Lower semicontinuity and Γ-convergence of a class of linear growth functionals. (November 2019) - Wunderli, T.
Abstract: We prove some new results for linear growth functionals ∫ Ω φ ( x, D u ), u ∈ B V Ω, where φ ( x, p ) = g ( x, | p | ) if | p | ≤ β ψ ( x ) | p | + k ( x ) if | p | > β, ψ ∈ C Ω ∩ L ∞ Ω, ψ ≥ 0 on Ω, and φ is convex in p . In particular, we give conditions on φ for which ∫ Ω φ ( x, D u ) is lower semicontinuous in L 1 Ω . Notably, we make no lower semicontinuity assumption for φ in ( x, p ) and no differentiability assumption for φ in p, as is done in earlier work. We also consider more general linear growth functionals ∫ Ω g ( x, | D u | ) with g ( x, | p | ) convex in | p | and prove Γ -convergence of functionals of the form ∫ Ω φ ( x, D u ) to ∫ Ω g ( x, | D u | ) . Finally, functionals with specified trace values for u are also considered.