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\title{Partial Differential Equations}
\author{Homework 2}
\date{due March 4, 2014}
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\begin{enumerate}

\item Let $U$ be open.  Find an exponent $\theta \in (0,1)$ such that 
\[
  \lVert u \rVert_{L^q}
  \leq \lVert u \rVert_{L^p}^\theta \, 
       \lVert u \rVert_{L^r}^{1-\theta}
\]
for all $u \in L^p(U) \cap L^r(U)$ and $1 \leq p \leq q \leq r \leq
\infty$. 

\item Evans, p.\ 290, Problem 6

\item Evans, p.\ 290, Problem 8

\item Evans, p.\ 290, Problem 9

\item Show that the Rellich--Kondrachov is sharp, i.e., that $W^{1,p}$
is not compactly embedded into $L^{p^*}$ where $p^*=np/(n-p)$ is the
Sobolev conjugate.

\emph{Hint:} Consider sequences of dilations
\[
  u_m (x) = m^\alpha \, u(mx)
\]
of a compactly supported function $u\in W^{1,p}$.

\item Evans, p.\ 291, Problem 10

\emph{Hint:} Look at the proof of Morrey's inequality in Evans (not
covered in class).

\end{enumerate}


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