Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?

Alain Damlamian; Patrizia Donato

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • Volume: 8, page 555-585
  • ISSN: 1292-8119

Abstract

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In this paper we give a general presentation of the homogenization of Neumann type problems in periodically perforated domains, including the case where the shape of the reference hole varies with the size of the period (in the spirit of the construction of self-similar fractals). We shows that H 0 -convergence holds under the extra assumption that there exists a bounded sequence of extension operators for the reference holes. The general class of Jones-domains gives an example where this result applies. When this assumption fails, another approach, using the Poincaré–Wirtinger inequality is presented. A corresponding class where it applies is that of John-domains, for which the Poincaré–Wirtinger constant is controlled. The relationship between these two kinds of assumptions is also clarified.

How to cite

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Damlamian, Alain, and Donato, Patrizia. "Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 555-585. <http://eudml.org/doc/245942>.

@article{Damlamian2002,
abstract = {In this paper we give a general presentation of the homogenization of Neumann type problems in periodically perforated domains, including the case where the shape of the reference hole varies with the size of the period (in the spirit of the construction of self-similar fractals). We shows that $H^0$-convergence holds under the extra assumption that there exists a bounded sequence of extension operators for the reference holes. The general class of Jones-domains gives an example where this result applies. When this assumption fails, another approach, using the Poincaré–Wirtinger inequality is presented. A corresponding class where it applies is that of John-domains, for which the Poincaré–Wirtinger constant is controlled. The relationship between these two kinds of assumptions is also clarified.},
author = {Damlamian, Alain, Donato, Patrizia},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {periodic homogenization; perforated domains; $H^0$-convergence; Poincaré–Wirtinger inequality; Jones domains; John domains; Neumann type problems; Sobolev spaces; -convergence; Poincaré-Wirtinger inequality},
language = {eng},
pages = {555-585},
publisher = {EDP-Sciences},
title = {Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?},
url = {http://eudml.org/doc/245942},
volume = {8},
year = {2002},
}

TY - JOUR
AU - Damlamian, Alain
AU - Donato, Patrizia
TI - Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 8
SP - 555
EP - 585
AB - In this paper we give a general presentation of the homogenization of Neumann type problems in periodically perforated domains, including the case where the shape of the reference hole varies with the size of the period (in the spirit of the construction of self-similar fractals). We shows that $H^0$-convergence holds under the extra assumption that there exists a bounded sequence of extension operators for the reference holes. The general class of Jones-domains gives an example where this result applies. When this assumption fails, another approach, using the Poincaré–Wirtinger inequality is presented. A corresponding class where it applies is that of John-domains, for which the Poincaré–Wirtinger constant is controlled. The relationship between these two kinds of assumptions is also clarified.
LA - eng
KW - periodic homogenization; perforated domains; $H^0$-convergence; Poincaré–Wirtinger inequality; Jones domains; John domains; Neumann type problems; Sobolev spaces; -convergence; Poincaré-Wirtinger inequality
UR - http://eudml.org/doc/245942
ER -

