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Ufimsk. Mat. Zh., 2017, Volume 9, Issue 3, Pages 138–147 (Mi ufa395)  

Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to limit equation

Yu. Z. Shaygardanov

Institute of Mathematics, Ufa Scientific Center, RAS, Chernyshevsky str. 112, 450077, Ufa, Russia

Abstract: 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, elliptic equation.

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English version:
Ufa Mathematical Journal, 2017, 9:3, 137–147 (PDF, 345 kB); https://doi.org/10.13108/2017-9-3-137

Bibliographic databases:

UDC: 517.928
MSC: 34E05, 35J25
Received: 09.06.2017

Citation: Yu. Z. Shaygardanov, “Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to limit equation”, Ufimsk. Mat. Zh., 9:3 (2017), 138–147; Ufa Math. J., 9:3 (2017), 137–147

Citation in format AMSBIB
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\pages 138--147
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