Geodesic
The method of orthogonal projections and the Dirichlet problem in domains with a fine-grained boundary
Sbornik. Mathematics, Tome 17 (1972) no. 1, pp. 37-59 
E. Ya. Khruslov. The method of orthogonal projections and the Dirichlet problem in domains with a fine-grained boundary. Sbornik. Mathematics, Tome 17 (1972) no. 1, pp. 37-59. http://geodesic.mathdoc.fr/item/SM_1972_17_1_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The Dirichlet problem is considered for an elliptic selfadjoint operator $L$ in a domain $D^{(s)}=D\setminus\bigcup_{i=1}^s F_i^{(s)}$, where $D$ is a bounded domain in $R_n$ and the $F_i^{(s)}$ are nonintersecting closed sets (grains). It is shown that, if the grain diameters tend to zero, and the number $s$ of grains tends to infinity, the solution of the problem reduces, under certain conditions, to the solution of another boundary value problem in the simpler domain $D$. Bibliography: 8 titles.

[1] H. Weyl, “Method of orthogonal projection in potential theory”, Duke Math. J., 7 (1940), 417–444 | DOI | MR

[2] M. I. Vishik, “Metod ortogonalnykh i pryamykh razlozhenii v teorii ellipticheskikh differentsialnykh razlozhenii”, Matem. sb., 25(67) (1949), 189–234 | Zbl

[3] K. Moren, Metody gilbertova prostranstva, Mir, Moskva, 1965 | MR

[4] S. G. Mikhlin, Problema minimuma kvadratichnogo funktsionala, Gostekhizdat, Moskva–Leningrad, 1952

[5] S. L. Sobolev, Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, LGU, 1950

[6] G. Polia, G. Sege, Izoperimetricheskie neravenstva v matematicheskoi fizike, Fizmatgiz, Moskva, 1962 | MR

[7] V. G. Mazya, “Poligarmonicheskaya emkost v teorii pervoi kraevoi zadachi”, Sib. matem. zh., VI:1 (1965), 127–148

[8] Yu. N. Berezanskii, Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, Naukova dumka, Kiev, 1965 | MR