A new nonautonomous chain rule in BV

Abstract

The aim of this note is to present a new nonautonomous chain rule formula for the distributional derivative of the composite function $v(x)=B(x,u(x))$, where $u:{\mathbb{R}}^N\to \mathbb{R}$ is a scalar function of bounded variation and $B$ admits a special integral form in terms of a locally bounded function $b(x,t)$, with $b(\cdot .t)$ of bounded variation. It is an useful tool especially in view to applications to semicontinuity results for integral functional (see [1, 8, 9, 10]) and to conservation laws (see [5, 6]).