Relaxation of free-discontinuity energies with obstacles

Given a Borel function $\psi$ defined on a bounded open set $\Omega$ with Lipschitz boundary and $\varphi \in L^1(\partial \Omega ,{\mathscr{H}}^{n-1})$, we prove an explicit representation formula for the $L^1$ lower semicontinuous envelope of Mumford-Shah type functionals with the obstacle constraint $u^{+}\ge \psi$ ${\mathscr{H}}^{n-1}$ a.e. on $\Omega$ and the Dirichlet boundary condition $u=\varphi$ on $\partial \Omega$.

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