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A class of interface problems for the Helmholtz equation in R n
Author(s) -
Speck F.O.
Publication year - 2015
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3386
Subject(s) - mathematics , factorization , helmholtz free energy , helmholtz equation , resolvent , interface (matter) , operator (biology) , space (punctuation) , pure mathematics , class (philosophy) , matrix (chemical analysis) , interpretation (philosophy) , mathematical analysis , algebra over a field , algorithm , quantum mechanics , philosophy , materials science , repressor , artificial intelligence , boundary value problem , linguistics , gibbs isotherm , chemistry , computer science , surface tension , composite material , biochemistry , transcription factor , programming language , physics , gene
Motivated by the Sommerfeld diffraction problem, we consider interface problems for the n‐dimensional Helmholtz equation in Ω = R + n ∪ R − n(due to x n >0 or x n <0, respectively, with main interest in n = 3) where the interface Γ = ∂ Ω is identified withR n − 1and divided into two parts, Σ and Σ ′ , with different transmission conditions of first and second kind. These two parts are half‐spaces ofR n − 1(half‐planes for n = 3) and more general sets in the first part of the paper. The aim of this work is to construct resolvent operators acting from the interface data into the energy space H 1 ( Ω ) and representing the solution explicitly. The approach is based upon a factorization conception for Wiener–Hopf operators (according to the interface equations), the so‐called Wiener–Hopf factorization through an intermediate space, that includes Simonenko's well‐known ‘generalized factorization of matrix functions in L p spaces’ and avoids an interpretation of the factors as unbounded operators. Copyright © 2015 John Wiley & Sons, Ltd.