Projects per year
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 Sept 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
Fingerprint
Dive into the research topics of 'The Helmholtz equation in heterogeneous media: a priori bounds, wellposedness, and resonances'. Together they form a unique fingerprint.Projects
 2 Finished

Fast solvers for frequencydomain wavescattering problems and applications
Graham, I., Gazzola, S. & Spence, E.
Engineering and Physical Sciences Research Council
1/01/19 → 31/12/22
Project: Research council

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council
Profiles

Euan Spence
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching