Abstract
A rigorous method for constructing asymptotic approximations in thin domains was first proposed by Gol’denveizer (The Theory of Elastic Thin Shells, Nauka, Moscow, 1976; Prikl. Mat. Meh. 26(4):668–686, 1962); it was further developed for thin domains of cylindrical type in (Dzhavadov, Differ. Urav. 4(10):1901–1909, 1968; Caillerie, Math. Math. Appl. Sci. 6:159–191, 1984; Vasil’eva and Butuzov, Asymptotic Methods in the Theory of Singular Perturbations, Vyssh. Shkola, Moscow, 1990; Nazarov, Vestn. Leningr. Univ. Ser. Mat. Mekh. Astron. 2:65–68, 1982). These authors considered thin domains of cylindrical type and the main approach to asymptotic analysis was to make a special change of coordinates after which the scaled domain was independent of the small parameter. Then a small parameter appeared in the higher derivatives of the differential equations and the Lyusternik–Vishik method (Vishik and Lyusternik, Usp. Mat. Nauk 12(5):3–192, 1957) was used.
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