Doubt about Pólya–Szegő inequality

Question

Pólya–Szegő inequality states that if $u \in W^{1,p}_0(\Omega)$, then $$ (1) \,\,\,\,\,\,\,\,\,\ \int_{\Omega} |\nabla u|^p dx \geq \int_{\Omega} |\nabla u^\ast|^p dx, $$ where $u^\ast$ denotes the Schwarz rearrangement of $u$. In all the books and papers I saw this inequality it is required the function $u$ to be in $W^{1,p}_0(\Omega)$ (see for example S. Kesavan - Symmetrization & applications-World Scientific (2006), Theorem 2.3.1 or here). However I wonder if it is enough there exists all weak derivatives of $u$ of first order to make the inequality true. I mean, if there exists $u_{x_1}, ..., u_{x_N}$ then inequality (1) is true. In particular, if $u \in W^{1,p}_0(\Omega)$, then $u^\ast \in W^{1,p}_0(\Omega)$.