Singularly Perturbed Elliptic Dirichlet Problem with a Multiple Root of the Degenerate Equation

Abstract

A singularly perturbed elliptic problem with Dirichlet boundary conditions is considered in the case of multiple roots of the degenerate equation. A complete asymptotic expansion of the solution is constructed and justified. It is qualitatively different from the known expansion in the case where the root of the degenerate equation is simple: the asymptotic expansion of the solution being in fractional powers of the small parameter, boundary-layer variables have a different scale, boundary-layer series is constructed using a non-standard algorithm, the boundary layer in the vicinity of the domain boundary consists of three zones with different behavior of the solution in different zones.

Keywords

• сингулярно возмущенное эллиптическое уравнение
• случай кратного корня вырожденного уравнения
• асимптотическое разложение решения погранслойного типа
• трехзонный пограничный слой