Relaxation of free-discontinuity energies with obstacles

Matteo Focardi; Maria Stella Gelli

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 4, page 879-896
  • ISSN: 1292-8119

Abstract

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Given a Borel function ψ defined on a bounded open set Ω with Lipschitz boundary and ϕ ∈ L 1 ( ∂ Ω , ℋ n - 1 ) , we prove an explicit representation formula for the L1 lower semicontinuous envelope of Mumford-Shah type functionals with the obstacle constraint u + ≥ ψ ℋ n - 1 a.e. on Ω and the Dirichlet boundary condition u = ϕ on ∂ Ω .

How to cite

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Focardi, Matteo, and Gelli, Maria Stella. "Relaxation of free-discontinuity energies with obstacles." ESAIM: Control, Optimisation and Calculus of Variations 14.4 (2008): 879-896. <http://eudml.org/doc/250308>.

@article{Focardi2008,
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 L1 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 $\partial\Omega$. },
author = {Focardi, Matteo, Gelli, Maria Stella},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Obstacle problems; Mumford-Shah energy; relaxation; obstacle problems},
language = {eng},
month = {2},
number = {4},
pages = {879-896},
publisher = {EDP Sciences},
title = {Relaxation of free-discontinuity energies with obstacles},
url = {http://eudml.org/doc/250308},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Focardi, Matteo
AU - Gelli, Maria Stella
TI - Relaxation of free-discontinuity energies with obstacles
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/2//
PB - EDP Sciences
VL - 14
IS - 4
SP - 879
EP - 896
AB - 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 L1 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 $\partial\Omega$.
LA - eng
KW - Obstacle problems; Mumford-Shah energy; relaxation; obstacle problems
UR - http://eudml.org/doc/250308
ER -

References

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