RELAXATION OF FREE DISCONTINUITY ENERGIES WITH OBSTACLES

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