Abstract
In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.
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References
Akcoglu, M.A., Krengel, U.: Ergodic theorems for superadditive processes. J. Reine Angew. Math. 323, 53–67 (1981)
Alicandro, R., Cicalese, M., Ruf, M.: Domain formation in magnetic polymer composites: an approach via stochastic homogenization. Arch. Rat. Mech. Anal. 218, 945–984 (2015)
Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. Springer, Berlin (2006)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variations and Free Discontinuity Problems. Clarendon Press, Oxford (2000)
Armstrong, S.N., Kuusi, T., Mourrat, J.-C.: Mesoscopic higher regularity and subadditivity in elliptic homogenization. Commun. Math. Phys. 347(2), 315–361 (2016)
Armstrong, S.N., Kuusi, T., Mourrat, J.-C.: The additive structure of elliptic homogenization. Invent. Math. 208(3), 999–1054 (2017)
Armstrong, S.N., Mourrat, J.-C.: Lipschitz regularity for elliptic equations with random coefficients. Arch. Ration. Mech. Anal. 219(1), 255–348 (2016)
Armstrong, S.N., Smart, C.K.: Quantitative stochastic homogenization of convex integral functionals. Ann. Sci. Éc. Norm. Supér. (4) 49(2), 423–481, 2016
Barchiesi, M., Dal Maso, G.: Homogenization of fiber reinforced brittle materials: the extremal cases. SIAM J. Math. Anal. 41(5), 1874–1889 (2009)
Barchiesi, M., Focardi, M.: Homogenization of the Neumann problem in perforated domains: an alternative approach. Calc. Var. Partial Differ. Equ. 42, 257–288 (2011)
Barchiesi, M., Lazzaroni, G., Zeppieri, C.I.: A bridging mechanism in the homogenisation of brittle composites with soft inclusions. SIAM J. Math. Anal. 48(2), 1178–1209 (2016)
Bhattacharya, R., Waymire, E.C.: A Basic Course in Probability Theory. Springer, Berlin (2007)
Braides, A., Defranceschi, A., Vitali, E.: Homogenization of free discontinuity problems. Arch. Ration. Mech. Anal. 135, 297–356 (1996)
Braides, A., Piatnitski, A.: Homogenization of surface and length energies for spin systems. J. Funct. Anal. 264, 1296–1328 (2013)
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. 95, 349–381 (2011)
Cagnetti, F., Dal Maso, G., Scardia L., Zeppieri, C. I.: \(\Gamma \)-Convergence of free-discontinuity problems. Ann. Inst. H. Poincaré Anal. Non Linéaire. Published online at https://doi.org/10.1016/j.anihpc.2018.11.003
Dal Maso, G., Modica, L.: Nonlinear stochastic homogenization. Ann. Mat. Pura Appl. 4(144), 347–389 (1986)
Dal Maso, G., Modica, L.: Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math. 368, 28–42 (1986)
Dal Maso, G., Francfort, G.A., Toader, R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176, 165–225 (2005)
Dal Maso, G., Zeppieri, C.I.: Homogenization of fiber reinforced brittle materials: the intermediate case. Adv. Calc. Var. 3(4), 345–370 (2010)
De Giorgi, E.: Free Discontinuity problems in calculus of variations. Frontiers in pure and applied Mathematics, a collection of papers dedicated to J.L. Lions on the occasion of his 60th birthday, R. Dautray ed., North Holland, 1991
De Giorgi, E., Ambrosio, L.: New functionals in the calculus of variations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 82(2), (1988) 199–210, 1989 (Italian)
Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. North-Holland Publishing Company, Amsterdam (1978)
Doob, J.L.: Stochastic Processes. John Wiley & Sons, Wiley Classics Library Edition, Hoboken (1990)
Focardi, M., Gelli, M.S., Ponsiglione, M.: Fracture mechanics in perforated domains: a variational model for brittle porous media. Math. Models Methods Appl. Sci. 19, 2065–2100 (2009)
Gloria, A., Otto, F.: An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39(3), 779–856 (2011)
Gloria, A., Otto, F.: An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22(1), 1–28 (2012)
Kozlov, S.M.: The averaging of random operators. Mat. Sb. 109, 188–2012 (1979)
Krengel, U.: Ergodic theorems. De Gruyter Studies in Mathematics, Vol. 6. Walter de Gruyter & Co., Berlin, 1985
Messaoudi, K., Michaille, G.: Stochastic homogenization of nonconvex integral functionals. RAIRO Modél. Math. Anal. Numér. 3(28), 329–356 (1991)
Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. Random Fields, Vol. I, II (Esztergom, : Colloquia Mathematica Societatis János Bolyai, 27, p. 1981. North-Holland, Amsterdam (1979)
Papanicolaou, G.C., Varadhan, S.R.S.: Diffusions with random coefficients. Essays in Honor of C. R. Rao. North-Holland, Amsterdam, Statistics and Probability (1982)
Pellet, X., Scardia, L., Zeppieri, C. I.: Homogenization of high-contrast Mumford–Shah energies, submitted 2018. Preprint version: arXiv:1807.08705
Scardia, L.: Damage as \(\Gamma \)-limit of microfractures in anti-plane linearized elasticity. Math. Models Methods Appl. Sci. 18, 1703–1740 (2008)
Scardia, L.: Damage as the \(\Gamma \)-limit of microfractures in linearized elasticity under the non-interpenetration constraint. Adv. Calc. Var. 3, 423–458 (2010)
Acknowledgements
F. Cagnetti wishes to thank Panagiotis E. Souganidis for suggesting to consider the stochastic counterpart of [15]. The authors are grateful to Marco Cicalese who drew their attention to the argument in [2, Proof of Theorem 5.5, Step 2] (see also the previous result by Braides and Piatnitski [14, Proposition 2.10]) which is crucial in the proof of Theorem 6.1. F. Cagnetti was supported by the EPSRC under the Grant EP/P007287/1 “Symmetry of Minimisers in Calculus of Variations”. The research of G. Dal Maso was partially funded by the European Research Council under Grant No. 290888 “Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture”. G. Dal Maso is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). L. Scardia acknowledges support by the EPSRC under the Grant EP/N035631/1 “Dislocation patterns beyond optimality”.
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Cagnetti, F., Dal Maso, G., Scardia, L. et al. Stochastic Homogenisation of Free-Discontinuity Problems. Arch Rational Mech Anal 233, 935–974 (2019). https://doi.org/10.1007/s00205-019-01372-x
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DOI: https://doi.org/10.1007/s00205-019-01372-x