Relaxation of free-discontinuity energies with obstacles

Focardi, Matteo ; Gelli, Maria Stella

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 879-896

Résumé

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 $\partial \Omega$.

Détail

DOI : 10.1051/cocv:2008014

Classification : 49J45, 74R10
Keywords: obstacle problems, Mumford-Shah energy, relaxation