Given a sequence $u\_k\in W^{1,p}(B\_1)\cap C^{\alpha}(B\_1)$ such that $\|u\_k\|\_{C^{\alpha}(B\_1)}\le 1$ for all $k\in \mathbb N$. Suppose we have
$$
u\_k \rightharpoonup u\;\;\mbox{weakly in $W^{1,p}(B\_1)$}
$$
and we also have
$$
u\_k \rightarrow u\_0\;\;\mbox{in $C^{\alpha}(B\_1)$}.
$$


Can I imply from the above two informations that $u\_k\rightarrow u\_0$ strongly in $W^{1,p}(B\_1)$?



Does a weakly convergent sequence in $W^{1,p}(B_1)$ which also converges in $C^{0,\alpha}(B_1)$ converges strongly in $W^{1,p}(B_1)$?
I don't think so. Define a sequence of 'zig-zag' functions $(f\_n)$ on $[0,1]$ as follows. Let $f\_n$ be $\frac{2}{n}$-periodic with $f\_n(x) = x$ when $x \in [0,\frac{1}{n}]$ and $f\_n(x) = \frac{2}{n}- x$ when $x \in [\frac{1}{n},\frac{2}{n}]$. Every function is Lipschitz with $\lvert f\_n \rvert\_\infty \leq 1/n$ and $\lvert f\_n' \rvert\_\infty = 1$.


Moreover the sequence converges to zero weakly in $W^{1,p}$ for every $p \in (1,\infty)$ and strongly in $C^{0,\alpha}$ for every $\alpha \in (0,1)$, while $\lvert f\_n \rvert\_{1,p} \geq 1$ for every $n$.


