Abstract
We consider the Helmholtz transmission problem with one penetrable star-shaped 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 resonance-free 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 star-shapedness 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 much-more 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.
| Language | English |
|---|---|
| Journal | Mathematical Models & Methods in Applied Sciences |
| Early online date | 18 Jan 2019 |
| DOIs | |
| Status | E-pub ahead of print - 18 Jan 2019 |
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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
Cite this
Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions. / Spence, Euan; Moiola, Andrea.
In: Mathematical Models & Methods in Applied Sciences, 18.01.2019.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions
AU - Spence, Euan
AU - Moiola, Andrea
PY - 2019/1/18
Y1 - 2019/1/18
N2 - We consider the Helmholtz transmission problem with one penetrable star-shaped 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 resonance-free 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 star-shapedness 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 much-more 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.
AB - We consider the Helmholtz transmission problem with one penetrable star-shaped 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 resonance-free 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 star-shapedness 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 much-more 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.
KW - Helmholtz equation
KW - Lipschitz domain
KW - Morawetz identity
KW - Transmission problem
KW - acoustic
KW - frequency explicit
KW - resonance
KW - semiclassical
KW - wavenumber explicit
UR - http://www.scopus.com/inward/record.url?scp=85060254762&partnerID=8YFLogxK
U2 - 10.1142/S0218202519500106
DO - 10.1142/S0218202519500106
M3 - Article
JO - Mathematical Models & Methods in Applied Sciences
T2 - Mathematical Models & Methods in Applied Sciences
JF - Mathematical Models & Methods in Applied Sciences
SN - 0218-2025
ER -