References

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  1. [1] E. Acerbi, V. Chiado’ Piat, G. Dal Maso and D. Percivale, An extension theorem for connected sets, and homogenization in general periodic domains. Nonlinear Anal. TMA 18 (1992) 481-495. Zbl0779.35011
  2. [2] G. Allaire and F. Murat, Homogenization of the Neumann problem with non-isolated holes. Asymptot. Anal. 7 (1993) 81-95. Zbl0823.35014MR1225439
  3. [3] H. Attouch, Variational convergence for functions and operators. Pitman, Boston, Appl. Math. Ser. (1984). Zbl0561.49012MR773850
  4. [4] N.S. Bakhvalov and G.P. Panasenko, Homogenization: Averaging Processes in Periodic Media. Kluwer, Dordrecht (1989). Zbl0692.73012
  5. [5] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978). Zbl0404.35001MR503330
  6. [6] B. Bojarski, Remarks on Sobolev imbedding inequalities, in Complex Analysis. Springer-Verlag, Lecture Notes in Math. 1351 (1988) 257-324. Zbl0662.46037MR982072
  7. [7] M. Briane, Poincare’–Wirtinger’s inequality for the homogenization in perforated domains. Boll. Un. Mat. Ital. B 11 (1997) 53-82. Zbl0901.35004
  8. [8] M. Briane, A. Damlamian and P. Donato, H -convergence in perforated domains, in Nonlinear Partial Differential Equations Appl., Collège de France Seminar, Vol. XIII, edited by D. Cioranescu and J.-L. Lions. Longman, New York, Pitman Res. Notes in Math. Ser. 391 (1998) 62-100. Zbl0943.35005MR1773075
  9. [9] S. Buckley and P. Koskela, Sobolev–Poincaré implies John. Math. Res. Lett. 2 (1995) 577-593. Zbl0847.30012
  10. [10] D. Chenais, On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975) 189-219. Zbl0317.49005MR385666
  11. [11] D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford University Press, Oxford Lecture Ser. in Math. Appl. 17 (1999). Zbl0939.35001MR1765047
  12. [12] D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979) 590-607. Zbl0427.35073MR548785
  13. [13] D. Cioranescu and J. Saint Jean Paulin, Homogenization of reticulated structures. Springer-Verlag, Berlin, New York (1999). Zbl0929.35002MR1676922
  14. [14] C. Conca and P. Donato, Non-homogeneous Neumann problems in domains with small holes. ESAIM: M2AN 22 (1988) 561-608. Zbl0669.35028
  15. [15] A. Damlamian and P. Donato, Homogenization with small shape-varying perforations. SIAM J. Math. Anal. 22 (1991) 639-652. Zbl0762.35025MR1091674
  16. [16] F.W. Gehring and O. Martio, Quasiextremal distance domains and extension of quasiconformal mappings. J. Anal. Math. 45 (1985) 181-206. Zbl0596.30031MR833411
  17. [17] F.W. Gehring and B.G. Osgood, Uniform domains and the quasi-hyperbolic metric. J. Anal. Math. 36 (1979) 50-74. Zbl0449.30012MR581801
  18. [18] E. Hruslov, The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain. Maths. USSR Sbornik 35 (1979). Zbl0421.35019
  19. [19] P. Jones, Quasiconformal mappings and extensions of functions in Sobolev spaces. Acta Math. 1-2 (1981) 71-88. Zbl0489.30017MR631089
  20. [20] O. Martio, Definitions for uniform domains. Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980) 179-205. Zbl0469.30017MR595191
  21. [21] O. Martio, John domains, bilipschitz balls and Poincaré inequality. Rev. Roumaine Math. Pures Appl. 33 (1988) 107-112. Zbl0652.30012MR948443
  22. [22] O. Martio and J. Sarvas, Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979) 383-401. Zbl0406.30013MR565886
  23. [23] V.G. Maz’ja, Sobolev spaces. Springer-Verlag, Berlin (1985). 
  24. [24] F. Murat, H -Convergence, Séminaire d’Analyse Fonctionnelle et Numérique (1977/1978). Université d’Alger, Multigraphed. 
  25. [25] F. Murat and L. Tartar, H -Convergence, in Topics in the Mathematical Modelling of Composite Materials, edited by A. Cherkaev and R. Kohn. Birkhäuser, Boston (1997) 21-43. Zbl0920.35019MR1493039
  26. [26] E. Sanchez–Palencia, Non homogeneous Media and Vibration Theory. Springer-Verlag, Lecture Notes in Phys. 127 (1980). Zbl0432.70002
  27. [27] W. Smith and D.A. Stegenga, Hölder domains and Poincaré domains. Trans. Amer. Math. Soc. 319 (1990) 67-100. Zbl0707.46028MR978378
  28. [28] S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Sc. Norm. Sup. Pisa 22 (1968) 571-597. Zbl0174.42101MR240443
  29. [29] E.M. Stein, Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, N.J. (1970). Zbl0207.13501MR290095
  30. [30] L. Tartar, Cours Peccot au Collège de France (1977). 
  31. [31] J. Väisalä, Uniform domains. Tohoku Math. J. 40 (1988) 101-118. Zbl0627.30017MR927080
  32. [32] H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake. Manuscripta Math. 73 (1991) 117-125. Zbl0793.46015MR1128682
  33. [33] V.V. Zhikov, Connectedness and Homogenization. Examples of fractal conductivity. Sbornik Math. 187 (1196) 1109-1147. Zbl0874.35011MR1418340

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