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

Previous Article
Asymptotic analysis of an elastic material reinforced with thin fractal strips
NHM Home
This Issue
Next Article
An explicit finite volume algorithm for vanishing viscosity solutions on a network

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

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

Lorenza D'Elia ^,

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.

Keywords: Quadratic functionals, homogenization, $ \Gamma $-convergence, two-scale convergence, non-local functional.

Mathematics Subject Classification: Primary: 35B27, 35B40, 49J45; Secondary: 74Q05.

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.

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. 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.

[1]	Martin Heida, Stefan Neukamm, Mario Varga. Stochastic two-scale convergence and Young measures. Networks and Heterogeneous Media, 2022, 17 (2) : 227-254. doi: 10.3934/nhm.2022004
[2]	Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485
[3]	Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223
[4]	Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems and Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036
[5]	Antonio De Rosa, Domenico Angelo La Manna. A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2101-2116. doi: 10.3934/cpaa.2021059
[6]	Stig-Olof Londen, Hana Petzeltová. Convergence of solutions of a non-local phase-field system. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 653-670. doi: 10.3934/dcdss.2011.4.653
[7]	Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks and Heterogeneous Media, 2014, 9 (2) : 353-382. doi: 10.3934/nhm.2014.9.353
[8]	Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016
[9]	Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071
[10]	Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. On the $\Gamma$-limit for a non-uniformly bounded sequence of two-phase metric functionals. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 411-426. doi: 10.3934/dcds.2015.35.411
[11]	Patrick Henning. Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems. Networks and Heterogeneous Media, 2012, 7 (3) : 503-524. doi: 10.3934/nhm.2012.7.503
[12]	Julián Fernández Bonder, Analía Silva, Juan F. Spedaletti. Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2125-2140. doi: 10.3934/dcds.2020355
[13]	Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017
[14]	Alexander Mielke. Deriving amplitude equations via evolutionary $\Gamma$-convergence. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2679-2700. doi: 10.3934/dcds.2015.35.2679
[15]	Fang Liu, Aihui Zhou. Localizations and parallelizations for two-scale finite element discretizations. Communications on Pure and Applied Analysis, 2007, 6 (3) : 757-773. doi: 10.3934/cpaa.2007.6.757
[16]	Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058
[17]	Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Two-Scale numerical simulation of sand transport problems. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 151-168. doi: 10.3934/dcdss.2015.8.151
[18]	Grigor Nika, Bogdan Vernescu. Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1553-1564. doi: 10.3934/dcdss.2016062
[19]	Junhao Hu, Chenggui Yuan. Strong convergence of neutral stochastic functional differential equations with two time-scales. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5831-5848. doi: 10.3934/dcdsb.2019108
[20]	Tahir Bachar Issa, Rachidi Bolaji Salako. Asymptotic dynamics in a two-species chemotaxis model with non-local terms. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3839-3874. doi: 10.3934/dcdsb.2017193

2020 Impact Factor: 1.213

Tools

Metrics

Other articles
by authors

on AIMS
Lorenza D'Elia
on Google Scholar
Lorenza D'Elia

2020 Impact Factor: 1.213

Tools

Article outline

2020 Impact Factor: 1.213

Tools

Metrics

Other articles
by authors

on AIMS
Lorenza D'Elia
on Google Scholar
Lorenza D'Elia

[Back to Top]