Let $U$ be a bounded, connected open subset of $\mathbb R^n$ with $C^1$ boundary $\partial U$. Asume $|\beta| \leq k-1$ and $k$ is a integer. Show that for each $\epsilon >0$ there exists a constant $C_\epsilon$, such that
$||\partial^\beta u||_{L^p(U)} \leq \epsilon \|u\|_{W^{k,p}(U)} + C_\epsilon\|u\|_{L^{p}(U)}$
For each function $u$ in $W^{k,p}(U)$ (sobolev space)
Hint: Argue by contradiction and use Rellich-Kondrachov theorem.
Any ideas?
Best Answer
We have $\|\partial^\beta u\|_{L^p(U)} \leq \|u\|_{W^{k-1,p}(U)}$
Also, by the theorem, $W^{k,p}(U) \rightarrow W^{k-1,p}(U)$ is compact.
Now, suppose there is no $C_\epsilon$. Then we get a sequence $u_n \in W^{k,p}(U)$ (normalize so these are all norm 1) violating it with $n$ in place of $C_\epsilon$. That is, $\|u_n\|_{W^{k-1,p}} \geq \epsilon + n \|u_n\|_{L^p(U)}$.
There's a subsequence which is convergent in $W^{k-1,p}(U)$. Call the limit $u$.
Now, what can you say about $\|u\|_{L^p(U)}$?