Σ -convergence of nonlinear monotone operators in perforated domains with holes of small size

Jean Louis Woukeng

Applications of Mathematics (2009)

  • Volume: 54, Issue: 6, page 465-489
  • ISSN: 0862-7940

Abstract

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This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in ℝ N with isolated holes of size ε 2 ( ε > 0 a small parameter). The order of the size of the holes is twice that of the oscillations of the coefficients of the operator, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The usual periodic perforation of the domain and the classical periodicity hypothesis on the coefficients of the operator are here replaced by an abstract assumption covering a great variety of behaviors such as the periodicity, the almost periodicity and many more besides. We illustrate this abstract setting by working out a few concrete homogenization problems. Our main tool is the recent theory of homogenization structures.

How to cite

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Woukeng, Jean Louis. "$\Sigma $-convergence of nonlinear monotone operators in perforated domains with holes of small size." Applications of Mathematics 54.6 (2009): 465-489. <http://eudml.org/doc/37833>.

@article{Woukeng2009,
abstract = {This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in $\mathbb \{R\}^N$ with isolated holes of size $\varepsilon ^2$ ($\varepsilon >0$ a small parameter). The order of the size of the holes is twice that of the oscillations of the coefficients of the operator, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The usual periodic perforation of the domain and the classical periodicity hypothesis on the coefficients of the operator are here replaced by an abstract assumption covering a great variety of behaviors such as the periodicity, the almost periodicity and many more besides. We illustrate this abstract setting by working out a few concrete homogenization problems. Our main tool is the recent theory of homogenization structures.},
author = {Woukeng, Jean Louis},
journal = {Applications of Mathematics},
keywords = {perforated domains; homogenization; reiterated; perforated domains; homogenization; reiterated},
language = {eng},
number = {6},
pages = {465-489},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$\Sigma $-convergence of nonlinear monotone operators in perforated domains with holes of small size},
url = {http://eudml.org/doc/37833},
volume = {54},
year = {2009},
}

TY - JOUR
AU - Woukeng, Jean Louis
TI - $\Sigma $-convergence of nonlinear monotone operators in perforated domains with holes of small size
JO - Applications of Mathematics
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 6
SP - 465
EP - 489
AB - This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in $\mathbb {R}^N$ with isolated holes of size $\varepsilon ^2$ ($\varepsilon >0$ a small parameter). The order of the size of the holes is twice that of the oscillations of the coefficients of the operator, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The usual periodic perforation of the domain and the classical periodicity hypothesis on the coefficients of the operator are here replaced by an abstract assumption covering a great variety of behaviors such as the periodicity, the almost periodicity and many more besides. We illustrate this abstract setting by working out a few concrete homogenization problems. Our main tool is the recent theory of homogenization structures.
LA - eng
KW - perforated domains; homogenization; reiterated; perforated domains; homogenization; reiterated
UR - http://eudml.org/doc/37833
ER -

