Given a Lebesgue measurable function u:\mathbb{R}^n\rightarrow \mathbb{R}^+,the symmetric decreasing rearrangement u^*:\mathbb{R}^n\rightarrow \mathbb{R}^+, is the unique function such that for every t \in \mathbb{R}, the sublevel set u^*{}^{-1}((t, +\infty)) is a open ball centred at the origin 0 \in \mathbb{R}^nthat has the same Lebesgue measure as u^{-1}((t, +\infty)).
The Pólya-Szegő inequality states that if moreover u \in W^{1,p}(\mathbb{R}^n), then u^* \in W^{1,p}(\mathbb{R}^n) and \int_{\mathbb{R}^n} |\nabla u^*|^p \leq \int_{\mathbb{R}^n} |\nabla u|^p.