Geodesic
The first boundary value problem in domains with a complicated boundary for higher order equations
Sbornik. Mathematics, Tome 32 (1977) no. 4, pp. 535-549 
E. Ya. Khruslov. The first boundary value problem in domains with a complicated boundary for higher order equations. Sbornik. Mathematics, Tome 32 (1977) no. 4, pp. 535-549. http://geodesic.mathdoc.fr/item/SM_1977_32_4_a8/
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     title = {The first boundary value problem in~domains with a~complicated boundary for higher order equations},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The first boundary value problem is considered for an elliptic selfadjoint operator $L$ of order $2m$ in a domain $\Omega^{(s)}$ of complicated structure of the form $\Omega^{(s)}=\Omega\setminus F^{(s)}$, where $\Omega$ is a comparatively simple domain in $\mathbf R_n$ ($n\geqslant2$) and $F^{(s)}$ is a closed, connected, highly fragmented set in $\Omega$. The asymptotic behavior of the resolvent $R^{(s)}$ of this problem is studied for $s\to\infty$ when the set $F^{(s)}$ becomes ever more fragmented and is disposed volumewise in $\Omega$ so that the distance from $F^{(s)}$ to any point $x\in\Omega$ tends to zero. It is shown that $R^{(s)}$ converges in norm to the resolvent $R^c$ of an operator $L+c(x)$, which is considered in the simple domain $\Omega$ under null conditions in $\partial\Omega$. A massivity characteristic of the sets $F^{(s)}$ (of capacity type) is introduced, which is used to formulate necessary and sufficient conditions for convergence, and the function $c(x)$ is described. Bibliography: 7 titles.

[1] E. Ya. Khruslov, “Metod ortogonalnykh proektsii i zadacha Dirikhle v oblastyakh s melkozernistoi granitsei”, Matem. sb., 88 (130) (1972), 38–60 | Zbl

[2] V. G. Mazya, “Poligarmonicheskaya emkost v teorii pervoi kraevoi zadachi”, Sib. matem. zh., 6:1 (1965), 127–148

[3] E. Ya. Khruslov, “Ob usloviyakh skhodimosti posledovatelnosti reshenii pervoi kraevoi zadachi”, Teoriya funktsii, funkts. analiz i ikh prilozheniya, 12, izd-vo KhGU, Kharkov, 1971, 103–110

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

[5] E. Ya. Khruslov, “Kraevye zadachi v oblastyakh s melkozernistoi granitsei”, Tezisy dokladov Vsesoyuznoi konferentsii po uravneniyam s chastnymi proizvodnymi, posvyaschennoi 75-letiyu so dnya rozhdeniya akademika I. G. Petrovskogo, 1976 | Zbl

[6] V. A. Marchenko, E. Ya. Khruslov, Kraevye zadachi v oblastyakh s melkozernistoi granitsei, izd-vo «Naukova dumka», Kiev, 1974 | MR

[7] V. A. Kondratev, “O razreshimosti pervoi kraevoi zadachi dlya silno ellipticheskikh uravnenii”, Trudy Mosk. matem. ob-va, XVI:1 (1967), 293–318 | MR