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June  2017, 6(2): 299-318. doi: 10.3934/eect.2017016

A note on dimension reduction for unbounded integrals with periodic microstructure via the unfolding method for slender domains

Elvira Zappale

Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno, via Giovanni Paolo Ⅱ, 132,84084 Fisciano (SA), Italy

Received  December 2016 Revised  January 2017 Published  April 2017

A 3D-2D dimension reduction for a nonhomogeneous constrained energy is performed in the realm of $Γ$-convergence, and two-scale convergence for slender domains, providing an integral representation for the limit functional. Applications to supremal functionals are also given.

Keywords: Dimensional reduction, homogenization, unbounded functionals, gradient constrained problems, supremal functionals.
Mathematics Subject Classification: Primary: 49J45; Secondary: 35B27, 74Q05.
Citation: Elvira Zappale. A note on dimension reduction for unbounded integrals with periodic microstructure via the unfolding method for slender domains. Evolution Equations and Control Theory, 2017, 6 (2) : 299-318. doi: 10.3934/eect.2017016
References:
[1]

E. Acerbi, G. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string, Elasticity J., 25 (1991), 137-148.  doi: 10.1007/BF00042462.

[2]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.

[3]

J.-F. Babadjian, F. Prinari and E. Zappale, Dimensional reduction for supremal functionals, Discrete Contin. Dyn. Syst., 32 (2012), 1503-1535.  doi: 10.3934/dcds.2012.32.1503.

[4]

E. N. Barron, R. R. Jensen and C. Y. Wang, Lower semicontinuity of L1 functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 495-517.  doi: 10.1016/S0294-1449(01)00070-1.

[5]

A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J., 49 (2000), 1367-1404.  doi: 10.1512/iumj.2000.49.1822.

[6]

A. Briani, F. Prinari and A. Garroni, Homogenization of L1 functionals, Math. Models Methods Appl. Sci., 14 (2004), 1784.  doi: 10.1142/S0218202504003817.

[7]

L. Carbone and R. De Arcangelis, Unbounded Functionals in the Calculus of Variations. Representation, Relaxation and Homogenization, Chapman and Hall/CRC Monogr. Surv. Pure Appl. Math. 125, Chapman and Hall/CRC Boca Raton, FL, 2001.

[8]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977.

[9]

D. Cioranescu, A. Damlamian and R. De Arcangelis, Homogenization of integrals with pointwise gradient constraints via the periodic unfolding method, Ric. Mat., 44 (2006), 31-53.  doi: 10.1007/s11587-006-0003-0.

[10]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  doi: 10.1137/080713148.

[11]

G. Dal Maso, An Introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston, MA, 1983. doi: 10.1007/978-1-4612-0327-8.

[12]

A. Davini and M. Ponsiglione, Homogenization of two-phase metrics and applications, J. Anal. Math., 103 (2007), 157-196.  doi: 10.1007/s11854-008-0005-9.

[13]

C. Kreisbeck and S. Kroemer, Heterogeneous thin films: Combining homogenization and dimension reduction with directors, SIAM J. Math. Anal., 48 (2016), 785-820.  doi: 10.1137/15M1032557.

[14]

H. Le Dret and A. Raoult, The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., 74 (1995), 548-578. 

[15]

S. Neukamm, Homogenization, Linearization and Dimension Reduction in Elasticity with Variational Methods, Dissertation Technische Universität München, 2010.

[16]

A. M. Ribeiro and E. Zappale, Existence of minimizers for nonlevel convex supremal functionals, SIAM J. Control Optim., 52 (2014), 3341-3370.  doi: 10.1137/13094390X.

[17]

E. Zappale, A remark on dimension reduction for supremal functionals: The case with convex domains, Differential Integral Equations, 26 (2013), 1077-1090. 

show all references

References:
[1]

E. Acerbi, G. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string, Elasticity J., 25 (1991), 137-148.  doi: 10.1007/BF00042462.

[2]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.

[3]

J.-F. Babadjian, F. Prinari and E. Zappale, Dimensional reduction for supremal functionals, Discrete Contin. Dyn. Syst., 32 (2012), 1503-1535.  doi: 10.3934/dcds.2012.32.1503.

[4]

E. N. Barron, R. R. Jensen and C. Y. Wang, Lower semicontinuity of L1 functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 495-517.  doi: 10.1016/S0294-1449(01)00070-1.

[5]

A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J., 49 (2000), 1367-1404.  doi: 10.1512/iumj.2000.49.1822.

[6]

A. Briani, F. Prinari and A. Garroni, Homogenization of L1 functionals, Math. Models Methods Appl. Sci., 14 (2004), 1784.  doi: 10.1142/S0218202504003817.

[7]

L. Carbone and R. De Arcangelis, Unbounded Functionals in the Calculus of Variations. Representation, Relaxation and Homogenization, Chapman and Hall/CRC Monogr. Surv. Pure Appl. Math. 125, Chapman and Hall/CRC Boca Raton, FL, 2001.

[8]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977.

[9]

D. Cioranescu, A. Damlamian and R. De Arcangelis, Homogenization of integrals with pointwise gradient constraints via the periodic unfolding method, Ric. Mat., 44 (2006), 31-53.  doi: 10.1007/s11587-006-0003-0.

[10]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  doi: 10.1137/080713148.

[11]

G. Dal Maso, An Introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston, MA, 1983. doi: 10.1007/978-1-4612-0327-8.

[12]

A. Davini and M. Ponsiglione, Homogenization of two-phase metrics and applications, J. Anal. Math., 103 (2007), 157-196.  doi: 10.1007/s11854-008-0005-9.

[13]

C. Kreisbeck and S. Kroemer, Heterogeneous thin films: Combining homogenization and dimension reduction with directors, SIAM J. Math. Anal., 48 (2016), 785-820.  doi: 10.1137/15M1032557.

[14]

H. Le Dret and A. Raoult, The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., 74 (1995), 548-578. 

[15]

S. Neukamm, Homogenization, Linearization and Dimension Reduction in Elasticity with Variational Methods, Dissertation Technische Universität München, 2010.

[16]

A. M. Ribeiro and E. Zappale, Existence of minimizers for nonlevel convex supremal functionals, SIAM J. Control Optim., 52 (2014), 3341-3370.  doi: 10.1137/13094390X.

[17]

E. Zappale, A remark on dimension reduction for supremal functionals: The case with convex domains, Differential Integral Equations, 26 (2013), 1077-1090. 

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