Suppose that $\Omega \subset \Bbb R^d$ is a sufficiently nice domain. From the examples of orthogonal bases in Hilbert space cases (or looking at a wavelets basis), it seems natural to me that one may expect the elements of a (normalized) Schauder basis $\{u_n\}_{n=1}^\infty$ of $W^{1,p}_0(\Omega)$ (for $p>1$) to be more and more "oscillatory" as $n\to\infty$. I wonder if that is true in the sense that
$$
\lim_{n\to\infty} \mathscr L^d(\{ x\in\Omega : |\nabla u_n(x)| \le \varepsilon \}) = 0
$$
for any fixed $\varepsilon>0 $? 

If the above is not true, is it possible to construct such a basis that satisfies the above property?