The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances

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Abstract

We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation ∇⋅(A∇u)+k 2nu=−f where both A and n are functions of position. We prove new a priori bounds on the solution under conditions on A, n, and the domain that ensure nontrapping of rays; the novelty is that these bounds are explicit in k, A, n, and geometric parameters of the domain. We then show that these a priori bounds hold when A and n are L and satisfy certain monotonicity conditions, and thereby obtain new results both about the well-posedness of such problems and about the resonances of acoustic transmission problems (i.e. A and n discontinuous) where the transmission interfaces are only assumed to be C 0 and star-shaped; the novelty of this latter result is that until recently the only known results about resonances of acoustic transmission problems were for C convex interfaces with strictly positive curvature.

Original languageEnglish
Pages (from-to)2869-2923
Number of pages55
JournalJournal of Differential Equations
Volume266
Issue number6
Early online date12 Sep 2018
DOIs
Publication statusPublished - 5 Mar 2019

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Keywords

  • Helmholtz equation
  • Heterogeneous
  • High frequency
  • Nontrapping
  • Resolvent
  • Resonance
  • Semiclassical
  • Transmission problem
  • Uniqueness
  • Variable wave speed

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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