# American Institute of Mathematical Sciences

February  2022, 17(1): 15-45. doi: 10.3934/nhm.2021022

## $\Gamma$-convergence of quadratic functionals with non uniformly elliptic conductivity matrices

 Dipartimento di Matematica, Università di Roma Tor Vergata, via della ricerca scientifica 1, Roma, 00133, Italy

* Corresponding author: Lorenza D'Elia

Received  June 2021 Revised  July 2021 Published  February 2022 Early access  September 2021

We investigate the homogenization through $\Gamma$-convergence for the $L^2({\Omega})$-weak topology of the conductivity functional with a zero-order term where the matrix-valued conductivity is assumed to be non strongly elliptic. Under proper assumptions, we show that the homogenized matrix $A^\ast$ is provided by the classical homogenization formula. We also give algebraic conditions for two and three dimensional $1$-periodic rank-one laminates such that the homogenization result holds. For this class of laminates, an explicit expression of $A^\ast$ is provided which is a generalization of the classical laminate formula. We construct a two-dimensional counter-example which shows an anomalous asymptotic behaviour of the conductivity functional.

Citation: Lorenza D'Elia. $\Gamma$-convergence of quadratic functionals with non uniformly elliptic conductivity matrices. Networks and Heterogeneous Media, 2022, 17 (1) : 15-45. doi: 10.3934/nhm.2021022
##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084. [2] G. Allaire, Shape Optimization by the Homogenization Method, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6. [3] A. Beurling and J. Deny, Espaces de dirichlet, Acta Math., 99 (1958), 203-224.  doi: 10.1007/BF02392426. [4] A. Braides, $\Gamma$-Convergence for Beginners, Oxford University Press, Oxford, 2002.  doi: 10.1093/acprof:oso/9780198507840.001.0001. [5] A. Braides, A handbook of $\Gamma$-convergence, Handbook of Differential Equations: Stationary Partial Differential Equations Vol. 3, Elsevier, (2006), 101–213. doi: 10.1016/S1874-5733(06)80006-9. [6] A. Braides, V. C. Piat and A. Piatnitski, A variational approach to double-porosity problems, Asymptot. Anal., 39 (2004), 281-308. [7] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext series, Springer, New York, 2010. [8] M. Briane, Correctors for the homogenization of a laminate, Adv. Math. Sci. Appl., 4 (1994), 357-379. [9] M. Briane, Non-Markovian quadratic forms obtained by homogenization, Boll. Uni. Mate. Ital. Sez. B Artic. Ric. Mat., 6 (2003), 323-337. [10] M. Briane and G. A. Francfort, Loss of ellipticity through homogenization in linear elasticity, Math. Mod. Met. Appl. Sci., 25 (2015), 905-928.  doi: 10.1142/S0218202515500220. [11] M. Briane and G. A. Francfort, A two-dimensional labile aether through homogenization, Commun. Math. Phys., 367 (2019), 599-628.  doi: 10.1007/s00220-019-03333-7. [12] M. Briane and A. J. Pallares Martín, Homogenization of weakly coercive integral functionals in three-dimensional linear elasticity, J. Éc. Polytech. Math., 4 (2017), 483–514. doi: 10.5802/jep.49. [13] G. Dal Maso, An Introduction to $\Gamma$-Convergence, Volume 8 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8. [14] S. Gutiérrez, Laminations in linearized elasticity: The isotropic non-very strongly elliptic case, Q. J. Mech. Appl. Math, 57 (2004), 571-582.  doi: 10.1093/qjmam/57.4.571. [15] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093. [16] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043. [17] L. Tartar, Estimations fines de coefficients homogénéisés, Ennio De Giorgi Colloquium, Ed. P. Krée, Pitman Research Notes in Mathematics, 125 (1985), 168-187.

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##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084. [2] G. Allaire, Shape Optimization by the Homogenization Method, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6. [3] A. Beurling and J. Deny, Espaces de dirichlet, Acta Math., 99 (1958), 203-224.  doi: 10.1007/BF02392426. [4] A. Braides, $\Gamma$-Convergence for Beginners, Oxford University Press, Oxford, 2002.  doi: 10.1093/acprof:oso/9780198507840.001.0001. [5] A. Braides, A handbook of $\Gamma$-convergence, Handbook of Differential Equations: Stationary Partial Differential Equations Vol. 3, Elsevier, (2006), 101–213. doi: 10.1016/S1874-5733(06)80006-9. [6] A. Braides, V. C. Piat and A. Piatnitski, A variational approach to double-porosity problems, Asymptot. Anal., 39 (2004), 281-308. [7] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext series, Springer, New York, 2010. [8] M. Briane, Correctors for the homogenization of a laminate, Adv. Math. Sci. Appl., 4 (1994), 357-379. [9] M. Briane, Non-Markovian quadratic forms obtained by homogenization, Boll. Uni. Mate. Ital. Sez. B Artic. Ric. Mat., 6 (2003), 323-337. [10] M. Briane and G. A. Francfort, Loss of ellipticity through homogenization in linear elasticity, Math. Mod. Met. Appl. Sci., 25 (2015), 905-928.  doi: 10.1142/S0218202515500220. [11] M. Briane and G. A. Francfort, A two-dimensional labile aether through homogenization, Commun. Math. Phys., 367 (2019), 599-628.  doi: 10.1007/s00220-019-03333-7. [12] M. Briane and A. J. Pallares Martín, Homogenization of weakly coercive integral functionals in three-dimensional linear elasticity, J. Éc. Polytech. Math., 4 (2017), 483–514. doi: 10.5802/jep.49. [13] G. Dal Maso, An Introduction to $\Gamma$-Convergence, Volume 8 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8. [14] S. Gutiérrez, Laminations in linearized elasticity: The isotropic non-very strongly elliptic case, Q. J. Mech. Appl. Math, 57 (2004), 571-582.  doi: 10.1093/qjmam/57.4.571. [15] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093. [16] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043. [17] L. Tartar, Estimations fines de coefficients homogénéisés, Ennio De Giorgi Colloquium, Ed. P. Krée, Pitman Research Notes in Mathematics, 125 (1985), 168-187.
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