Approximation by difference schemes of fracture energies: the vectorial case

We provide a variational approximation, in the sense of De Giorgi’s Γ-convergence, by finite-difference schemes of functionals of the type g(u+ − u− , νu ) dH2 ψ(∇u) dx + Ω Ju defined for u ∈ SBV (Ω; RN ), where Ω is an open set in R3 , ψ and g are assigned. More precisely, ψ is a quasi-convex function with p-growth, p > 1, and g satisfies standard lower semicontinuity conditions. The approximating functionals are of the form ψε (∇u(x)) dx Tε ∩Ω where ψε is an interaction potential taking into account a separation of scales, Tε is a suitable regular triangulation of R3 and u is affine on each element of the assigned triangulation.