The goal of this paper is to investigate the stability of the Helmholtz equation in the high-frequency regime with non-smooth and rapidly oscillating coefficients on bounded domains. Existence and uniqueness of the problem can be proved using the unique continuation principle in Fredholm’s alternative. However, this approach does not give directly a coefficient-explicit energy estimate. We present a new theoretical approach for the one-dimensional problem and find that for a new class of coefficients, including coefficients with an arbitrary number of discontinuities, the stability constant (i.e. the norm of the solution operator) is bounded by a term independent of the number of jumps. We emphasize that no periodicity of the coefficients is required. By selecting the wave speed function in a certain “resonant” way, we construct a class of oscillatory configurations, such that the stability constant grows exponentially in the frequency. This shows that our estimates are sharp.
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The authors gratefully acknowledge support by the Swiss National Science Foundation under Grant No. 172803. The authors are also grateful to the Applied Mathematics Department at ENSTA ParisTech for the kind hospitality in the fall semester 2016 and the Hausdorff Research Institute for Mathematics in Bonn for Visiting Fellowships in their 2017 Trimester Programme on Multiscale Methods, during which part of this work was carried out.
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Sauter, S., Torres, C. Stability estimate for the Helmholtz equation with rapidly jumping coefficients. Z. Angew. Math. Phys. 69, 139 (2018). https://doi.org/10.1007/s00033-018-1031-9
Mathematics Subject Classification
- Primary: 65N80
- Secondary: 35J05
- Helmholtz equation
- High frequency
- Heterogeneous media
- Stability estimates