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

We prove lower semicontinuity in L 1 Ω for a class of functionals G : B V Ω ⟶ ℝ of the form G u = ∫ Ω g x , ∇ u d x + ∫ Ω ψ x d D s u where g : Ω × ℝ N ⟶ ℝ , Ω ⊂ ℝ N is open and bounded, g · , p ∈ L 1 Ω for each p , satisfies the linear growth condition lim p ⟶ ∞ g x , p / p = ψ x ∈ C Ω ∩ L ∞ Ω , and is convex in p depending only on p for a.e. x . Here, we recall for u ∈ B V Ω ; the gradient measure D u = ∇ u   d x + d D s u x is decomposed into mutually singular measures ∇ u   d x and d D s u x . As an example, we use this to prove that ∫ Ω ψ x α 2 x + ∇ u 2   d x + ∫ Ω ψ x d D s u is lower semicontinuous in L 1 Ω for any bounded continuous ψ and any α ∈ L 1 Ω . Under minor addtional assumptions on g , we then have the existence of minimizers of functionals to variational problems of the form G u + u − u 0 L 1 for the given u 0 ∈ L 1 Ω , due to the compactness of B V Ω in L 1 Ω .