On the gradient rearrangement of functions

In this paper, we introduce a symmetrization technique for the gradient of a \(\text {BV}\) $\text{BV}$ function, which separates its absolutely continuous part from its singular part (sum of jump and Cantorian part). In particular, we prove a \(\text {L }^{\text {1}}\) $\text{L}{}^{\text{1}}$ comparison between the function and the symmetrization just mentioned. Furthermore, we apply this result to obtain Saint-Venant type inequalities for some geometric functionals.