$\Gamma$-convergence of free-discontinuity problems

Cagnetti, Filippo; Dal Maso, Gianni; Scardia, Lucia; Zeppieri, Caterina Ida

doi:10.1016/j.anihpc.2018.11.003

Γ

-convergence of free-discontinuity problems

Cagnetti, Filippo ¹ ; Dal Maso, Gianni ² ; Scardia, Lucia ³ ; Zeppieri, Caterina Ida ⁴

¹ Department of Mathematics, University of Sussex, Brighton, United Kingdom
² SISSA, Trieste, Italy
³ Department of Mathematics, Heriot-Watt University, Edinburgh, United Kingdom
⁴ Institut für numerische und angewandte Mathematik, WWU Münster, Germany

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 1035-1079.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé

We study the Γ-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u.

We obtain three main results: compactness with respect to Γ-convergence, representation of the Γ-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.

MR Zbl

DOI : 10.1016/j.anihpc.2018.11.003

Classification : 49J45, 49Q20, 74Q05
Mots-clés : Free-discontinuity problems, Γ-convergence, Homogenisation

Affiliations des auteurs :

Cagnetti, Filippo ¹ ; Dal Maso, Gianni ² ; Scardia, Lucia ³ ; Zeppieri, Caterina Ida ⁴

@article{AIHPC_2019__36_4_1035_0,
     author = {Cagnetti, Filippo and Dal Maso, Gianni and Scardia, Lucia and Zeppieri, Caterina Ida},
     title = {$\Gamma$-convergence of free-discontinuity problems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1035--1079},
     publisher = {Elsevier},
     volume = {36},
     number = {4},
     year = {2019},
     doi = {10.1016/j.anihpc.2018.11.003},
     mrnumber = {3955110},
     zbl = {1417.49010},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2018.11.003/}
}

TY  - JOUR
AU  - Cagnetti, Filippo
AU  - Dal Maso, Gianni
AU  - Scardia, Lucia
AU  - Zeppieri, Caterina Ida
TI  - $\Gamma$-convergence of free-discontinuity problems
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2019
SP  - 1035
EP  - 1079
VL  - 36
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2018.11.003/
DO  - 10.1016/j.anihpc.2018.11.003
LA  - en
ID  - AIHPC_2019__36_4_1035_0
ER  -

%0 Journal Article
%A Cagnetti, Filippo
%A Dal Maso, Gianni
%A Scardia, Lucia
%A Zeppieri, Caterina Ida
%T $\Gamma$-convergence of free-discontinuity problems
%J Annales de l'I.H.P. Analyse non linéaire
%D 2019
%P 1035-1079
%V 36
%N 4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2018.11.003/
%R 10.1016/j.anihpc.2018.11.003
%G en
%F AIHPC_2019__36_4_1035_0

Cagnetti, Filippo; Dal Maso, Gianni; Scardia, Lucia; Zeppieri, Caterina Ida. $\Gamma$-convergence of free-discontinuity problems. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 1035-1079. doi : 10.1016/j.anihpc.2018.11.003. http://www.numdam.org/articles/10.1016/j.anihpc.2018.11.003/

Bibliographie
Cité par

[1] Ambrosio, L. Existence theory for a new class of variational problems, Arch. Ration. Mech. Anal., Volume 111 (1990), pp. 291–322 | DOI | MR | Zbl

[2] Ambrosio, L. On the lower semicontinuity of quasi-convex integrals in $S B V (Ω; R^{k})$ , Nonlinear Anal., Volume 23 (1994), pp. 405–425 | DOI | MR | Zbl

[3] Ambrosio, L.; Fusco, N.; Pallara, D. Functions of Bounded Variations and Free Discontinuity Problems, Clarendon Press, Oxford, 2000 | DOI | MR | Zbl

[4] Barchiesi, M.; Dal Maso, G. Homogenization of fiber reinforced brittle materials: the extremal cases, SIAM J. Math. Anal., Volume 41 (2009) no. 5, pp. 1874–1889 | DOI | MR | Zbl

[5] Barchiesi, M.; Focardi, M. Homogenization of the Neumann problem in perforated domains: an alternative approach, Calc. Var. Partial Differ. Equ., Volume 42 (2011), pp. 257–288 | DOI | MR | Zbl

[6] Barchiesi, M.; Lazzaroni, G.; Zeppieri, C.I. A bridging mechanism in the homogenisation of brittle composites with soft inclusions, SIAM J. Math. Anal., Volume 48 (2016) no. 2, pp. 1178–1209 | DOI | MR | Zbl

[7] Bouchitté, G.; Fonseca, I.; Leoni, G.; Mascarenhas, L. A global method for relaxation in $W^{1, p}$ and in $S B V^{p}$ , Arch. Ration. Mech. Anal., Volume 165 (2002), pp. 187–242 | MR | Zbl

