A New Extension of Serrin's Lower Semicontinuity Theorem

In this paper, we present a new extension of the famous Serrin's lower semicontinuity theorem for the variational functional $\int_Ωf(x,u,u')dx$,we prove its lower semicontinuity in $W_{loc}^{1,1}(Ω)$ with respect to the strong $L_{loc}^{1}$ topology assuming that the integrand $f(x,s,ξ)$ has the usual continuity on all the three variables and the convexity property on the variable $ξ$ and the local absolute continuity on the variable $x$.

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