Geodesic
Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to
Ufa mathematical journal, Tome 9 (2017) no. 3, pp. 137-147 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In a bounded domain $Q\subset\mathbb{R}^3$ with a smooth boundary $\Gamma$ we consider the boundary value problem $$\varepsilon Au-\frac{ \partial u}{\partial x_3}=f(x),\quad u|_{\Gamma}=0.$$ Here $A$ is a second order elliptic operator, $\varepsilon$ is a small parameter. The limiting equation, as $\varepsilon=0$, is the first order equation. Its characteristics are the straight lines parallel to the axis $Ox_3$. For the domain $\overline{Q}$ we assume that the characteristic either intersects $\Gamma$ at two points or touches $\Gamma$ from outside. The set of touching point forms a closed smooth curve. In the paper we construct the asymptotics as $\varepsilon\to 0$ for the solutions to the studied problem in the vicinity of this curve. For constructing the asymptotics we employ the method of matching asymptotic expansions.
Keywords: small parameter, asymptotic
Mots-clés : elliptic equation. 
@article{UFA_2017_9_3_a13,
     author = {Yu. Z. Shaygardanov},
     title = {Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to},
     journal = {Ufa mathematical journal},
     pages = {137--147},
     year = {2017},
     volume = {9},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a13/}
}
TY  - JOUR
AU  - Yu. Z. Shaygardanov
TI  - Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to
JO  - Ufa mathematical journal
PY  - 2017
SP  - 137
EP  - 147
VL  - 9
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a13/
LA  - en
ID  - UFA_2017_9_3_a13
ER  - 
%0 Journal Article
%A Yu. Z. Shaygardanov
%T Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to
%J Ufa mathematical journal
%D 2017
%P 137-147
%V 9
%N 3
%U http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a13/
%G en
%F UFA_2017_9_3_a13
Yu. Z. Shaygardanov. Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to. Ufa mathematical journal, Tome 9 (2017) no. 3, pp. 137-147. http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a13/

[1] M. I. Vishik, L. A. Lyusternik, “Regular degeneration and boundary layer for linear differential equations with small parameter”, Uspekhi Matem. Nauk, 12:5, 3–122 (in Russian) | Zbl

[2] A. M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems, Amer. Math. Soc., Providence, RI, 1992 | MR | MR | Zbl

[3] E. F. Lelikova, “On asymptotics for a solution to second order elliptic equation with a small parameter at higher derivatives”, Differ. Uravn., 12:10 (1976), 1852–1865 (in Russian) | MR | Zbl