# American Institute of Mathematical Sciences

• Previous Article
Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity
• CPAA Home
• This Issue
• Next Article
Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials
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., ().
 [1] Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503 [2] Patrizia Donato, Florian Gaveau. Homogenization and correctors for the wave equation in non periodic perforated domains. Networks and Heterogeneous Media, 2008, 3 (1) : 97-124. doi: 10.3934/nhm.2008.3.97 [3] Y. Efendiev, B. Popov. On homogenization of nonlinear hyperbolic equations. Communications on Pure and Applied Analysis, 2005, 4 (2) : 295-309. doi: 10.3934/cpaa.2005.4.295 [4] Zhongyi Huang. Tailored finite point method for the interface problem. Networks and Heterogeneous Media, 2009, 4 (1) : 91-106. doi: 10.3934/nhm.2009.4.91 [5] 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 [6] Luisa Faella, Carmen Perugia. Optimal control for a hyperbolic problem in composites with imperfect interface: A memory effect. Evolution Equations and Control Theory, 2017, 6 (2) : 187-217. doi: 10.3934/eect.2017011 [7] Ben Schweizer, Marco Veneroni. The needle problem approach to non-periodic homogenization. Networks and Heterogeneous Media, 2011, 6 (4) : 755-781. doi: 10.3934/nhm.2011.6.755 [8] Rémi Goudey. A periodic homogenization problem with defects rare at infinity. Networks and Heterogeneous Media, 2022  doi: 10.3934/nhm.2022014 [9] Hongsong Feng, Shan Zhao. A multigrid based finite difference method for solving parabolic interface problem. Electronic Research Archive, 2021, 29 (5) : 3141-3170. doi: 10.3934/era.2021031 [10] Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems and Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025 [11] Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks and Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014 [12] Madalina Petcu, Roger Temam. An interface problem: The two-layer shallow water equations. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5327-5345. doi: 10.3934/dcds.2013.33.5327 [13] Sara Monsurrò, Carmen Perugia. Homogenization and exact controllability for problems with imperfect interface. Networks and Heterogeneous Media, 2019, 14 (2) : 411-444. doi: 10.3934/nhm.2019017 [14] Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190 [15] Antonin Chambolle, Gilles Thouroude. Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem. Networks and Heterogeneous Media, 2009, 4 (1) : 127-152. doi: 10.3934/nhm.2009.4.127 [16] Eugenia Pérez. On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 859-883. doi: 10.3934/dcdsb.2007.7.859 [17] Fanghua Lin, Xiaodong Yan. A type of homogenization problem. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 1-30. doi: 10.3934/dcds.2003.9.1 [18] Laura Sigalotti. Homogenization of pinning conditions on periodic networks. Networks and Heterogeneous Media, 2012, 7 (3) : 543-582. doi: 10.3934/nhm.2012.7.543 [19] Champike Attanayake, So-Hsiang Chou. An immersed interface method for Pennes bioheat transfer equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 323-337. doi: 10.3934/dcdsb.2015.20.323 [20] Jian Hao, Zhilin Li, Sharon R. Lubkin. An augmented immersed interface method for moving structures with mass. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1175-1184. doi: 10.3934/dcdsb.2012.17.1175

2020 Impact Factor: 1.916