Lower Semicontinuity in L1 of a Class of Functionals Defined on BV with Carathéodory Integrands

We prove lower semicontinuity in L1(Ω) for a class of functionals G:BVΩ⟶ℝ of the form Gu=∫Ωgx,∇udx+∫ΩψxdDsu where g : Ω × ℝN⟶ℝ, Ω ⊂ ℝN is open and bounded, g(·, p) ∈ L1(Ω) for each p, satisfies the linear growth condition limp⟶∞gx,p/p=ψx∈CΩ∩L∞Ω, and is convex in p depending only on |p| for a.e. x. Here, we recall for u ∈ BV(Ω); the gradient measure Du = ∇u dx + d(Dsu)(x) is decomposed into mutually singular measures ∇u dx and d(Dsu)(x). As an example, we use this to prove that ∫Ωψxα2x+∇u2 dx+∫ΩψxdDsu is lower semicontinuous in L1(Ω) for any bounded continuous ψ and any α ∈ L1(Ω). Under minor addtional assumptions on g, we then have the existence of minimizers of functionals to variational problems of the form Gu+u−u0L1 for the given u0 ∈ L1(Ω), due to the compactness of BV(Ω) in L1(Ω).