An Extension of the Serrin's Lower Semicontinuity Theorem

We present a new extension of a celebrated Serrin's lower semicontinuity theorem. We consider an integral of the calculus of variation $\int_{\Omega }f\left( x,u,Du\right) dx\,$ and we prove its lower semicontinuity in $W_{loc}^{1,1}\left( \Omega \right)$ 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 $x\in \Bbb{R}^{n}$, say local Lipschitz continuity, which - we show with a specific counterexample - cannot be replaced, in general, by local Hölder continuity.