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

Abstract

We prove lower semicontinuity in L1Ω 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∈L1Ω 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∈BVΩ; the gradient measure Du=∇u dx+dDsux is decomposed into mutually singular measures ∇u dx and dDsux. As an example, we use this to prove that ∫Ωψxα2x+∇u2 dx+∫ΩψxdDsu is lower semicontinuous in L1Ω for any bounded continuous ψ and any α∈L1Ω. 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∈L1Ω, due to the compactness of BVΩ in L1Ω.