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mumford-shah energy

ESAIM: Control, Optimisation and Calculus of Variations (1)

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Relaxation of free-discontinuity energies with obstacles
Matteo Focardi, Maria Stella Gelli
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 14 / Issue 4 / October 2008

Published online by Cambridge University Press:

07 February 2008, pp. 879-896

Print publication:

October 2008
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Given a Borel function ψ defined on a bounded open set Ω 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 $u^+\ge\psi$ ${\mathcal H}^{n-1}$ a.e. on Ω and the Dirichlet boundary condition $u=\varphi$ on $\partial\Omega$ .