Geodesic
Connectedness and homogenization. Examples of fractal conductivity
Sbornik. Mathematics, Tome 187 (1996) no. 8, pp. 1109-1147 
V. V. Zhikov. Connectedness and homogenization. Examples of fractal conductivity. Sbornik. Mathematics, Tome 187 (1996) no. 8, pp. 1109-1147. http://geodesic.mathdoc.fr/item/SM_1996_187_8_a0/
@article{SM_1996_187_8_a0,
     author = {V. V. Zhikov},
     title = {Connectedness and homogenization. {Examples} of fractal conductivity},
     journal = {Sbornik. Mathematics},
     pages = {1109--1147},
     year = {1996},
     volume = {187},
     number = {8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1996_187_8_a0/}
}
TY  - JOUR
AU  - V. V. Zhikov
TI  - Connectedness and homogenization. Examples of fractal conductivity
JO  - Sbornik. Mathematics
PY  - 1996
SP  - 1109
EP  - 1147
VL  - 187
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/SM_1996_187_8_a0/
LA  - en
ID  - SM_1996_187_8_a0
ER  - 
%0 Journal Article
%A V. V. Zhikov
%T Connectedness and homogenization. Examples of fractal conductivity
%J Sbornik. Mathematics
%D 1996
%P 1109-1147
%V 187
%N 8
%U http://geodesic.mathdoc.fr/item/SM_1996_187_8_a0/
%G en
%F SM_1996_187_8_a0

Voir la notice de l'article provenant de la source Math-Net.Ru

A detailed study of the concept of $p$-connectedness is carried out; in particular, a criterion for the $p$-connectedness of two disjoint domains with Lipschitz boundaries and with fractal contact is formulated. New examples of open periodic sets with positive effective conductivity are constructed on the basis of this analysis. A new class of objects, elliptic operators in a Euclidean space with measure, is introduced; the corresponding concept of $p$-connectedness is introduced and a generalized theory of homogenization is developed.

[1] Zhikov V. V., Kozlov S. M., Oleinik O. A., Usrednenie differentsialnykh operatorov, Nauka, M., 1993 | MR | Zbl

[2] Oleinik O. A., Iosifyan G. A., Shamaev A. S., Matematicheskie zadachi teorii silno neodnorodnykh sred, MGU, M., 1990

[3] Zhikov V. V., “Ob usrednenii v perforirovannykh sluchainykh oblastyakh obschego vida”, Matem. zametki, 53:1 (1993), 41–58 | MR | Zbl

[4] Zhikov V. V., “On the Homogenization of Nonlinear Variational Problems in perforated Domains”, Russian J. Math. Phys., 2:3 (1994), 393–408 | MR | Zbl

[5] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1976 | MR | Zbl

[6] Stein I., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR

[7] Mazya V. G., Poborchii S. V., “Prodolzhenie iz klassov S. L. Soboleva vo vneshnosti oblasti s vershinoi pika na granitse, I”, Czechoslovak Math. J., 36 (111) (1986), 634–661 | MR | Zbl

[8] Evans L. C., Gariepy R. F., Measure theory and fine properties of functions, CRC PRESS, London, 1992 | MR | Zbl

[9] Barlow M. T., Bass R. F., “On the resistans of the Sierpinski carpet”, Proc. Roy. Soc. London Ser. A, 431:1882 (1990), 345–360 | DOI | MR | Zbl

[10] Kozlov S. M., “Harmonization and homogenization on Fractals”, Commun. Math. Phys., 153 (1993), 339–357 | DOI | MR | Zbl

[11] Gelbaum B., Olmsted Dzh., Kontrprimery v analize, Mir, M., 1967 | Zbl