On the gradient rearrangement of functions

Abstract

In this paper, we introduce a symmetrization technique for the gradient of a \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 \text{\emph{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