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

The Helmholtz equation $-\nabla\cdot (a\nabla u) - \omega^2 u = f$ is considered in an unbounded wave guide $\Omega := \mathbb{R} \times S \subset \mathbb{R}^d$ , $S\subset \mathbb{R}^{d-1}$ a bounded domain. coefficient strictly elliptic and either periodic the direction $x_1 \in \mathbb{R}$ or outside compact subset; latter case, two different media can be used directions. For non-singular frequencies $\omega$ we show existence of solution . While previous proofs such results were based on analyticity arguments within operator theory, here, only energy methods are used.