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January  2014, 13(1): 249-272. doi: 10.3934/cpaa.2014.13.249

Homogenization and correctors for the hyperbolic problems with imperfect interfaces via the periodic unfolding method

1. 

Department of Mathematics, South-Central University for Nationalities, Wuhan 430074, China

Received  December 2012 Revised  June 2013 Published  July 2013

In this paper, we study the homogenization and corrector results for the hyperbolic problem in a two-component composite with $\varepsilon$-periodic connected inclusions. The condition prescribed on the interface is that a jump of the solution is proportional to the conormal derivatives via a function of order $\varepsilon^\gamma$ ($\gamma < -1$). The main ingredient of the proof of our main theorems is the time-dependent periodic unfolding method in two-component domains. Our homogenization results recover those of the corresponding case in [Donato, Faella and Monsurrò, J. Math. Pures Appl. 87 (2007), pp. 119-143]. We also derive the corresponding corrector results.
Citation: Zhanying Yang. Homogenization and correctors for the hyperbolic problems with imperfect interfaces via the periodic unfolding method. Communications on Pure and Applied Analysis, 2014, 13 (1) : 249-272. doi: 10.3934/cpaa.2014.13.249
References:
[1]

S. Brahim-Otsman, G. Francfort and F. Murat, Correctors for the homogenization of the wave and heat equations, J. Math. Pures Appl., 71 (1992), 197-231.

[2]

D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44 (2002), 718-760. doi: 10.1137/100817942.

[3]

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.

[4]

D. Cioranescu and P. Donato, "An Introduction to Homogenization," Oxford Univ. Press, Oxford, 1999.

[5]

H. Carslaw and J. Jaeger, "Conduction of Heat in Solids," Clarendon Press, Oxford, 1947.

[6]

P. Donato, Some corrector results for composites with imperfect interface, Rend. Mat. Appl., VII. Ser., 26 (2006), 189-209.

[7]

P. Donato, L. Faella and S. Monsurrò, Homogenization of the wave equation in composites with imperfect interface: A memory effect, J. Math. Pures Appl., 87 (2007), 119-143. doi: 10.1016/j.matpur.2006.11.004.

[8]

P. Donato, L. Faella and S. Monsurrò, Correctors for the homogenization of a class of hyperbolic equations with imperfect interfaces, SIAM J. Math. Anal., 40 (2009), 1952-1978. doi: 10.1137/080712684.

[9]

P. Donato and E. Jose, Corrector results for a parabolic problem with a memory effect, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), 421-454. doi: 10.1051/m2an/2010008.

[10]

P. Donato and S. Monsurrò, Homogenization of two heat conductors with an interfacial contact resistance, Analysis and Applications, 2 (2004), 247-273. doi: 10.1142/S0219530504000345.

[11]

P. Donato, K. Le Nguyen and R. Tardieu, The periodic unfolding method for a class of imperfect transmission problems, J. Math. Sci., 176 (2011), 891-927.

[12]

P. Donato and Z. Yang, The periodic unfolding method for the wave equation in domains with holes, Advances in Mathematical Sciences and Applications, 22 (2012), 521-551.

[13]

E. Jose, Homogenization of a parabolic problem with an imperfect interface, Rev. Rouma. Math. Pures Appl., 54 (2009), 189-222.

[14]

S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 13 (2003), 43-63.

[15]

S. Monsurrò, Erratum for the paper Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 14 (2004), 375-377.

[16]

A. Nabil, A corrector result for the wave equations in perforated domains, Gakuto Internat. Ser., Math. Sci. Appl., 9 (1997), 309-321.

[17]

L. Tartar, Quelques remarques sur l'homogénéisation, in "Functional Analysis and Numerical Analysis" (eds. H. Fujita), Proc. Japan-France Seminar, (1976), 468-482.

[18]

Z. Yang, The periodic unfolding method for a class of parabolic problems with imperfect interfaces, Submitted.

show all references

References:
[1]

S. Brahim-Otsman, G. Francfort and F. Murat, Correctors for the homogenization of the wave and heat equations, J. Math. Pures Appl., 71 (1992), 197-231.

[2]

D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44 (2002), 718-760. doi: 10.1137/100817942.

[3]

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.

[4]

D. Cioranescu and P. Donato, "An Introduction to Homogenization," Oxford Univ. Press, Oxford, 1999.

[5]

H. Carslaw and J. Jaeger, "Conduction of Heat in Solids," Clarendon Press, Oxford, 1947.

[6]

P. Donato, Some corrector results for composites with imperfect interface, Rend. Mat. Appl., VII. Ser., 26 (2006), 189-209.

[7]

P. Donato, L. Faella and S. Monsurrò, Homogenization of the wave equation in composites with imperfect interface: A memory effect, J. Math. Pures Appl., 87 (2007), 119-143. doi: 10.1016/j.matpur.2006.11.004.

[8]

P. Donato, L. Faella and S. Monsurrò, Correctors for the homogenization of a class of hyperbolic equations with imperfect interfaces, SIAM J. Math. Anal., 40 (2009), 1952-1978. doi: 10.1137/080712684.

[9]

P. Donato and E. Jose, Corrector results for a parabolic problem with a memory effect, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), 421-454. doi: 10.1051/m2an/2010008.

[10]

P. Donato and S. Monsurrò, Homogenization of two heat conductors with an interfacial contact resistance, Analysis and Applications, 2 (2004), 247-273. doi: 10.1142/S0219530504000345.

[11]

P. Donato, K. Le Nguyen and R. Tardieu, The periodic unfolding method for a class of imperfect transmission problems, J. Math. Sci., 176 (2011), 891-927.

[12]

P. Donato and Z. Yang, The periodic unfolding method for the wave equation in domains with holes, Advances in Mathematical Sciences and Applications, 22 (2012), 521-551.

[13]

E. Jose, Homogenization of a parabolic problem with an imperfect interface, Rev. Rouma. Math. Pures Appl., 54 (2009), 189-222.

[14]

S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 13 (2003), 43-63.

[15]

S. Monsurrò, Erratum for the paper Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 14 (2004), 375-377.

[16]

A. Nabil, A corrector result for the wave equations in perforated domains, Gakuto Internat. Ser., Math. Sci. Appl., 9 (1997), 309-321.

[17]

L. Tartar, Quelques remarques sur l'homogénéisation, in "Functional Analysis and Numerical Analysis" (eds. H. Fujita), Proc. Japan-France Seminar, (1976), 468-482.

[18]

Z. Yang, The periodic unfolding method for a class of parabolic problems with imperfect interfaces, Submitted.

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