Homogenization of nonconvex unbounded singular integrals

Omar Anza Hafsa; Nicolas Clozeau; Jean-Philippe Mandallena

doi:10.5802/ambp.367

Omar Anza Hafsa ^1,² ; Nicolas Clozeau ³ ; Jean-Philippe Mandallena ²

¹ LMGC, UMR-CNRS 5508 Place Eugène Bataillon 34095 Montpellier, France
² Université de Nîmes Laboratoire MIPA, Site des Carmes Place Gabriel Péri 30021 Nîmes, France
³ École Normale Supérieure de Cachan 61 avenue du président Wilson 94230 Cachan, France

Annales mathématiques Blaise Pascal, Tome 24 (2017) no. 2, pp. 135-193.

Résumé

We study periodic homogenization by $Γ$ -convergence of integral functionals with integrands $W (x, ξ)$ having no polynomial growth and which are both not necessarily continuous with respect to the space variable and not necessarily convex with respect to the matrix variable. This allows to deal with homogenization of composite hyperelastic materials consisting of two or more periodic components whose the energy densities tend to infinity as the volume of matter tends to zero, i.e., $W (x, ξ) = \sum_{j \in J} 1_{V_{j}} (x) H_{j} (ξ)$ where ${V_{j}}_{j \in J}$ is a finite family of open disjoint subsets of $ℝ^{N}$ , with $| \partial V_{j} | = 0$ for all $j \in J$ and $| ℝ^{N} ∖ ⋃_{j \in J} V_{j} | = 0$ , and, for each $j \in J$ , $H_{j} (ξ) \to \infty$ as $det ξ \to 0$ . In fact, our results apply to integrands of type $W (x, ξ) = a (x) H (ξ)$ when $H (ξ) \to \infty$ as $det ξ \to 0$ and $a \in L^{\infty} (ℝ^{N}; [0, \infty [)$ is $1$ -periodic and is either continuous almost everywhere or not continuous. When $a$ is not continuous, we obtain a density homogenization formula which is a priori different from the classical one by Braides–Müller. Although applications to hyperelasticity are limited due to the fact that our framework is not consistent with the constraint of noninterpenetration of the matter, our results can be of technical interest to analysis of homogenization of integral functionals.

Publié le : 2017-11-20

DOI : 10.5802/ambp.367

Mots clés : Homogenization, $\Gamma $-convergence, Unbounded integrand, Singular growth, Determinant constraint type, hyperelasticity

Affiliations des auteurs :

Omar Anza Hafsa ^{1, 2} ; Nicolas Clozeau ³ ; Jean-Philippe Mandallena ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Homogenization of nonconvex unbounded singular integrals},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {135--193},
     publisher = {Annales math\'ematiques Blaise Pascal},
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Omar Anza Hafsa; Nicolas Clozeau; Jean-Philippe Mandallena. Homogenization of nonconvex unbounded singular integrals. Annales mathématiques Blaise Pascal, Tome 24 (2017) no. 2, pp. 135-193. doi : 10.5802/ambp.367. https://ambp.centre-mersenne.org/articles/10.5802/ambp.367/

Bibliographie
Cité par

[1] Emilio Acerbi; Nicola Fusco Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal., Volume 86 (1984) no. 2, pp. 125-145 | DOI | Zbl

[2] Omar Anza Hafsa; Mohamed Lamine Leghmizi; Jean-Philippe Mandallena On a homogenization technique for singular integrals, Asymptotic Anal., Volume 74 (2011) no. 3-4, pp. 123-134 | Zbl

[3] Omar Anza Hafsa; Jean-Philippe Mandallena Relaxation of variational problems in two-dimensional nonlinear elasticity, Ann. Mat. Pura Appl., Volume 186 (2007) no. 1, pp. 185-196 | DOI | Zbl

[4] Omar Anza Hafsa; Jean-Philippe Mandallena Relaxation theorems in nonlinear elasticity, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 25 (2008) no. 1, pp. 135-148 | DOI | Zbl

[5] Omar Anza Hafsa; Jean-Philippe Mandallena Relaxation et passage 3D-2D avec contraintes de type déterminant (2009) (https://arxiv.org/abs/0901.3688)

