Show $u_n\rightharpoonup u$ in $W^{1,p}(a,b)\implies u_n\to u$ in $L^p(a,b)$

Question

Let $p\in ]1,\infty [$. Show that $$u_n\rightharpoonup u\text{ in }W^{1,p}(a,b)\implies u_n\to u\text{ in }L^p(a,b).$$

I tried by contradiction, but it didn't really work...

Answer 1

Let us assume that $-\infty<a<b<\infty$.

Remember that a compact operator maps weakly convergent sequences into strongly convergent sequences:

Theorem 8.1-7 in Kreyszig's book. Let $X$ and $Y$ be normed spaces and $T: X \to Y$ a compact linear operator. If $x_n\rightharpoonup x $ in $X$ then $Tx_n\to Tx$ in $Y$.

Remember also that, according to the one dimensional version of Rellich-Kondrachov Theorem, the embedding $\iota$ of $W^{1,p}(a,b)$ into $L^p(a,b)$ is compact for $p\in]1,\infty[$:

Theorem 8.8 in Brezis book.
(a) The injection $W^{1,p}(a,b)\ni u\mapsto u\in C([a,b])$ is compact for $1 < p \leq \infty$;
(b) The injection $W^{1,1} (a,b)\ni u\mapsto u\in L^p (a,b)$ is compact for $1 \leq p < \infty$.

So, the desired result follows from the first theorem above by taking $X=W^{1,p}(a,b)$, $Y=L^p(a,b)$ and $T=\iota$.

Answer 2

You have to use the Ascoli Arzela' theorem. If a sequence converges weakly, it is bounded in $W^{1,p}$. Using the fundamental theorem of calculus and Holder's inequality you can prove that you have equi-continuity and equi-boundedness.