Abstract
We consider Helmholtz problems with a perturbed wave speed, where the single-signed perturbation is linear in a parameter z. Both the wave speed and the perturbation are allowed to be discontinuous (modelling a penetrable obstacle). We show that there exists a polynomial function of frequency such that, for any frequency, for most values of z, the norm of the solution operator is bounded by that function.
This solution-operator bound is most interesting for Helmholtz problems with strong trapping; recall that here there exists a sequence of real frequencies, tending to infinity, through which the solution operator grows super algebraically, with these frequencies often called quasi-resonances. The result of this paper then shows that, at every fixed frequency in the quasi-resonance, the norm of the solution operator becomes much smaller for most single-signed perturbations of the wave speed, i.e., quasi-resonances are unstable under most such perturbations.
This solution-operator bound is most interesting for Helmholtz problems with strong trapping; recall that here there exists a sequence of real frequencies, tending to infinity, through which the solution operator grows super algebraically, with these frequencies often called quasi-resonances. The result of this paper then shows that, at every fixed frequency in the quasi-resonance, the norm of the solution operator becomes much smaller for most single-signed perturbations of the wave speed, i.e., quasi-resonances are unstable under most such perturbations.
| Original language | English |
|---|---|
| Article number | 113441 |
| Journal | Journal of Differential Equations |
| Volume | 440 |
| Early online date | 27 May 2025 |
| DOIs | |
| Publication status | E-pub ahead of print - 27 May 2025 |
Data Availability Statement
No data was used for the research described in the article.Funding
The authors thank Jeffrey Galkowski (University College London) for useful discussions about the penetrable-obstacle problem and Stephen Shipman (Louisiana State University) for useful discussions about the literature on quasi-resonances. Figs. 1.1 and 1.2 were produced using code originally written by Andrea Moiola (University of Pavia) for the paper [39] . EAS acknowledges support from EPSRC grant EP/R005591/1 . JW acknowledges partial support from NSF grant DMS-2054424 .
| Funders | Funder number |
|---|---|
| Engineering and Physical Sciences Research Council | EP/R005591/1 |
| National Science Foundation | DMS-2054424 |
| Università degli Studi di Pavia | |
| Louisiana State University | |
| National Science Foundation | DMS-2054424 |
