Lower Semicontinuity in L 1 of a Class of Functionals Defined on

Abstract

We prove lower semicontinuity in $L^1\left(\Omega \right)$ for a class of functionals $\mathcal{G}:BV\left(\Omega \right)⟶ℝ$ of the form $\mathcal{G}\left(u\right)={\int }_{\Omega }g\left(x,\nabla u\right)dx+{\int }_{\Omega }\psi \left(x\right)d\left|D^{s}u\right|$ where $g:\Omega \times {ℝ}^N⟶ℝ$ , $\Omega \subset {ℝ}^N$ is open and bounded, $g\left(·,p\right)\in L^1\left(\Omega \right)$ for each $p,$ satisfies the linear growth condition $\underset{\left|p\right|⟶\infty }{\lim }g\left(x,p\right)/\left|p\right|=\psi \left(x\right)\in C\left(\Omega \right)\cap L^{\infty }\left(\Omega \right),$ and is convex in $p$ depending only on $\left|p\right|$ for a.e. $x.$ Here, we recall for $u\in BV\left(\right)$