In the paper a class of elliptic boundary value problems is considered. The main operator in the domain depends on a small parameter, the boundary conditions contain additional functions defined on the boundary of the domain. The formal asymptotic solution is constructed by means of the Vishik-Lyusternik method. For these problems the definition of ellipticity with small parameter is introduced.

[1] Kozlov V.A., Maz’ya V.G., Rossmann J. Elliptic boundary value problems in domains with point singularities // Mathematical surveys and monographs 52. – American mathematical society, 1997. – 414 р.

[2] Roitberg, Y. Boundary Value Problems in the Spaces of Distributions // Mathematics and Its Applications. – Hardcover, 2000. – 498 . – 288 p.

[3] Gindikin S. G., Volevich L.R. Mixed problem for partial differential equations with quasi-uniform senior part. -Moscow: URSS, 1999. - 272 p.

[4] Vishik M.I., Lusternik L.A. Regular degeneration and boundary layer for linear differential equations with small parameter // Uspekhi mat. nauk. - 1957. -12, № 5. - P. 3-122.

[5] Volevich L.R., Vishik-Lusternik method in elliptic problems with a small parameter // Proc. of Mosc.Mat. Society, 2006. -67. - P. 104-147.

[6] Agranovich M.S., Vishik M.I. Elliptic boundary value problems with parameter and parabolic problems of general form// Uspekhi matem. nauk. - 1964. -19, № 3. - С. 43-161.

[7] Ilyin A.M. Coordination of asymptotic decompositions of solutions of boundary value problems. - Moscow: Nauka, 1989.

[8] Denk R., Mennicken R., Volevich L.R. On Elliptic Operator Pencils with General Boundary Conditions// Integ. Eq. Operator Th. – 2001. – 9 . – P. 25-40.

[9] Demidenko G.V., Uspensky S.V. Equations and systems not solved with respect to the senior derivative. - Novosibirsk: Nauchnaya Kniga, 1998. -436 p.