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

Full-text PDF

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.**Milton Van Dyke,*Perturbation methods in fluid mechanics*, Applied Mathematics and Mechanics, Vol. 8, Academic Press, New York-London, 1964. MR**0176702****2.**A. M. Il′in,*\cyr Soglasovanie asimptoticheskikh razlozheniĭ resheniĭ kraevykh zadach*, “Nauka”, Moscow, 1989 (Russian). With an English summary. MR**1007834****3.**A. M. Il′in and E. F. Lelikova,*The method of matching asymptotic expansions for the equation 𝜖Δ𝑢-𝑎(𝑥,𝑦)𝑢_{𝑦}=𝑓(𝑥,𝑦) in a rectangle*, Mat. Sb. (N.S.)**96(138)**(1975), no. 4, 568–583, 645 (Russian). MR**0382824****4.**A. M. Il′in, Ju. P. Gor′kov, and E. F. Lelikova,*Asymptotic behavior of the solution of an elliptic equation with a small parameter multiplying the highest derivatives in the neighborhood of a singular characteristic of the limit equation*, Trudy Sem. Petrovsk.**Vyp. 1**(1975), 75–133 (Russian). MR**0440177****5.**E. F. Lelikova,*The asymptotic expansion of the solution of a second order elliptic equation with a small parameter multiplying the highest derivatives*, Differencial′nye Uravnenija**12**(1976), no. 10, 1852–1865, 1919 (Russian). MR**0445100****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 value problems for elliptic equations in domains with conical or angular points*, Trudy Moskov. Mat. Obšč.**16**(1967), 209–292 (Russian). MR**0226187****8.**M. A. Lavrent′ev and B. V. Šabat,*Metody teorii funkciĭ kompleksnogo peremennogo*, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1951 (Russian). MR**0051293**

Retrieve articles in *Transactions of the Moscow Mathematical Society*
with MSC (2000):
35B40,
35J25,
35B25,
35C20

Retrieve articles in all journals with MSC (2000): 35B40, 35J25, 35B25, 35C20

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