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

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.