{in $\Omega_\alpha$},\\
&u = 0\qquad&\mbox{when $\theta = 0, \alpha$},\\
&u = \sin\left(\frac{\pi \theta}{\alpha}\right)
\qquad&\mbox{when $r = 1$}.
\eeann
Show that $u \in H^1(\Omega_\alpha)$, as required by the general
existence theorem, but that $u \notin H^2(\Omega_\alpha)$
when $\alpha > \pi$. (This example shows that global regularity may fail
when the boundary is not sufficiently smooth.)
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