A Density Result in GSBDp with Applications to the Approximation of Brittle Fracture Energies

We prove that any function in GSBD p (Ω) , with Ω a n-dimensional open bounded set with finite perimeter, is approximated by functions u k ∈ SBV(Ω ; R n ) ∩ L ∞ (Ω ; R n ) whose jump is a finite union of C 1 hypersurfaces. The approximation takes place in the sense of Griffith-type energies ∫ Ω W(e(u)) dx + H n-1 (J u ) , e(u) and J u being the approximate symmetric gradient and the jump set of u, and W a nonnegative function with p-growth, p > 1. The difference between u k and u is small in L p outside a sequence of sets E k ⊂ Ω whose measure tends to 0 and if | u| r ∈ L 1 (Ω) with r∈ (0 , p] , then | u k - u| r → 0 in L 1 (Ω). Moreover, an approximation property for the (truncation of the) amplitude of the jump holds. We apply the density result to deduce Γ -convergence approximation à la Ambrosio-Tortorelli for Griffith-type energies with either Dirichlet boundary condition or a mild fidelity term, such that minimisers are a priori not even in L 1 (Ω ; R n ).