A chain rule in L^1(div;Omega) and its applications to lower semicontinuity

A chain rule in the space L^1(div;A) is obtained under weak regularity conditions. This chain rule has important applications in the study of lower semicontinuity problems for functionals of the form \int_A (a(x,u)+b(x,u)Du)dx, in W1,1(A) with respect to strong convergence in L^1(A) and in turn for general functionals of the form F(u,A):=\int_A f(x,u(x),Du(x))dx, in W^{1,1}(A). Classical results of Serrin and of De Giorgi, Buttazzo and Dal Maso are extended and generalized.