Lower Semicontinuity in L 1 of a Class of Functionals Defined on B V with Carathéodory Integrands

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 $u\in BV\left(\Omega \right)$ ; the gradient measure $Du=\nabla udx+d\left(D^{s}u\right)\left(x\right)$ is decomposed into mutually singular measures $\nabla udx$ and $d\left(D^{s}u\right)\left(x\right)$ . As an example, we use this to prove that ${\int }_{\Omega }\psi \left(x\right)\sqrt{{\alpha }^2\left(x\right)+{\left|\nabla u\right|}^2}dx+{\int }_{\Omega }\psi \left(x\right)d\left|D^{s}u\right|$ is lower semicontinuous in $L^1\left(\Omega \right)$ for any bounded continuous $\psi$ and any $\alpha \in L^1\left(\Omega \right).$ Under minor addtional assumptions on $g$ , we then have the existence of minimizers of functionals to variational problems of the form $\mathcal{G}\left(u\right)+{‖u-u_0‖}_{L^1}$ for the given $u_0\in L^1\left(\Omega \right),$ due to the compactness of $BV\left(\Omega \right)$ in $L^1\left(\Omega \right).$