Abstract
We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation ∇⋅(A∇u)+k ^{2}nu=−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 wellposedness 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 starshaped; 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 language  English 

Pages (fromto)  28692923 
Number of pages  55 
Journal  Journal of Differential Equations 
Volume  266 
Issue number  6 
Early online date  12 Sep 2018 
DOIs  
Publication status  Published  5 Mar 2019 
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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Euan Spence
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching