Consider an open bounded set $\Omega \subset \mathbb{R}^n$ and the trace-zero Sobolev space $W^{1,p}_{0}(\Omega)$, i.e. the completion of the space of $C^{\infty}$ functions compactly supported in $\Omega$ with respect to the norm$$\lVert u \rVert = \left(\int_{\Omega} |\nabla u|^{p} \ dx\right)^{\frac{1}{p}}.$$Take a nonzero function $u \in W^{1,p}_{0}(\Omega)$, since it is zero on the boundary, intuitively I think it should have, with positive measure, all combinations of signs for its partial derivatives. If you consider this problem with $n=1$, the function is absolutely continuous and the sets where $u'> 0$ and $u'< 0$ have positive measure, this is easy to prove in this case, my question is if this extends to higher dimensions (the case where the functions are $C^{1}$ is true, but I tried to use this and density to prove it and got stuck). Explaining what I want to know in the case of dimension two, define the sets$$E_{i} = \left\{x \in \Omega \colon \nabla u(x) \ \text{is in the i-th quadrant}\right\}$$(for $\mathbb{R}^{n}$ you can exchange quadrant with hyperoctant). Is it true that for all $i$ you have $E_{i}$ with positive measure?
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