[6] Omar Anza Hafsa; Jean-Philippe Mandallena Homogenization of nonconvex integrals with convex growth, J. Math. Pures Appl., Volume 96 (2011) no. 2, pp. 167-189 | DOI | Zbl

[7] Omar Anza Hafsa; Jean-Philippe Mandallena Homogenization of unbounded singular integrals in $W^{1, \infty}$ , Ric. Mat., Volume 61 (2012) no. 2, pp. 185-217 | DOI | Zbl

[8] Omar Anza Hafsa; Jean-Philippe Mandallena $Γ$ -limits of functionals determined by their infima, J. Convex Anal., Volume 23 (2016) no. 1, pp. 103-137 | Zbl

[9] Omar Anza Hafsa; Jean-Philippe Mandallena Relaxation of nonconvex unbounded intergals with general growth conditions in Cheeger-Sobolev spaces (2016) (à paraître dans Bull. Sci. Math.)

[10] Omar Anza Hafsa; Jean-Philippe Mandallena; Hamdi Zorgati Homogenization of unbounded integrals with quasiconvex growth, Ann. Mat. Pura Appl., Volume 194 (2015) no. 6, pp. 1619-1648 | DOI | Zbl

[11] Hafedh Ben Belgacem Modélisation de structures minces en élasticité non linéaire, Université Pierre et Marie Curie (France) (1996) (Ph. D. Thesis)

[12] Guy Bouchitté; Michel Bellieud Regularization of a set function – application to integral representation, Ric. Mat., Volume 49 (suppl.) (2000), pp. 79-93 | Zbl

[13] Guy Bouchitté; Irene Fonseca; Luisa Mascarenhas A global method for relaxation, Arch. Ration. Mech. Anal., Volume 145 (1998) no. 1, pp. 51-98 | DOI | Zbl

[14] Andrea Braides Homogenization of some almost periodic coercive functional, Rend. Accad. Naz. Sci. Detta XL, Mem. Mat., Volume 9 (1985) no. 1, pp. 313-322 | Zbl

[15] Andrea Braides $Γ$ -convergence for beginners, Oxford Lecture Series in Mathematics and its Applications, 22, Oxford University Press, 2002, xii+2118 pages | Zbl

[16] Andrea Braides; Anneliese Defranceschi Homogenization of multiple integrals, Oxford Lecture Series in Mathematics and its Applications, 12, Clarendon Press, 1998, xiv+298 pages | Zbl

[17] Bernard Dacorogna Quasiconvexity and relaxation of nonconvex problems in the calculus of variations, J. Funct. Anal., Volume 46 (1982), pp. 102-118 | DOI | Zbl

[18] Bernard Dacorogna Direct methods in the calculus of variations, Applied Mathematical Sciences, 78, Springer, 2008, xii+619 pages | Zbl

[19] Bernard Dacorogna; Ana Margarida Ribeiro Existence of solutions for some implicit partial differential equations and applications to variational integrals involving quasi-affine functions, Proc. R. Soc. Edinb., Sect. A, Math., Volume 134 (2004) no. 5, pp. 907-921 | DOI | Zbl

[20] Gianni Dal Maso An introduction to $Γ$ -convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuse, 1993, xiv+340 pages | Zbl

[21] Mitia Duerinckx; Antoine Gloria Stochastic Homogenization of Nonconvex Unbounded Integral Functionals with Convex Growth, Arch. Ration. Mech. Anal., Volume 221 (2016) no. 3, pp. 1511-1584 | DOI | Zbl

[22] Irene Fonseca The lower quasiconvex envelope of the stored energy function for an elastic crystal, J. Math. Pures Appl., Volume 67 (1988) no. 2, pp. 175-195 | Zbl

[23] Charles B.jun. Morrey Quasi-convexity and the lower semicontinuity of multiple integrals, Pac. J. Math., Volume 2 (1952), pp. 25-53 | DOI | Zbl

[24] Stefan Müller Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Ration. Mech. Anal., Volume 99 (1987) no. 3, pp. 189-212 | DOI | Zbl

Cité par Sources :