A New Extension of Serrin's Lower Semicontinuity Theorem

Abstract

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