Asymptotics of Solutions to General Elliptic Boundary Value Problems in Domains Perturbed Near Cone Vertices

Abstract

In this chapter general elliptic boundary value problems depending on a small parameter α will be considered. Here the domain Ω(ε) arises as a small singular perturbation of the limit domain Ω whose boundary contains a finite number of cone vertices. The complete asymptotic expansions will be constructed and justified. The present chapter provides the basis for further study of special singularly perturbed boundary value problems. The general results contained here allow us to restrict ourselves, concerning the solution of a number of problems in the following chapters, to the construction of the formal asymptotics. The methods developed in Chapter 2 for the Laplace operator will be obtained as a special case of the general scheme developed here.