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 $\int_{\Omega}f(x,u,u')dx$,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 $x$.