The range of weak gradient of a function in $W^{1,p}_{0}(\Omega)$

Question

Let $\Omega$ be an open and bounded subset of $\mathbb{R}^n$ and $u \in W_{0}^{1,p}(\Omega)$ (sobolev functions with trace zero) and $u \neq 0$. I was wondering what one can say about the range of the weak gradient $\nabla u$. When $u \in C_{0}^{1}(\Omega)$ then there is a ball $B_{r}(0)$ in the range of $\nabla u$ (here the derivatives are the classical ones, see Lemma 1 in here), maybe for functions in $W_{0}^{1,p}(\Omega)$ one can have that the range is dense in the ball? Maybe with positive measure? When $u \in C_{0}^{1}(\Omega)$, $(\nabla u)^{-1}(B_{r}(0))$ is an open set and therefore with positive measure, for functions in sobolev space could we hope for positive measure too? I was thinking on using density arguments, but I got stuck. Do you think any of this is true or at least have counterexamples?

Answer 1

Take $\Omega=(-1,1)$ and $u(x) = 1-|x|$. Then $u\in W^{1,p}_0(\Omega)$ for all $p<\infty$ but the range of $\nabla u$ is equal to $\{-1,+1\}$.