| Title:
|
$\Sigma $-convergence of nonlinear monotone operators in perforated domains with holes of small size (English) |
| Author:
|
Woukeng, Jean Louis |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
54 |
| Issue:
|
6 |
| Year:
|
2009 |
| Pages:
|
465-489 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in $\Bbb R^N$ 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. (English) |
| Keyword:
|
perforated domains |
| Keyword:
|
homogenization |
| Keyword:
|
reiterated |
| MSC:
|
35B27 |
| MSC:
|
35B40 |
| MSC:
|
35J60 |
| MSC:
|
46J10 |
| MSC:
|
46N20 |
| idZBL:
|
Zbl 1212.35023 |
| idMR:
|
MR2563121 |
| DOI:
|
10.1007/s10492-009-0030-8 |
| . |
| Date available:
|
2010-07-20T13:24:37Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140379 |
| . |
| Reference:
|
[1] Allaire, G., Murat, F.: Homogenization of the Neumann problem with nonisolated holes.Asymptotic Anal. 7 (1993), 81-95. Zbl 0823.35014, MR 1225439, 10.3233/ASY-1993-7201 |
| Reference:
|
[2] Amaziane, B., Goncharenko, M., Pankratov, L.: Homogenization of a convection-diffusion equation in perforated domains with a weak adsorption.Z. Angew. Math. Phys. 58 (2007), 592-611. Zbl 1124.35006, MR 2341687, 10.1007/s00033-006-5070-2 |
| Reference:
|
[3] Besicovitch, A. S.: Almost Periodic Functions.Dover Publications New York (1955). Zbl 0065.07102, MR 0068029 |
| Reference:
|
[4] Bourbaki, N.: Éléments de mathématique. Intégration, Chap. 1-4.Hermann Paris (1966), French. |
| Reference:
|
[5] Bourbaki, N.: Éléments de matématique. Topologie générale, Chap. 1-4.Hermann Paris (1971), French. MR 0358652 |
| Reference:
|
[6] Cardone, G., Donato, P., Gaudiello, A.: A compactness result for elliptic equations with subquadratic growth in perforated domains.Nonlin. Anal., Theory Methods Appl. 32 (1998), 335-361. Zbl 0988.35060, MR 1610578, 10.1016/S0362-546X(97)00486-0 |
| Reference:
|
[7] V. Chiadò Piat, Defranceschi, A.: Asymptotic behaviour of quasi-linear problems with Neumann boundary conditions on perforated domains.Appl. Anal. 36 (1990), 65-87. MR 1040879, 10.1080/00036819008839922 |
| Reference:
|
[8] Cioranescu, D., J. Saint Jean Paulin: Homogenization in open sets with holes.J. Math. Anal. Appl. 71 (1979), 590-607. Zbl 0427.35073, MR 0548785, 10.1016/0022-247X(79)90211-7 |
| Reference:
|
[9] Cioranescu, D., Donato, P.: An introduction to homogenization.Oxford Lecture Series in Mathematics and its Applications, 17 Oxford University Press Oxford (1999). Zbl 0939.35001, MR 1765047 |
| Reference:
|
[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. |
| Reference:
|
[11] Conca, C., Donato, P.: Non-homogeneous Neumann problems in domains with small holes.RAIRO, Modél. Math. Anal. Numér. 22 (1988), 561-607. Zbl 0669.35028, MR 0974289, 10.1051/m2an/1988220405611 |
| Reference:
|
[12] Donato, P., Sgambati, L.: Homogenization for some nonlinear problems in perforated domains.Rev. Mat. Apl. 15 (1994), 17-38. Zbl 0806.35011, MR 1289135 |
| Reference:
|
[13] Fournier, J. F. F., Stewart, J.: Amalgams of $L^p$ and $l^q$.Bull. Am. Math. Soc. 13 (1985), 1-21. MR 0788385, 10.1090/S0273-0979-1985-15350-9 |
| Reference:
|
[14] Guichardet, A.: Analyse harmonique commutative.Dunod Paris (1968), French. Zbl 0159.18601, MR 0240554 |
| Reference:
|
[15] Huang, C.: Homogenization of biharmonic equations in domains perforated with tiny holes.Asymptotic Anal. 15 (1997), 203-227. Zbl 0898.35009, MR 1487711, 10.3233/ASY-1997-153-401 |
| Reference:
|
[16] Larsen, R.: Banach Algebras. An Introduction.Marcel Dekker New York (1973). Zbl 0264.46042, MR 0487369 |
| Reference:
|
[17] Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires.Dunod/Gauthier-Vilars Paris (1969), French. Zbl 0189.40603, MR 0259693 |
| Reference:
|
[18] Lukkassen, D., Nguetseng, G., Nnang, H., Wall, P.: Reiterated homogenization of nonlinear elliptic operators in a general deterministic setting.J. Funct. Spaces Appl. 7 (2009), 121-152. MR 2541229, 10.1155/2009/102486 |
| Reference:
|
[19] Nguetseng, G.: Homogenization structures and applications I.Z. Anal. Anwend. 22 (2003), 73-107. Zbl 1045.46031, MR 1962077, 10.4171/ZAA/1133 |
| Reference:
|
[20] Nguetseng, G.: Homogenization structures and applications II.Z. Anal. Anwend. 23 (2004), 483-508. Zbl 1116.46064, MR 2094594, 10.4171/ZAA/1208 |
| Reference:
|
[21] Nguetseng, G.: Mean value on locally compact abelian groups.Acta Sci. Math. 69 (2003), 203-221. Zbl 1047.43003, MR 1991666 |
| Reference:
|
[22] Nguetseng, G.: Almost periodic homogenization: Asymptotic analysis of a second order elliptic equation.Preprint. |
| Reference:
|
[23] Nguetseng, G.: Homogenization in perforated domains beyond the periodic setting.J. Math. Anal. Appl. 289 (2004), 608-628. Zbl 1037.35020, MR 2026928, 10.1016/j.jmaa.2003.08.045 |
| Reference:
|
[24] Nguetseng, G., Nnang, H.: Homogenization of nonlinear monotone operators beyond the periodic setting.Electron. J. Differ. Equ. 2003 (2003). Zbl 1032.35031, MR 1971022 |
| Reference:
|
[25] Nguetseng, G., Woukeng, J. L.: Deterministic homogenization of parabolic monotone operators with time dependent coefficients.Electron. J. Differ. Equ. 2004 (2004). Zbl 1058.35025, MR 2075421 |
| Reference:
|
[26] Nguetseng, G., Woukeng, J. L.: $\Sigma $-convergence of nonlinear parabolic operators.Nonlinear Anal., Theory Methods Appl. 66 (2007), 968-1004. Zbl 1116.35011, MR 2288445, 10.1016/j.na.2005.12.035 |
| Reference:
|
[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. MR 1444937 |
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