Bounding acoustic layer potentials via oscillatory integral techniques

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Abstract

We consider the Helmholtz single-layer operator (the trace of the single-layer potential) as an operator on (Formula presented.) where (Formula presented.) is the boundary of a 3-d obstacle. We prove that if (Formula presented.) is (Formula presented.) and has strictly positive curvature then the norm of the single-layer operator tends to zero as the wavenumber (Formula presented.) tends to infinity. This result is proved using a combination of (1) techniques for obtaining the asymptotics of oscillatory integrals, and (2) techniques for obtaining the asymptotics of integrals that become singular in the appropriate parameter limit. This paper is the first time such techniques have been applied to bounding norms of layer potentials. The main motivation for proving this result is that it is a component of a proof that the combined-field integral operator for the Helmholtz exterior Dirichlet problem is coercive on such domains in the space (Formula presented.).
Original languageEnglish
Pages (from-to)279-318
JournalBIT Numerical Mathematics
Volume55
Issue number1
Early online date19 Jul 2014
DOIs
Publication statusPublished - Mar 2015

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