Inhomogeneous Helmholtz equations in wave guides – existence and uniqueness results with energy methods

Abstract

The Helmholtz equation −∇⋅(a∇u)−ω2u=f is considered in an unbounded wave guide Ω:=R×S⊂Rd , S⊂Rd−1 a bounded domain. The coefficient a is strictly elliptic and either periodic in the unbounded direction x1∈R or periodic outside a compact subset; in the latter case, two different periodic media can be used in the two unbounded directions. For non-singular frequencies ω , we show the existence of a solution u. While previous proofs of such results were based on analyticity arguments within operator theory, here, only energy methods are used.