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 ⊂ R 3 with a smooth boundary Γ we consider the boundary value problem Here 𝐴 is a second order elliptic operator, 𝜀 is a small parameter. The limiting equation, as 𝜀 = 0, is the first order equation. Its characteristics are the straight lines parallel to the axis 𝑂𝑥 3 . For the domain 𝑄 we assume that the characteristic either intersects Γ at two points or touches Γ from outside. The set of touching point forms a closed smooth curve. In the paper we construct the asymptotics as 𝜀 → 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.