### 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.

Original language | English |
---|---|

Pages (from-to) | 317-354 |

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 |

### Fingerprint

### 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.** / Moiola, Andrea; Spence, Euan.

Research output: Contribution to journal › Article

*Mathematical Models & Methods in Applied Sciences*, vol. 29, no. 2, pp. 317-354. https://doi.org/10.1142/S0218202519500106

}

TY - JOUR

T1 - Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions

AU - Moiola, Andrea

AU - Spence, Euan

PY - 2019/2/28

Y1 - 2019/2/28

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

VL - 29

SP - 317

EP - 354

JO - Mathematical Models & Methods in Applied Sciences

JF - Mathematical Models & Methods in Applied Sciences

SN - 0218-2025

IS - 2

ER -