Abstract
We consider the Helmholtz transmission problem with one penetrable starshaped Lipschitz obstacle. Under a natural assumption about the ratio of the wavenumbers, we prove bounds on the solution in terms of the data, with these bounds explicit in all parameters. In particular, the (weighted) H1 norm of the solution is bounded by the L2 norm of the source term, independently of the wavenumber. These bounds then imply the existence of a resonancefree strip beneath the real axis. The main novelty is that the only comparable results currently in the literature are for smooth, convex obstacles with strictly positive curvature, while here we assume only Lipschitz regularity and starshapedness with respect to a point. Furthermore, our bounds are obtained using identities first introduced by Morawetz (essentially integration by parts), whereas the existing bounds use the muchmore sophisticated technology of microlocal analysis and propagation of singularities. We also adapt existing results to show that if the assumption on the wavenumbers is lifted, then no bound with polynomial dependence on the wavenumber is possible.
Original language  English 

Pages (fromto)  317354 
Number of pages  38 
Journal  Mathematical Models & Methods in Applied Sciences 
Volume  29 
Issue number  2 
Early online date  18 Jan 2019 
DOIs  
Publication status  Published  28 Feb 2019 
Keywords
 Helmholtz equation
 Lipschitz domain
 Morawetz identity
 Transmission problem
 acoustic
 frequency explicit
 resonance
 semiclassical
 wavenumber explicit
ASJC Scopus subject areas
 Modelling and Simulation
 Applied Mathematics
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Profiles

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