An extension of the Serrin's lower semicontinuity theorem

In this paper we present a new extension of the celebrated Serrin's lower semicontinuity theorem. We consider an integral of the calculus of variation integral(Omega) f (x, u, Du)dx and we prove its lower semicontinuity in W-loc(1,1) (Omega) with respect to the strong L-loc(1) norm topology, under the usual continuity and convexity property of the integrand f (x, s, xi), only assuming a mild (more precisely, local) condition on the independent variable x in R-n, say local Lipschitz continuity, which - we show with a specific counterexample - cannot be replaced, in general, by local Holder continuity.