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, 141174
MSC (2000):
Primary 35B40, 35J25; Secondary 35B25, 35C20
Published electronically:
December 21, 2010
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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 YorkLondon, 1964.
MR
0176702 (31 #974)
 2.
A.
M. Il′in, Soglasovanie asimptoticheskikh razlozhenii reshenii
kraevykh zadach, “Nauka”, Moscow, 1989 (Russian). With an
English summary. MR 1007834
(90i:35062)
 3.
A.
M. Il′in and E.
F. Lelikova, The method of matching asymptotic expansions for the
equation
𝜖Δ𝑢𝑎(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
(52 #3706)
 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
(55 #13056)
 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
(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, 107119. (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
(37 #1777)
 8.
M.
A. Lavrent′ev and B.
V. Šabat, Metody teorii funkciĭ\ kompleksnogo
peremennogo, Gosudarstv. Izdat. Tehn.Teor. Lit., MoscowLeningrad,
1951 (Russian). MR 0051293
(14,457e)
 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), 568583. (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), 75133. (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, 18521865. (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, 107119. (Russian)
 7.
 V. A. Kondrat'ev, Boundary problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obva 16 (1967), 209292; English transl., Trans. Moscow Math. Soc. 16 (1967), 227313. MR 0226187 (37:1777)
 8.
 M. A. Lavrent'ev and B. V. Shabat, Methods of the theory of functions of a complex variable, Gostekhizdat, MoscowLeningrad, 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:
http://dx.doi.org/10.1090/S007715542010001859
Keywords:
Asymptotic behaviour of solutions,
boundaryvalue 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
