### 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 well-posedness 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 star-shaped; 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 |
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Pages (from-to) | 2869-2923 |

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 |

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### Keywords

- Helmholtz equation
- Heterogeneous
- High frequency
- Nontrapping
- Resolvent
- Resonance
- Semiclassical
- Transmission problem
- Uniqueness
- Variable wave speed

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics