Given a Borel function defined on a bounded open set with Lipschitz boundary and , we prove an explicit representation formula for the lower semicontinuous envelope of Mumford-Shah type functionals with the obstacle constraint a.e. on and the Dirichlet boundary condition on .
Mots clés : obstacle problems, Mumford-Shah energy, relaxation
@article{COCV_2008__14_4_879_0, author = {Focardi, Matteo and Gelli, Maria Stella}, title = {Relaxation of free-discontinuity energies with obstacles}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {879--896}, publisher = {EDP-Sciences}, volume = {14}, number = {4}, year = {2008}, doi = {10.1051/cocv:2008014}, mrnumber = {2451801}, zbl = {1148.49011}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2008014/} }
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 PY - 2008 SP - 879 EP - 896 VL - 14 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008014/ DO - 10.1051/cocv:2008014 LA - en ID - COCV_2008__14_4_879_0 ER -
%0 Journal Article %A Focardi, Matteo %A Gelli, Maria Stella %T Relaxation of free-discontinuity energies with obstacles %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 879-896 %V 14 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008014/ %R 10.1051/cocv:2008014 %G en %F COCV_2008__14_4_879_0
Focardi, Matteo; Gelli, Maria Stella. Relaxation of free-discontinuity energies with obstacles. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 879-896. doi : 10.1051/cocv:2008014. http://www.numdam.org/articles/10.1051/cocv:2008014/
[1] Energies in SBV and variational models in fracture mechanics1997) 1-22. | MR | Zbl
and ,[2] Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000). | MR | Zbl
, and ,[3] The Euler equation for functionals with linear growth. Trans. Amer. Math. Soc. 290 (1985) 483-501. | MR | Zbl
,[4] Approximation of Free-Discontinuity Problems, Lecture Notes in Mathematics. Springer-Verlag, Berlin (1998). | MR | Zbl
,[5] -convergence for beginners. Oxford University Press, Oxford (2002). | MR | Zbl
,[6] Relaxation of the non-parametric Plateau problem with an obstacle. J. Math. Pures Appl. 67 (1988) 359-396. | MR | Zbl
, , and ,[7] 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. | MR | Zbl
, , and ,[8] Una definizione alternativa per una misura usata nello studio di ipersuperfici minimali. Boll. Un. Mat. Ital. 8 (1973) 159-173. | MR | Zbl
,[9] An Introduction to -convergence. Birkhäuser, Boston (1993). | MR | Zbl
,[10] Variational problems in Fracture Mechanics. Preprint S.I.S.S.A. (2006). | MR
,[11] Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176 (2005) 165-225. | MR | Zbl
, and ,[12] Problemi di superfici minime con ostacoli: forma non cartesiana. Boll. Un. Mat. Ital. 8 (1973) 80-88. | MR | Zbl
,[13] Un nuovo funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82 (1988) 199-210. | MR | Zbl
and ,[14] Frontiere orientate di misura minima e questioni collegate. Quaderno della Scuola Normale Superiore di Pisa, Editrice Tecnico Scientifica, Pisa (1972). | MR | Zbl
, and ,[15] Asymptotic analysis of Mumford-Shah type energies in periodically perforated domains. Interfaces and Free Boundaries 9 (2007) 107-132. | MR | Zbl
and ,[16] A measure of De Giorgi and others does not equal twice the Hausdorff measure. Notices Amer. Math. Soc. 24 (1977) A-240.
,[17] On the relationship between Hausdorff measure and a measure of De Giorgi, Colombini, Piccinini. Boll. Un. Mat. Ital. 18-B (1981) 619-628. | MR | Zbl
,[18] Optimal approximation by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 17 (1989) 577-685. | MR | Zbl
and ,[19] 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). | MR | Zbl
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