Relaxation of free-discontinuity energies with obstacles,

Given a Borel function ψ defined on a bounded open set Ω with Lipschitz boundary and φ ∈ L1 (∂Ω, Hn−1 ), we prove an explicit representation formula for the L1 lower semicontinuous envelope of Mumford-Shah type functionals with the obstacle constraint u+ ≥ ψ H^{n−1} a.e. on Ω and the Dirichlet boundary condition u = φ on ∂Ω