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

Alain Damlamian; Patrizia Donato

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • 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 H0-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 (2010): 555-585. <http://eudml.org/doc/90660>.

@article{Damlamian2010,
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 H0-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; H0-convergence; Poincaré–Wirtinger inequality; Jones domains; John domains.; Neumann type problems; Sobolev spaces; -convergence; Poincaré-Wirtinger inequality},
language = {eng},
month = {3},
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/90660},
volume = {8},
year = {2010},
}

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
DA - 2010/3//
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 H0-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; H0-convergence; Poincaré–Wirtinger inequality; Jones domains; John domains.; Neumann type problems; Sobolev spaces; -convergence; Poincaré-Wirtinger inequality
UR - http://eudml.org/doc/90660
ER -

References

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  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. TMA18 (1992) 481-495.  
  2. G. Allaire and F. Murat, Homogenization of the Neumann problem with non-isolated holes. Asymptot. Anal.7 (1993) 81-95.  
  3. H. Attouch, Variational convergence for functions and operators. Pitman, Boston, Appl. Math. Ser. (1984).  
  4. N.S. Bakhvalov and G.P. Panasenko, Homogenization: Averaging Processes in Periodic Media. Kluwer, Dordrecht (1989).  
  5. A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978).  
  6. B. Bojarski, Remarks on Sobolev imbedding inequalities, in Complex Analysis. Springer-Verlag, Lecture Notes in Math. 1351 (1988) 257-324.  
  7. M. Briane, Poincare'-Wirtinger's inequality for the homogenization in perforated domains. Boll. Un. Mat. Ital. B11 (1997) 53-82.  
  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.  
  9. S. Buckley and P. Koskela, Sobolev-Poincaré implies John. Math. Res. Lett.2 (1995) 577-593.  
  10. D. Chenais, On the existence of a solution in a domain identification problem. J. Math. Anal. Appl.52 (1975) 189-219.  
  11. D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford University Press, Oxford Lecture Ser. in Math. Appl. 17 (1999).  
  12. D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl.71 (1979) 590-607.  
  13. D. Cioranescu and J. Saint Jean Paulin, Homogenization of reticulated structures. Springer-Verlag, Berlin, New York (1999).  
  14. C. Conca and P. Donato, Non-homogeneous Neumann problems in domains with small holes. ESAIM: M2AN22 (1988) 561-608.  
  15. A. Damlamian and P. Donato, Homogenization with small shape-varying perforations. SIAM J. Math. Anal.22 (1991) 639-652.  
  16. F.W. Gehring and O. Martio, Quasiextremal distance domains and extension of quasiconformal mappings. J. Anal. Math.45 (1985) 181-206.  
  17. F.W. Gehring and B.G. Osgood, Uniform domains and the quasi-hyperbolic metric. J. Anal. Math.36 (1979) 50-74.  
  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).  
  19. P. Jones, Quasiconformal mappings and extensions of functions in Sobolev spaces. Acta Math.1-2 (1981) 71-88.  
  20. O. Martio, Definitions for uniform domains. Ann. Acad. Sci. Fenn. Ser. A I Math.5 (1980) 179-205.  
  21. O. Martio, John domains, bilipschitz balls and Poincaré inequality. Rev. Roumaine Math. Pures Appl.33 (1988) 107-112.  
  22. O. Martio and J. Sarvas, Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A I Math.4 (1979) 383-401.  
  23. V.G. Maz'ja, Sobolev spaces. Springer-Verlag, Berlin (1985).  
  24. F. Murat, H-Convergence, Séminaire d'Analyse Fonctionnelle et Numérique (1977/1978). Université d'Alger, Multigraphed.  
  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.  
  26. E. Sanchez-Palencia, Non homogeneous Media and Vibration Theory. Springer-Verlag, Lecture Notes in Phys. 127 (1980).  
  27. W. Smith and D.A. Stegenga, Hölder domains and Poincaré domains. Trans. Amer. Math. Soc.319 (1990) 67-100.  
  28. S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Sc. Norm. Sup. Pisa22 (1968) 571-597.  
  29. E.M. Stein, Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, N.J. (1970).  
  30. L. Tartar, Cours Peccot au Collège de France (1977).  
  31. J. Väisalä, Uniform domains. Tohoku Math. J.40 (1988) 101-118.  
  32. H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake. Manuscripta Math.73 (1991) 117-125.  
  33. V.V. Zhikov, Connectedness and Homogenization. Examples of fractal conductivity. Sbornik Math.187 (1196) 1109-1147.  

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