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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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References

L. Ambrosio and A. Braides, Energies in SBV and variational models in fracture mechanics, in Homogenization and Applications to Material Sciences, D. Cioranescu, A. Damlamian and P. Donato Eds., GAKUTO, Gakk $\rm\bar o$ tosho, Tokio, Japan (1997) 1–22.
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000).
Anzellotti, G., The Euler equation for functionals with linear growth. Trans. Amer. Math. Soc. 290 (1985) 483–501. CrossRef
A. Braides, Approximation of Free-Discontinuity Problems, Lecture Notes in Mathematics. Springer-Verlag, Berlin (1998).
A. Braides, Γ-convergence for beginners. Oxford University Press, Oxford (2002).
Carriero, M., Dal Maso, G., Leaci, A. and Pascali, E., Relaxation of the non-parametric Plateau problem with an obstacle. J. Math. Pures Appl. 67 (1988) 359–396.
M. Carriero, G. Dal Maso, A. Leaci and E. Pascali, Limits of obstacle problems for the area functional, in Partial Differential Equations and the Calculus of Variations, Vol. I, PNDEA 1, Birkhäuser Boston, Boston (1989) 285–309.
Colombini, F., Una definizione alternativa per una misura usata nello studio di ipersuperfici minimali. Boll. Un. Mat. Ital. 8 (1973) 159–173.
G. Dal Maso, An Introduction to Γ-convergence. Birkhäuser, Boston (1993).
G. Dal Maso, Variational problems in Fracture Mechanics. Preprint S.I.S.S.A. (2006).
Dal Maso, G., Francfort, G. and Toader, R., Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176 (2005) 165–225. CrossRef
De Giorgi, E., Problemi di superfici minime con ostacoli: forma non cartesiana. Boll. Un. Mat. Ital. 8 (1973) 80–88.
De Giorgi, E. and Ambrosio, L., Un nuovo funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82 (1988) 199–210.
E. De Giorgi, F. Colombini and L.C. Piccinini, Frontiere orientate di misura minima e questioni collegate. Quaderno della Scuola Normale Superiore di Pisa, Editrice Tecnico Scientifica, Pisa (1972).
Focardi, M. and Gelli, M.S., Asymptotic analysis of Mumford-Shah type energies in periodically perforated domains. Interfaces and Free Boundaries 9 (2007) 107–132. CrossRef
J.E. Hutchinson, A measure of De Giorgi and others does not equal twice the Hausdorff measure. Notices Amer. Math. Soc. 24 (1977) A–240.
J.E. Hutchinson, On the relationship between Hausdorff measure and a measure of De Giorgi, Colombini, Piccinini. Boll. Un. Mat. Ital. 18-B (1981) 619–628.
Mumford, D. and Shah, J., Optimal approximation by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 17 (1989) 577–685. CrossRef
L.C. Piccinini, De Giorgi's measure and thin obstacles, in Geometric measure theory and minimal surfaces, C.I.M.E. III Ciclo, Varenna (1972) 221–230; Edizioni Cremonese, Rome (1973).