Unilateral gradient flow of the Ambrosio–Tortorelli functional by minimizing movements
Babadjian, Jean-François ; Millot, Vincent
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 779-822 / Harvested from Numdam
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Motivated by models of fracture mechanics, this paper is devoted to the analysis of a unilateral gradient flow of the Ambrosio–Tortorelli functional, where unilaterality comes from an irreversibility constraint on the fracture density. Solutions of such evolution are constructed by means of an implicit Euler scheme. An asymptotic analysis in the Mumford–Shah regime is then carried out. It shows the convergence towards a generalized heat equation outside a time increasing crack set. In the spirit of gradient flows in metric spaces, a notion of curve of maximal unilateral slope is also investigated, and analogies with the unilateral slope of the Mumford–Shah functional are also discussed.

@article{AIHPC_2014__31_4_779_0,
     author = {Babadjian, Jean-Fran\c cois and Millot, Vincent},
     title = {Unilateral gradient flow of the Ambrosio--Tortorelli functional by minimizing movements},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {31},
     year = {2014},
     pages = {779-822},
     doi = {10.1016/j.anihpc.2013.07.005},
     zbl = {1302.35051},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2014__31_4_779_0}
}

Babadjian, Jean-François; Millot, Vincent. Unilateral gradient flow of the Ambrosio–Tortorelli functional by minimizing movements. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) pp. 779-822. doi : 10.1016/j.anihpc.2013.07.005. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_4_779_0/

[1] L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995), 191 -246 | MR 1387558 | Zbl 0957.49029

[2] L. Ambrosio, A. Braides, Energies in SBV and variational models in fracture mechanics, Homogenization and Applications to Material Sciences, Nice, 1995, GAKUTO Internat. Ser. Math. Sci. Appl. vol. 9 , Gakkotosho (1995), 1 -22 | MR 1473974 | Zbl 0904.73045

[3] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press (2000) | MR 1857292 | Zbl 0957.49001

[4] L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich , Birkhäuser Verlag, Basel (2008) | MR 2401600 | Zbl 1145.35001

[5] L. Ambrosio, V.M. Tortorelli, On the approximation of free discontinuity problems, Boll. Unione Mat. Ital. 7 (1992), 105 -123 | MR 1164940 | Zbl 0776.49029

[6] L. Ambrosio, V.M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals by Γ-convergence, Comm. Pure Appl. Math. 43 (1990), 999 -1036 | MR 1075076 | Zbl 0722.49020

[7] J.-F. Babadjian, V. Millot, Unilateral gradient flow of the Ambrosio–Tortorelli functional by minimizing movements, http://fr.arxiv.org/abs/1207.3687 (2012) | MR 3249813 | Zbl 1302.35051

[8] B. Bourdin, G.A. Francfort, J.-J. Marigo, Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids 48 (2000), 797 -826 | MR 1745759 | Zbl 0995.74057

[9] B. Bourdin, G.A. Francfort, J.-J. Marigo, The variational approach to fracture, J. Elasticity 9 (2008), 5 -148 | MR 2390547 | Zbl 1176.74018

[10] A. Braides, A handbook of Gamma-convergence, M. Chipot, P. Quittner (ed.), Handbook of Differential Equations. Stationary Partial Differential Equations, vol. 3, Elsevier (2006) | MR 2179018

[11] A. Braides, A. Chambolle, M. Solci, A relaxation result for energies defined on pairs set-function and applications, ESAIM Control Optim. Calc. Var. 13 (2007), 717 -734 | Numdam | MR 2351400 | Zbl 1149.49017

[12] H. Brézis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, American Elsevier, Amsterdam–London, New York (1973) | MR 348562 | Zbl 0252.47055

[13] A. Chambolle, F. Doveri, Minimizing movements of the Mumford–Shah functional, Discrete Contin. Dyn. Syst. A 3 (1997), 153 -174 | MR 1432071 | Zbl 0948.35073

[14] D. Cioranescu, F. Murat, Un terme étrange venu d'ailleurs, Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, vol. III, Paris, 1980/1981, Res. Notes in Math. vol. 70 , Pitman, Boston, MA (1982), 154 -178 | MR 670272 | Zbl 0498.35034 | Zbl 0496.35030

[15] G. Dal Maso, G. Francfort, R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Rational Mech. Anal. 176 (2005), 165 -225 | MR 2186036 | Zbl 1064.74150

[16] G. Dal Maso, C.J. Larsen, Existence for wave equations on domains with arbitrary growing cracks, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, (9) Mat. Appl. 22 (2011), 387 -408 | MR 2847479 | Zbl 1239.35086

