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\centerline{{\sc Problem set 1}}
\centerline{Math 218B, Spring 2001}

\bigskip\noindent
{\bf 1.} Prove that $C_c^\infty(\Rl^n)$ is dense in $W^{k,p}(\Rl^n)$
for $1\le p < \infty$. (You can assume that $C_c(\Rl^n)$ is dense
in $L^{p}(\Rl^n)$.)


\bigskip\noindent
{\bf 2.} If $n > 1$, show that the unbounded function
\[
u(x) = \log \log \left(1 + \frac{1}{|x|}\right)
\]
belongs to $W^{1,n}(B)$, where
\[
B = \left\{x\in \Rl^n \mid |x| < 1\right\}.
\]
This example shows that Sobolev functions need not be
continuous when $p=p^\ast$.

\bigskip\noindent
{\bf 3.} Show that there is no continuous linear map
\[
T : L^p(\Rl^n) \to L^p(\Rl^{n-1}),
\]
where $1\le p < \infty$,
such that for all $\phi\in C_c^\infty(\Rl^n)$,
\[
(T\phi)\left(x_1,\dots, x_{n-1}\right) = 
\phi\left(x_1,\dots, x_{n-1}, 0\right).
\]

\bigskip\noindent
{\bf 4.} Prove that any $u\in H^2(\Omega)\cap H^1_0(\Omega)$ satisfies
the interpolation inequality
\[
\int_\Omega |Du|^2 \,dx \le C \left(\int_\Omega u^2
\,dx\right)^{1/2}
\left(\int_\Omega |D^2 u|^2 \,dx\right)^{1/2}.
\]
Hint: first use integration by parts
to prove this for $u\in C_c^\infty(\Omega)$, then use
approximation. 


\bigskip\noindent
{\bf 5.} (Nikod\'ym, 1933) Let $\Omega\subset \Rl^2$
be the union of the rectangles
\beann
A_n &=& \left\{ (x,y) \mid 2^{-(n-1)} - 2^{-(n+1)}< x < 2^{-(n-1)},
\quad 2/3 < y < 1\right\},\\
B_n &=& \left\{ (x,y) \mid 2^{-(n-1)} - \epsilon_n< x < 2^{-(n-1)},
\quad 1/3 \le y \le 2/3\right\},\\
C &=& \left\{ (x,y) \mid 0< x < 1,
\quad 0 < y < 1/3\right\},
\eeann
where $0 < \epsilon_n < 2^{-(n+1)}$ and $n\in \Nl$. Let $u$ be a
continuous function on $\Omega$
that is equal to $\alpha_n$ on $A_n$, equal to $0$ on $C$, and linear
on $B_n$. Show that for a suitable choice of the constants
$\alpha_n$ and $\epsilon_n$ we have $Du\in L^2(\Omega)$ but
$u\notin L^2(\Omega)$. (It is possible to prove that
if $Du \in L^p(\Omega)$,
then $u\in L^p_{\mathrm{loc}}(\Omega)$.)

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