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

Abstract

Abstract. In a bounded domain ⊂ R 3 with a smooth boundary Γ we consider the boundary value problemHere is a second order elliptic operator, is a small parameter. The limiting equation, as = 0, is the first order equation. Its characteristics are the straight lines parallel to the axis 3 . For the domain 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 → 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. Mathematics Subject Classification: 34E05, 35J25 Formulation of problemIn a bounded simply-connected domain ⊂ R 3 with piecewise smooth boundary Γ we consider the boundary valu problemis an elliptic differentiation operator with a positive definite quadratic formAssume that the data of problem (0.1)-(0.2) are smooth (belong to ∞ ), then for each > 0 there exists the unique solution ( , ) ∈ ∞ ( ).Yu.Z. Shaygardanov, Asymptotics in parameter of solution to elliptic boundary value problem in vicinity of outer touching of characteristics to limit equation.