On functionals with convex Carathéodory integrands with a linear growth condition

In this paper we provide sufficient conditions for Carathéodory integrands φ(x,p) on which the formula∫Ωφ(x,Du):=supϕ∈V⁡{−∫Ωudivϕ+φ⁎(x,ϕ(x))dx}=∫Ωφ(x,∇u)dx+∫Ωψ(x)|Dsu| holds for each u∈BV(Ω). Here φ:Ω×RN→R, Ω⊂RN open and bounded, φ convex in p for a.e. x, φ satisfies the linear growth assumption φ(x,p)≤ψ(x)|p|+c for each |p|≥β for a.e. x, V is defined asV={ϕ∈C01(Ω,RN):|ϕ(x)|≤ψ(x) for all x∈Ω}, and φ⁎ is the conjugate function of φ. Lower semicontinuity of ∫Ωφ(x,Du) in L1(Ω) then easily follows, whereas in contrast to earlier work by other authors we do not assume continuity in the x variable for φ.