A Lower Semicontinuity Theorem for the Integral of the Calculus of Variations

We prove the lower semicontinuity of a an integral functional of the kind \int_{\Omega}L(x,u(x),Du(x))dx with respect to the convercence induced by L^1_{loc} on W^{1,1}: we suppose that L(x,u,v) has an at least linear growth with respect to v and fulfils a particular property which includes some well-known cases.