A New Extension of Serrin's Lower Semicontinuity Theorem

TL;DR

This paper extends Serrin's lower semicontinuity theorem to a broader class of variational functionals, proving lower semicontinuity in $W_{loc}^{1,1}$ under less restrictive conditions.

Contribution

It introduces a new extension of Serrin's theorem, establishing lower semicontinuity for functionals with integrands that are continuous, convex in the gradient, and locally absolutely continuous in the spatial variable.

Findings

Proves lower semicontinuity in $W_{loc}^{1,1}$ with respect to strong $L_{loc}^{1}$ topology.

Extends applicability of Serrin's theorem to more general integrands.

Demonstrates the importance of convexity and absolute continuity conditions.

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 $x$.