Geodesic
The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain
Sbornik. Mathematics, Tome 35 (1979) no. 2, pp. 266-282 
E. Ya. Khruslov. The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain. Sbornik. Mathematics, Tome 35 (1979) no. 2, pp. 266-282. http://geodesic.mathdoc.fr/item/SM_1979_35_2_a7/
@article{SM_1979_35_2_a7,
     author = {E. Ya. Khruslov},
     title = {The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain},
     journal = {Sbornik. Mathematics},
     pages = {266--282},
     year = {1979},
     volume = {35},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1979_35_2_a7/}
}
TY  - JOUR
AU  - E. Ya. Khruslov
TI  - The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain
JO  - Sbornik. Mathematics
PY  - 1979
SP  - 266
EP  - 282
VL  - 35
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1979_35_2_a7/
LA  - en
ID  - SM_1979_35_2_a7
ER  - 
%0 Journal Article
%A E. Ya. Khruslov
%T The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain
%J Sbornik. Mathematics
%D 1979
%P 266-282
%V 35
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1979_35_2_a7/
%G en
%F SM_1979_35_2_a7

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

The second boundary value problem is considered for the equation $\Delta u-cu=f$ in a domain $G^{(s)}$ of complicated structure of the form $G^{(s)}=\mathbf R_n\setminus F^{(s)}$, where $F^{(s)}$ is a closed finely partitioned set lying in a domain $\Omega\subset\mathbf R_n$ ($n\geqslant 2$) for all $s=1,2,\dots$. The asymptotic behavior of a solution $u^{(s)}(x)$ of this problem is studied as $s\to\infty$, when $F^{(s)}$ becomes more and more finely divided and is situated in $\Omega$ so that the distance from $F^{(s)}$ to any point $x\in\Omega$ tends to zero. It is proved that under specific conditions $u^{(s)}(x)$ converges in $\mathbf R_n\setminus\overline\Omega$ to a function $u(x)$ that is a solution of a conjugation problem. Sufficient conditions for convergence are formulated. Bibliography: 9 titles.

[1] S. L. Sobolev, Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, izd-vo LGU, Leningrad, 1950

[2] E. Ya. Khruslov, “O kraevoi zadache Neimana v oblasti so slozhnoi granitsei”, Matem. sb., 83(125) (1970), 556–574 | MR | Zbl

[3] E. Ya. Khruslov, “Pervaya kraevaya zadacha v oblastyakh so slozhnoi granitsei dlya uravnenii vysshikh poryadkov”, Matem. sb., 103(145) (1977), 614–629 | Zbl

[4] V. L. Berdichevskii, “Prostranstvennoe osrednenie periodicheskikh struktur”, DAN SSSR, 222:3 (1975), 565–567 | MR | Zbl

[5] E. de Giordi, S. Spagnolo, “Sulla convergergenza degli integrati dell"enegria per operatori ellitichi dell secondo ordine”, Boll. Unione. Mat. Ital. ser. 4, 8, 1973, no. 3

[6] H. S. Bakhvalov, “Osrednennye kharakteristiki tel s periodicheskoi strukturoi”, DAN SSSR, 218:5 (1974), 1046–1048

[7] O. A. Oleinik, “O rasprostranenii tepla v mnogomernykh dispersnykh sredakh”, Zadachi mekhaniki i matematicheskoi fiziki, izd-vo “Nauka”, Moskva, 1976, 224–236 | MR

[8] Zh–L. Lions, “Operatory s silno ostsilliruyuschimi koeffitsientami”, Uspekhi matem. nauk, XXXII:1(193) (1977), 239–246

[9] A. A. Samarskii, “O vliyanii zakrepleniya na sobstvennye chastoty zamknutykh ob'emov”, DAN SSSR, 63:6 (1948), 631–634 | MR | Zbl