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?

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?

Edit: Thanks to a comment, I realized that the proposed notion of oscillation is not true (almost trivially). I want to modify it a little bit to capture what I originally had in mind. For any fixed $\varepsilon>0$, is it true that $$ \lim_{n\to\infty} \int_{\{|\nabla u_n| \le \varepsilon \}} |\nabla u_n|^p \,dx = 0 \ ? $$

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 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?

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?