Abstract
In this paper, we study the asymptotic behaviour of a family of random free-discontinuity energies
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: ZE 1186/1-1
Funding source: Engineering and Physical Sciences Research Council
Award Identifier / Grant number: EP/V00204X/1
Award Identifier / Grant number: EP/V008897/1
Funding statement: The work of C. I. Zeppieri was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project number ZE 1186/1-1 and under the Germany Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics–Geometry–Structure. L. Scardia acknowledges support by the EPSRC under the grants EP/V00204X/1 and EP/V008897/1.
References
[1] E. Acerbi, V. Chiadò Piat, G. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal. 18 (1992), no. 5, 481–496. 10.1016/0362-546X(92)90015-7Suche in Google Scholar
[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., The Clarendon Press, Oxford, 2000. 10.1093/oso/9780198502456.001.0001Suche in Google Scholar
[3] M. Barchiesi and M. Focardi, Homogenization of the Neumann problem in perforated domains: An alternative approach, Calc. Var. Partial Differential Equations 42 (2011), no. 1–2, 257–288. 10.1007/s00526-010-0387-2Suche in Google Scholar
[4] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford Lecture Ser. Math. Appl. 12, Oxford University, Oxford, 1998. 10.1093/oso/9780198502463.001.0001Suche in Google Scholar
[5] A. Braides, A. Defranceschi and E. Vitali, Homogenization of free discontinuity problems, Arch. Ration. Mech. Anal. 135 (1996), no. 4, 297–356. 10.1007/BF02198476Suche in Google Scholar
[6] A. Braides and M. Solci, Multi-scale free-discontinuity problems with soft inclusions, Boll. Unione Mat. Ital. (9) 6 (2013), no. 1, 29–51. Suche in Google Scholar
[7] F. Cagnetti, G. Dal Maso, L. Scardia and C. I. Zeppieri, Γ-convergence of free-discontinuity problems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 36 (2019), no. 4, 1035–1079. 10.1016/j.anihpc.2018.11.003Suche in Google Scholar
[8] F. Cagnetti, G. Dal Maso, L. Scardia and C. I. Zeppieri, Stochastic homogenisation of free-discontinuity problems, Arch. Ration. Mech. Anal. 233 (2019), no. 2, 935–974. 10.1007/s00205-019-01372-xSuche in Google Scholar
[9] F. Cagnetti and L. Scardia, An extension theorem in SBV and an application to the homogenization of the Mumford–Shah functional in perforated domains, J. Math. Pures Appl. (9) 95 (2011), no. 4, 349–381. 10.1016/j.matpur.2010.03.002Suche in Google Scholar
[10] C. Calvo-Jurado, J. Casado-Díaz and M. Luna-Laynez, Homogenization of the Poisson equation with Dirichlet conditions in random perforated domains, J. Comput. Appl. Math. 275 (2015), 375–381. 10.1016/j.cam.2014.07.006Suche in Google Scholar
[11] C. Calvo-Jurado, J. Casado-Díaz and M. Luna-Laynez, Homogenization of nonlinear Dirichlet problems in random perforated domains, Nonlinear Anal. 133 (2016), 250–274. 10.1016/j.na.2015.12.008Suche in Google Scholar
[12] D. Cioranescu and F. Murat, Un terme étrange venu d’ailleurs, Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. II (Paris 1979/1980), Res. Notes in Math. 60, Pitman, Boston (1982), 98–138, 389–390. Suche in Google Scholar
[13] D. Cioranescu and J. S. J. Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl. 71 (1979), no. 2, 590–607. 10.1016/0022-247X(79)90211-7Suche in Google Scholar
[14] G. Congedo and I. Tamanini, On the existence of solutions to a problem in multidimensional segmentation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 8 (1991), no. 2, 175–195. 10.1016/s0294-1449(16)30271-2Suche in Google Scholar
[15] G. Dal Maso, J.-M. Morel and S. Solimini, A variational method in image segmentation: Existence and approximation results, Acta Math. 168 (1992), no. 1–2, 89–151. 10.1007/BF02392977Suche in Google Scholar
[16] E. De Giorgi, M. Carriero and A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Arch. Ration. Mech. Anal. 108 (1989), no. 3, 195–218. 10.1007/BF01052971Suche in Google Scholar
[17] M. Focardi and M. S. Gelli, Asymptotic analysis of Mumford–Shah type energies in periodically perforated domains, Interfaces Free Bound. 9 (2007), no. 1, 107–132. 10.4171/ifb/158Suche in Google Scholar
[18] M. Focardi, M. S. Gelli and M. Ponsiglione, Fracture mechanics in perforated domains: A variational model for brittle porous media, Math. Models Methods Appl. Sci. 19 (2009), no. 11, 2065–2100. 10.1142/S0218202509004042Suche in Google Scholar
