Can we find one solution $\omega \in W_{0}^{1,\infty}(\Omega;\mathbb R^3)$, where $\Omega$ is an open subset of $\mathbb R^3$ such that the following holds:
$$\frac{\partial \omega_2}{\partial x_3}=\frac{\partial \omega_3}{\partial \omega_2}$$ $$and\ \ \frac{\partial \omega_3}{\partial \omega_1}=\frac{\partial \omega_1}{\partial \omega_3}.$$ Here $\omega \in W_{0}^{1,\infty}(\Omega; \mathbb R^3)$ means that $\omega \in L^{\infty}$ and all the (weak) partial derivatives of $\omega$ are in $L^{\infty}$ and $\omega|_{\partial \Omega}=0.$
I actually can't find any solution like this right now or it's not coming into my mind. Or probably it needs vital needs of Partial Differential Equation.
Any types of help or hints are appreciated.
Thank you.