[8] Bourdin, B.; Francfort, G.A.; Marigo, J.-J. The Variational Approach to Fracture, J. Elast., Volume 91 (2008), pp. 5–148 (Reprinted from, 2008) | DOI | MR | Zbl

[9] Braides, A. $Γ$ -Convergence for Beginners , Oxford Lecture Series in Mathematics and its Applications, vol. 22, Oxford University Press, Oxford, 2002 | DOI | MR | Zbl

[10] Braides, A.; Defranceschi, A. Homogenization of Multiple Integrals, Oxford University Press, New York, 1998 | DOI | MR | Zbl

[11] Braides, A.; Defranceschi, A.; Vitali, E. Homogenization of free discontinuity problems, Arch. Ration. Mech. Anal., Volume 135 (1996), pp. 297–356 | DOI | MR | Zbl

[12] F. Cagnetti, G. Dal Maso, L. Scardia, C.I. Zeppieri, Stochastic homogenisation of free discontyinuity problems, Preprint SISSA, Trieste, 2017. | MR

[13] Cagnetti, F.; Scardia, L. An extension theorem in SBV and an application to the homogenization of the Mumford–Shah functional in perforated domains, J. Math. Pures Appl., Volume 95 (2011), pp. 349–381 | DOI | MR | Zbl

[14] Celada, P.; Dal Maso, G. Further remarks on the lower semicontinuity of polyconvex integrals, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 11 (1994), pp. 661–691 | DOI | Numdam | MR | Zbl

[15] Chambolle, A. A density result in two-dimensional linearized elasticity, and applications, Arch. Ration. Mech. Anal., Volume 167 (2003), pp. 211–233 | DOI | MR | Zbl

[16] Dal Maso, G. An Introduction to $Γ$ -Convergence, Birkhäuser, Boston, 1993 | DOI | MR | Zbl

[17] Dal Maso, G.; Francfort, G.A.; Toader, R. Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., Volume 176 (2005), pp. 165–225 | MR | Zbl

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

[19] Dal Maso, G.; Zeppieri, C.I. Homogenization of fiber reinforced brittle materials: the intermediate case, Adv. Calc. Var., Volume 3 (2010) no. 4, pp. 345–370 | MR | Zbl

[20] De Giorgi, E.; Letta, G. Une notion générale de convergence faible pour des fonctions croissantes d'ensemble, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 4 (1977), pp. 61–99 | Numdam | MR | Zbl

[21] Evans, L.C.; Gariepy, R.F. Measure Theory and Fine Properties of Functions, CRC Press, 1992 | MR | Zbl

[22] Focardi, M.; Gelli, M.S.; Ponsiglione, M. Fracture mechanics in perforated domains: a variational model for brittle porous media, Math. Models Methods Appl. Sci., Volume 19 (2009), pp. 2065–2100 | DOI | MR | Zbl

[23] Fonseca, I.; Leoni, G. Modern Methods in the Calculus of Variations: $L^{p}$ Spaces, Springer, New York, 2007 | MR | Zbl

[24] Fonseca, I.; Müller, S.; Pedregal, P. Analysis of concentration and oscillation effects generated by gradients, SIAM J. Math. Anal., Volume 29 (1998), pp. 736–756 | DOI | MR | Zbl

[25] Giacomini, A.; Ponsiglione, M. A $Γ$ -convergence approach to stability of unilateral minimality properties, Arch. Ration. Mech. Anal., Volume 180 (2006), pp. 399–447 | DOI | MR | Zbl

[26] Francfort, G.A.; Larsen, C.J. Existence and convergence for quasi-static evolution in brittle fracture, Commun. Pure Appl. Math., Volume 56 (2003), pp. 1465–1500 | DOI | MR | Zbl

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

[28] Giusti, E. Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, vol. 80, Birkhäuser, Basel, 1984 | DOI | MR | Zbl

[29] Larsen, C.J. On the representation of effective energy densities, ESAIM Control Optim. Calc. Var., Volume 5 (2000), pp. 529–538 | DOI | Numdam | MR | Zbl

[30] Morse, A.P. Perfect blankets, Trans. Am. Math. Soc., Volume 61 (1947), pp. 418–442 | DOI | MR | Zbl

[31] Pellet, X.; Scardia, L.; Zeppieri, C.I. Homogenization of high-contrast Mumford–Shah energies (Arvix preprint) | arXiv | MR

[32] Scardia, L. Damage as $Γ$ -limit of microfractures in anti-plane linearized elasticity, Math. Models Methods Appl. Sci., Volume 18 (2008), pp. 1703–1740 | DOI | MR | Zbl

[33] Scardia, L. Damage as the $Γ$ -limit of microfractures in linearized elasticity under the non-interpenetration constraint, Adv. Calc. Var., Volume 3 (2010), pp. 423–458 | DOI | MR | Zbl

Cité par Sources :