Calculus of Variations and Geometric Measure Theory

M. Focardi - M. S. Gelli

Relaxation of free-discontinuity energies with obstacle

created by gelli on 21 Dec 2006
modified by focardi on 19 Dec 2011


Published Paper

Inserted: 21 dec 2006
Last Updated: 19 dec 2011

Volume: 14
Number: 4
Pages: 879--896
Year: 2008


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