The periodic unfolding method for a class of parabolic problems with imperfect interfaces
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 5, p. 1279-1302

In this paper, we use the adapted periodic unfolding method to study the homogenization and corrector problems for the parabolic problem in a two-component composite with ε-periodic connected inclusions. The condition imposed on the interface is that the jump of the solution is proportional to the conormal derivative via a function of order εγ with γ ≤ -1. We give the homogenization results which include those obtained by Jose in [Rev. Roum. Math. Pures Appl. 54 (2009) 189-222]. We also get the corrector results.

DOI : https://doi.org/10.1051/m2an/2013139
Classification:  35B27,  35K20,  82B24
Keywords: periodic unfolding method, heat equation, interface problems, homogenization, correctors
@article{M2AN_2014__48_5_1279_0,
     author = {Yang, Zhanying},
     title = {The periodic unfolding method for a class of parabolic problems with imperfect interfaces},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {5},
     year = {2014},
     pages = {1279-1302},
     doi = {10.1051/m2an/2013139},
     mrnumber = {3264354},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_5_1279_0}
}
Yang, Zhanying. The periodic unfolding method for a class of parabolic problems with imperfect interfaces. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 5, pp. 1279-1302. doi : 10.1051/m2an/2013139. http://www.numdam.org/item/M2AN_2014__48_5_1279_0/

[1] S. Brahim-Otsman, G.A. Francfort and F. Murat, Correctors for the homogenization of the wave and heat equations. J. Math. Pures Appl. 71 (1992) 197-231. | MR 1172450 | Zbl 0837.35016

[2] D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford University Press (1999). | MR 1765047 | Zbl 0939.35001

[3] D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes. SIAM J. Math. Anal. 44 (2012) 718-760. | MR 2914248 | Zbl 1250.49017

[4] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization. C.R. Acad. Sci., Paris, Sér. I, Math. 335 (2002) 99-104. | MR 1921004 | Zbl 1001.49016

[5] D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40 (2008) 1585-1620. | MR 2466168 | Zbl 1167.49013

[6] D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains. Port. Math. (N.S.) 63 (2006) 467-496. | MR 2287278 | Zbl 1119.49014

[7] H.S. Carslaw and J.C. Jaeger, Conduction of heat in solids. Clarendon Press, Oxford (1947). | MR 22294 | Zbl 0029.37801

[8] P. Donato, Some corrector results for composites with imperfect interface. Rend. Mat. Appl., VII. Ser. 26 (2006) 189-209. | MR 2275293 | Zbl 1129.35008

[9] 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. | MR 2296803 | Zbl 1112.35017

[10] 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. | MR 2471907 | Zbl 1197.35029

[11] P. Donato and E.C. Jose, Corrector results for a parabolic problem with a memory effect. ESAIM: M2AN 44 (2010) 421-454. | Numdam | MR 2666650 | Zbl 1195.35038

[12] P. Donato and S. Monsurrò, Homogenization of two heat conductors with an interfacial contact resistance. Anal. Appl. 2 (2004) 247-273. | MR 2070449 | Zbl 1083.35014

[13] P. Donato, K.H. Le Nguyen and R. Tardieu, The periodic unfolding method for a class of imperfect transmission problems. J. Math. Sci. 176 (2011) 891-927. | MR 2838982 | Zbl 1290.35018

[14] P. Donato and A. Nabil, Homogenization and correctors for the heat equation in perforated domains. Ricerche Mat. 50 (2001) 115-144. | MR 1941824 | Zbl 1102.35305

[15] P. Donato and Z. Yang, The periodic unfolding method for the wave equations in domains with holes. Adv. Math. Sci. Appl. 22 (2012) 521-551. | MR 3100008 | Zbl 1295.35043

[16] L. Faella and S. Monsurrò, Memory Effects Arising in the Homogenization of Composites with Inclusions, Topics on Mathematics for Smart Systems. World Sci. Publ., Hackensack, USA (2007) 107-121. | MR 2313177 | Zbl 1114.74048

[17] F. Gaveau, Homogénéisation et correcteurs pour quelques problèmes hyperboliques, Ph.D. Thesis, University of Paris VI, France (2009).

[18] E.C. Jose, Homogenization of a parabolic problem with an imperfect interface. Rev. Roum. Math. Pures Appl. 54 (2009) 189-222. | MR 2562269 | Zbl 1199.35015

[19] S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier. Adv. Math. Sci. Appl. 13 (2003) 43-63. | MR 2002395 | Zbl 1052.35022

[20] S. Monsurrò, Erratum for the paper Homogenization of a two-component composite with interfacial thermal barrier. Adv. Math. Sci. Appl. 14 (2004) 375-377. | MR 2083635 | Zbl 1069.35500

[21] L. Tartar, Quelques remarques sur l'homogénéisation, in Functional Analysis and Numerical Analysis, Proc. Japan-France Seminar, 1976. Jpn. Soc. Promot. Sci. (1978) 468-482.