On the gradient rearrangement of functions

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