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Relaxation of free-discontinuity energies with obstacles

Published online by Cambridge University Press:  07 February 2008

Matteo Focardi  and
Maria Stella Gelli
Matteo Focardi
Affiliation:
Dip. Mat. “U. Dini”, V.le Morgagni, 67/a, 50134 Firenze, Italy; focardi@math.unifi.it
Maria Stella Gelli
Affiliation:
Dip. Mat. “L. Tonelli”, L.go Bruno Pontecorvo, 5, 56127 Pisa, Italy; gelli@dm.unipi.it
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Abstract

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 $\partial\Omega$ .

Keywords

Obstacle problemsMumford-Shah energyrelaxation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 4 , October 2008 , pp. 879 - 896
Copyright
© EDP Sciences, SMAI, 2008

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