Abstract
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. The coefficient a is strictly elliptic and either periodic in the unbounded direction
$x_1 \in \mathbb{R}$
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
$\omega$
, 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.
Publisher
Cambridge University Press (CUP)