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?
On
Actually you can get, for a fixed $\epsilon>0$, that for any $u\in W^{m,p}(\Omega)$ $$ \|{u}\|_{W^{m,p}}\leq \epsilon\|{D^\alpha u}\|_{L^p}+C_{\epsilon}\|{u}\|_{L^q}, $$ where $|{\alpha}|=m$. Usually we just take $q=p$ but if domain is good enough and by embedding we could extend $q$ to where embedding would go.
This theorem states that the extreme term (The highest and lowest) in a sum often already suffice to control the intermediate terms. This is not only true is Sobolev space but also in $C^k$ spcase, Holder space...
Anyhow, the hint by @aes is good enough and here I just give you more information. If you want to find the prove of this, please go to check Adams book, theorem 5.2.
Also, this is actually gives you an equivalent norm for each fixed $\epsilon$.
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 $\|u\|_{L^p(U)}$?