[19] A. Giacomini and M. Ponsiglione, A Γ-convergence approach to stability of unilateral minimality properties in fracture mechanics and applications, Arch. Ration. Mech. Anal. 180 (2006), no. 3, 399–447. 10.1007/s00205-005-0392-3Suche in Google Scholar
[20] A. Giunti, R. Höfer and J. J. L. Velázquez, Homogenization for the Poisson equation in randomly perforated domains under minimal assumptions on the size of the holes, Comm. Partial Differential Equations 43 (2018), no. 9, 1377–1412. 10.1080/03605302.2018.1531425Suche in Google Scholar
[21] E. J. Hruslov, Asymptotic behaviour of solutions of the second boundary-value problem under fragmentation of the boundary of the domain, Math. USSR-Sb. 35 (1979), 266–282. 10.1070/SM1979v035n02ABEH001474Suche in Google Scholar
[22] V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, 1994. 10.1007/978-3-642-84659-5Suche in Google Scholar
[23] I. Tamanini and G. Congedo, Density theorems for local minimizers of area-type functionals, Rend. Semin. Mat. Univ. Padova 85 (1991), 217–248. Suche in Google Scholar
[24] L. Tartar, Cours Peccot au Collège de France, Paris, 1977 (unpublished). Suche in Google Scholar
[25] C. I. Zeppieri, Homogenisation of high-contrast brittle materials, Math. Eng. 2 (2020), no. 1, 174–202. 10.3934/mine.2020009Suche in Google Scholar
[26] V. V. Zhikov, Averaging in punctured random domains of general type, Mat. Zametki 53 (1993), no. 1, 41–58, 155. 10.1007/BF01208520Suche in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- A flow approach to the prescribed Gaussian curvature problem in ℍ𝑛+1
- A split special Lagrangian calibration associated with frame vorticity
- The homogeneous causal action principle on a compact domain in momentum space
- The Lp Minkowski problem for q-torsional rigidity
- A twist in sharp Sobolev inequalities with lower order remainder terms
- Least energy solutions for affine p-Laplace equations involving subcritical and critical nonlinearities
- Stochastic homogenisation of free-discontinuity functionals in randomly perforated domains
- Existence of minimizers for the SDRI model in 2d: Wetting and dewetting regime with mismatch strain
- Generalized minimizing movements for the varifold Canham–Helfrich flow
- A characterization of gauge balls in ℍ n by horizontal curvature
- Minimizers of 3D anisotropic interaction energies
- Regularity results for a class of widely degenerate parabolic equations
- On isosupremic vectorial minimisation problems in L ∞ with general nonlinear constraints
- Quasiconformal, Lipschitz, and BV mappings in metric spaces
- Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity
- Optimal transport with nonlinear mobilities: A deterministic particle approximation result
- On functions of bounded β-dimensional mean oscillation
- Relaxed many-body optimal transport and related asymptotics
- Minimizers of nonlocal polyconvex energies in nonlocal hyperelasticity
Artikel in diesem Heft
- Frontmatter
- A flow approach to the prescribed Gaussian curvature problem in ℍ𝑛+1
- A split special Lagrangian calibration associated with frame vorticity
- The homogeneous causal action principle on a compact domain in momentum space
- The Lp Minkowski problem for q-torsional rigidity
- A twist in sharp Sobolev inequalities with lower order remainder terms
- Least energy solutions for affine p-Laplace equations involving subcritical and critical nonlinearities
- Stochastic homogenisation of free-discontinuity functionals in randomly perforated domains
- Existence of minimizers for the SDRI model in 2d: Wetting and dewetting regime with mismatch strain
- Generalized minimizing movements for the varifold Canham–Helfrich flow
- A characterization of gauge balls in ℍ n by horizontal curvature
- Minimizers of 3D anisotropic interaction energies
- Regularity results for a class of widely degenerate parabolic equations
- On isosupremic vectorial minimisation problems in L ∞ with general nonlinear constraints
- Quasiconformal, Lipschitz, and BV mappings in metric spaces
- Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity
- Optimal transport with nonlinear mobilities: A deterministic particle approximation result
- On functions of bounded β-dimensional mean oscillation
- Relaxed many-body optimal transport and related asymptotics
- Minimizers of nonlocal polyconvex energies in nonlocal hyperelasticity