[17] G. Dal Maso, P. Longo, Γ-limits of obstacles, Ann. Mat. Pura Appl. 128 (1981), 1 -50 | MR 640775 | Zbl 0467.49004

[18] G. Dal Maso, R. Toader, A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Rational Mech. Anal. 162 (2002), 101 -135 | MR 1897378 | Zbl 1042.74002

[19] G. Dal Maso, R. Toader, A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci. 12 (2002), 1773 -1800 | MR 1946723 | Zbl 1205.74149

[20] G. Dal Maso, R. Toader, On a notion of unilateral slope for the Mumford–Shah functional, NoDEA 13 (2007), 713 -734 | MR 2329026 | Zbl 1119.49017

[21] E. De Giorgi, New problems on minimizing movements, C. Baiocchi, J.-L. Lions (ed.), Boundary Value Problems for PDE and Applications, Masson, Paris (1993), 81 -98 | Zbl 0851.35052

[22] E. De Giorgi, M. Carriero, A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Arch. Rational Mech. Anal. 108 (1989), 195 -218 | MR 1012174 | Zbl 0682.49002

[23] E. De Giorgi, A. Marino, M. Tosques, Problemi di evoluzione in spazi metrici e curve di massima pendenza, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 68 (1980), 180 -187 | MR 636814 | Zbl 0465.47041

[24] J. Diestel, J.J. Uhl, Vector Measures, Mathematical Surveys vol. 15 , American Mathematical Society, Providence, RI (1977) | MR 453964 | Zbl 0369.46039

[25] G.A. Francfort, C.J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Comm. Pure Appl. Math. 56 (2003), 1465 -1500 | MR 1988896 | Zbl 1068.74056

[26] G.A. Francfort, N.Q. Le, S. Serfaty, Critical Points of Ambrosio–Tortorelli converge to critical points of Mumford–Shah in the one-dimensional Dirichlet case, ESAIM Control Optim. Calc. Var. 15 (2009), 576 -598 | Numdam | MR 2542574 | Zbl 1168.49041

[27] G.A. Francfort, J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998), 1319 -1342 | MR 1633984 | Zbl 0966.74060

[28] X. Feng, A. Prohl, Analysis of gradient flow of a regularized Mumford–Shah functional for image segmentation and image inpainting, ESAIM: M2AN 38 (2004), 291 -320 | Numdam | MR 2069148 | Zbl 1074.65106

[29] M. Focardi, On the variational approximation of free discontinuity problems in the vectorial case, Math. Models Methods Appl. Sci. 11 (2001), 663 -684 | MR 1832998 | Zbl 1010.49010

[30] A. Giacomini, Ambrosio–Tortorelli approximation of quasi-static evolution of brittle fractures, Calc. Var. Partial Differential Equations 22 (2005), 129 -172 | MR 2106765 | Zbl 1068.35189

[31] M. Gobbino, Gradient flow for the one-dimensional Mumford–Shah functional, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), 145 -193 | Numdam | MR 1658873 | Zbl 0931.49010

[32] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics vol. 24 , Pitman (Advanced Publishing Program), Boston, MA (1985) | MR 775683 | Zbl 0695.35060

[33] C.J. Larsen, Epsilon-stable quasi-static brittle fracture evolution, Comm. Pure Appl. Math. 63 (2010), 630 -654 | MR 2583308 | Zbl 05689532

[34] C.J. Larsen, C. Ortner, E. Süli, Existence of solutions to a regularized model of dynamic fracture, Math. Models Methods Appl. Sci. 20 (2010), 1021 -1048 | MR 2673410 | Zbl 05781032

[35] N.Q. Le, Convergence results for critical points of the one-dimensional Ambrosio–Tortorelli functional with fidelity term, Adv. Differential Equations 15 (2010), 255 -282 | MR 2588770 | Zbl 1190.49048

[36] A. Mainik, A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations 22 (2005), 73 -99 | MR 2105969 | Zbl 1161.74387

[37] D. Mumford, J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 17 (1989), 577 -685 | MR 997568 | Zbl 0691.49036

[38] R. Rossi, G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var. 12 (2006), 564 -614 | Numdam | MR 2224826 | Zbl 1116.34048

[39] E. Sandier, S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg–Landau, Comm. Pure Appl. Math. 57 (2004), 1627 -1672 | MR 2082242 | Zbl 1065.49011

[40] S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. A 31 (2011), 1427 -1451 | MR 2836361 | Zbl 1239.35015