10.1 Introduction

In this chapter the exterior domain D' = D _e is denoted by ?. The material presented in this chapter is taken mainly from [90], [120], and the presentation follows closely [120].

Let us consider the behavior of the solution to the problems

(10.1)

(10.2)

where u for k > 0 always satisfies the radiation condition, D is a bounded domain with Liapunov's boundary (this means that ? ? C ^{1, ?}, ? > 0). We are also interested in the boundary conditions

(10.3)

and

(10.4)

where N is the unit normal on ? pointing into ?. The case when Lu = ? _i[ a _ij( x) ? _ju] stands in place of ? ² can be treated as well, provided that a _ij ? C ¹, a _ij( x) = ? _ij for x sufficiently large, and the matrix a _ij( x) satisfies the ellipticity condition c ₁ t _it _i ? a _ijt _it _j ? c ₂ t _it _i, c ₁ and c ₂ are positive constants, the bar stands for complex conjugate and over repeated indices one sums up. The function ?( s) ? C( ?) is assumed to satisfy conditions

(10.5)

Assumption (10.5) implies uniqueness of the solution to problem (10.1), (10.4). If equation (10.1) would...