A New Extension of Serrin's Lower Semicontinuity Theorem

Abstract

We present a new extension of Serrin's lower semicontinuity theorem. We prove that the variational functional${\int }_{\mathrm{\Omega}}^{}f(x,u,{u}^{\prime })dx$ defined on ${W}_{loc}^{1,1}(\mathrm{\Omega})$ is lower semicontinuous with respect to the strong convergence in ${L}_{loc}^{1}$, under the assumptions that the integrand $f(x,s,\xi )$ has the locally absolute continuity about the variable $x$.