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

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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},
     publisher = {mathdoc},
     volume = {9},
     number = {3},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2017_9_3_a13/}
}
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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/