The asymptotics of a solution of a second order elliptic equation with a small parameter multiplying one of the highest order derivatives

Author:
E. F. Lelikova

Translated by:
E. Khukhro

Original publication:
Trudy Moskovskogo Matematicheskogo Obshchestva, tom **71** (2010).

Journal:
Trans. Moscow Math. Soc. **2010**, 141-174

MSC (2000):
Primary 35B40, 35J25; Secondary 35B25, 35C20

DOI:
https://doi.org/10.1090/S0077-1554-2010-00185-9

Published electronically:
December 21, 2010

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Abstract | References | Similar Articles | Additional Information

Abstract: The asymptotic behaviour of a solution of the first boundary value problem for a second order elliptic equation is analysed in the case where a small parameter is involved as a factor multiplying only one of the highest order derivatives and the limit equation is an ordinary differential equation. In spite of the fact that the order of the limit equation is the same as that of the original equation, the problem under consideration is bisingular. The asymptotic behaviour of a solution of this problem is analysed using the method of matching asymptotic expansions.

**1.**M. van Dyke,*Perturbation methods in fluid mechanics*, Academic Press, New York, 1964. MR**0176702 (31:974)****2.**A. M. Il'in,*Matching of asymptotic expansions of solutions of boundary value problems*, Nauka, Moscow, 1989; English transl., Transl. Math. Monographs**102**, Amer. Math. Soc., Providence, RI, 1992. MR**1007834 (90i:35062)****3.**A. M. Il'in and E. F. Lelikova,*Method of matching of asymptotic expansions for the equation on a rectangle*, Mat. Sb.**96**(1975), 568-583. (Russian) MR**0382824 (52:3706)****4.**A. M. Il'in, Yu. P. Gor'kov, and E. F. Lelikova,*The asymptotics of the solution of an elliptic equation with a small parameter multiplying the highest order derivatives in a neighborhood of a singular characteristic of the limit equation*, Trudy Semin. Im. I. G. Petrovskogo**1**(1975), 75-133. (Russian) MR**0440177 (55:13056)****5.**E. F. Lelikova,*On the asymptotics of the solution of a second order elliptic equation with a small parameter multiplying the highest order derivatives*, Differentsial. Uravn.**12**(1976), no. 10, 1852-1865. (Russian) MR**0445100 (56:3445)****6.**E. F. Lelikova,*On the asymptotics of the solution of a second order elliptic equation with a small parameter at one of the highest derivatives*, Trudy Inst. Matem. Mekhan. Ural. Otd. Ross. Akad. Nauk**9**(2003), no. 1, 107-119. (Russian)**7.**V. A. Kondrat'ev,*Boundary problems for elliptic equations in domains with conical or angular points*, Trudy Moskov. Mat. Ob-va**16**(1967), 209-292; English transl., Trans. Moscow Math. Soc.**16**(1967), 227-313. MR**0226187 (37:1777)****8.**M. A. Lavrent'ev and B. V. Shabat,*Methods of the theory of functions of a complex variable*, Gostekhizdat, Moscow-Leningrad, 1951. (Russian) MR**0051293 (14:457e)**

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Additional Information

**E. F. Lelikova**

Affiliation:
Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, 16 Kovalevskaya Street, Ekaterinburg 620219, Russia

Email:
dar@imm.uran.ru

DOI:
https://doi.org/10.1090/S0077-1554-2010-00185-9

Keywords:
Asymptotic behaviour of solutions,
boundary-value problem,
second order elliptic equation,
matched asymptotic expansions.

Published electronically:
December 21, 2010

Additional Notes:
This research was carried out with the financial support of the Russian Foundation for Basic Research (grant nos. 08–01–00260, NSh–2215.2008.1).

Article copyright:
© Copyright 2010
American Mathematical Society