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. BraidesV. 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. 

show all references

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. BraidesV. 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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