A chain rule in $L^{1}\left({\operatorname*{div};\Omega}\right)$ and its applications to lower semicontinuity

Abstract

A chain rule in the space $L^{1}\left(\operatorname*{div};\Omega\right) $ is obtained under weak regularity conditions. This chain rule has important applications in the study of lower semicontinuity problems for general functionals of the form $\int_{\Omega}f(x,u,\nabla u) dx$ with respect to strong convergence in $L^{1}\left(\Omega\right) $ . Classical results of Serrin and of De Giorgi, Buttazzo and Dal Maso are extended and generalized.