Geodesic
A boundary value problem for the elliptic equation of second order in a domain with a narrow slit. 1. The two-dimensional case
Sbornik. Mathematics, Tome 28 (1976) no. 4, pp. 459-480 Cet article a éte moissonné depuis la source Math-Net.Ru

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An elliptic equation is considered in a domain $D_\varepsilon$ that is obtained from a two-dimensional domain $D$ by removing a neighborhood of a segment. It is assumed that this neighborhood contracts to the segment as $\varepsilon\to0$. An asymptotic expansion of the solution of the first boundary value problem in $D_\varepsilon$ is constructed and justified. Bibliography: 5 titles. 
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A. M. Il'in. A boundary value problem for the elliptic equation of second order in a domain with a narrow slit. 1. The two-dimensional case. Sbornik. Mathematics, Tome 28 (1976) no. 4, pp. 459-480. http://geodesic.mathdoc.fr/item/SM_1976_28_4_a1/

[1] M. Van-Daik, Metody vozmuschenii v mekhanike zhidkosti, izd-vo «Mir», Moskva, 1967

[2] V. A. Kondratev, “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Trudy Mosk. matem. ob-va, XVI (1967), 209–292

[3] A. M. Ilin, Yu. P. Gorkov, E. F. Lelikova, “Asimptotika resheniya ellipticheskogo uravneniya s malym parametrom pri starshikh proizvodnykh v okrestnosti osoboi kharakteristiki predelnogo uravneniya”, Trudy seminara im. I. G. Petrovskogo, 1975, no. 1, 75–133

[4] A. M. Ilin, E. F. Lelikova, “Metod sraschivaniya asimptoticheskikh razlozhenii dlya uravneniya $\varepsilon\Delta u+a(x,y)u_y=f(x,y)$ v pryamougolnike”, Matem. sb., 96(138) (1975), 568–583 | MR

[5] V. Yu. Novokshenov, “Postroenie asimptotiki resheniya odnogo ellipticheskogo uravneniya v polose”, Voprosy teorii i matematicheskie metody resheniya zadach, Bashk. filial AN SSSR, Ufa, 1975