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. The coefficient a is strictly elliptic and either periodic in the unbounded direction $x_1...

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Bibliographic Details
Published inEuropean journal of applied mathematics Vol. 34; no. 2; pp. 211 - 237
Main Author SCHWEIZER, BEN
Format Journal Article
LanguageEnglish
Published Cambridge Cambridge University Press 01.04.2023
Subjects
Applied mathematics
Energy
Energy methods
Helmholtz equations
Mathematical analysis
Partial differential equations
Radiation
Waveguides
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ISSN0956-7925
1469-4425
DOI10.1017/S0956792522000080

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Summary: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.
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ISSN:0956-7925
1469-4425
DOI:10.1017/S0956792522000080