References

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  1. Allaire, G., Murat, F., 10.3233/ASY-1993-7201, Asymptotic Anal. 7 (1993), 81-95. (1993) Zbl0823.35014MR1225439DOI10.3233/ASY-1993-7201
  2. Amaziane, B., Goncharenko, M., Pankratov, L., 10.1007/s00033-006-5070-2, Z. Angew. Math. Phys. 58 (2007), 592-611. (2007) Zbl1124.35006MR2341687DOI10.1007/s00033-006-5070-2
  3. Besicovitch, A. S., Almost Periodic Functions, Dover Publications New York (1955). (1955) Zbl0065.07102MR0068029
  4. Bourbaki, N., Éléments de mathématique. Intégration, Chap. 1-4, Hermann Paris (1966), French. (1966) 
  5. Bourbaki, N., Éléments de matématique. Topologie générale, Chap. 1-4, Hermann Paris (1971), French. (1971) MR0358652
  6. Cardone, G., Donato, P., Gaudiello, A., 10.1016/S0362-546X(97)00486-0, Nonlin. Anal., Theory Methods Appl. 32 (1998), 335-361. (1998) Zbl0988.35060MR1610578DOI10.1016/S0362-546X(97)00486-0
  7. V. Chiadò Piat, Defranceschi, A., 10.1080/00036819008839922, Appl. Anal. 36 (1990), 65-87. (1990) MR1040879DOI10.1080/00036819008839922
  8. Cioranescu, D., J. Saint Jean Paulin, 10.1016/0022-247X(79)90211-7, J. Math. Anal. Appl. 71 (1979), 590-607. (1979) Zbl0427.35073MR0548785DOI10.1016/0022-247X(79)90211-7
  9. Cioranescu, D., Donato, P., An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications, 17 Oxford University Press Oxford (1999). (1999) Zbl0939.35001MR1765047
  10. Cioranescu, D., Murat, F., Un terme étrange venu d'ailleurs. Nonlin. partial differential equations and their applications, Coll. de France Semin., Vol. II, Vol. III, Res. Notes Math. (1982), 98-138, 154-178 French. (1982) 
  11. Conca, C., Donato, P., 10.1051/m2an/1988220405611, RAIRO, Modél. Math. Anal. Numér. 22 (1988), 561-607. (1988) Zbl0669.35028MR0974289DOI10.1051/m2an/1988220405611
  12. Donato, P., Sgambati, L., Homogenization for some nonlinear problems in perforated domains, Rev. Mat. Apl. 15 (1994), 17-38. (1994) Zbl0806.35011MR1289135
  13. Fournier, J. F. F., Stewart, J., 10.1090/S0273-0979-1985-15350-9, Bull. Am. Math. Soc. 13 (1985), 1-21. (1985) MR0788385DOI10.1090/S0273-0979-1985-15350-9
  14. Guichardet, A., Analyse harmonique commutative, Dunod Paris (1968), French. (1968) Zbl0159.18601MR0240554
  15. Huang, C., 10.3233/ASY-1997-153-401, Asymptotic Anal. 15 (1997), 203-227. (1997) Zbl0898.35009MR1487711DOI10.3233/ASY-1997-153-401
  16. Larsen, R., Banach Algebras. An Introduction, Marcel Dekker New York (1973). (1973) Zbl0264.46042MR0487369
  17. Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod/Gauthier-Vilars Paris (1969), French. (1969) Zbl0189.40603MR0259693
  18. Lukkassen, D., Nguetseng, G., Nnang, H., Wall, P., 10.1155/2009/102486, J. Funct. Spaces Appl. 7 (2009), 121-152. (2009) MR2541229DOI10.1155/2009/102486
  19. Nguetseng, G., 10.4171/ZAA/1133, Z. Anal. Anwend. 22 (2003), 73-107. (2003) Zbl1045.46031MR1962077DOI10.4171/ZAA/1133
  20. Nguetseng, G., 10.4171/ZAA/1208, Z. Anal. Anwend. 23 (2004), 483-508. (2004) Zbl1116.46064MR2094594DOI10.4171/ZAA/1208
  21. Nguetseng, G., Mean value on locally compact abelian groups, Acta Sci. Math. 69 (2003), 203-221. (2003) Zbl1047.43003MR1991666
  22. Nguetseng, G., Almost periodic homogenization: Asymptotic analysis of a second order elliptic equation, Preprint. 
  23. Nguetseng, G., 10.1016/j.jmaa.2003.08.045, J. Math. Anal. Appl. 289 (2004), 608-628. (2004) Zbl1037.35020MR2026928DOI10.1016/j.jmaa.2003.08.045
  24. Nguetseng, G., Nnang, H., Homogenization of nonlinear monotone operators beyond the periodic setting, Electron. J. Differ. Equ. 2003 (2003). (2003) Zbl1032.35031MR1971022
  25. Nguetseng, G., Woukeng, J. L., Deterministic homogenization of parabolic monotone operators with time dependent coefficients, Electron. J. Differ. Equ. 2004 (2004). (2004) Zbl1058.35025MR2075421
  26. Nguetseng, G., Woukeng, J. L., 10.1016/j.na.2005.12.035, Nonlinear Anal., Theory Methods Appl. 66 (2007), 968-1004. (2007) Zbl1116.35011MR2288445DOI10.1016/j.na.2005.12.035
  27. Oleinik, O. A., Shaposhnikova, T. A., On the biharmonic equation in a domain perforated along manifolds of small dimension, Differ. Uravn. 32 (1996), 335-362. (1996) MR1444937

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