An extension of the Serrin’s lower semicontinuity theorem

In this paper we present a new extension of a celebrated Serrin’s lower semicontinuity theorem. We consider an integral of the calculus of varia-tion Ω f (x, u,Du) dx and we prove its lower semicontinuity in W 1,1loc (Ω) with respect to the strong L1loc norm topology, under the usual continuity and convexity property of the integrand f(x, s, ξ), only assuming a mild (more precisely, local) condition on the independent variable x ∈ Rn, say local Lipschitz continuity, which- we show with a specific counterexample- cannot be replaced, in general, by local Hölder continuity