Acoustic transmission problems: Wavenumber-explicit bounds and resonance-free regions
Acoustic transmission problems: Wavenumber-explicit bounds and resonance-free regions
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DOI：
10.1142/s0218202519500106
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发表时间：
2017-02
影响因子：
3.5
通讯作者：
A. Moiola;E. Spence
中科院分区：
文献类型：
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作者：
A. Moiola;E. Spence
考虑具有一个可穿透星形利普希茨障碍物的亥姆霍兹传输问题。在一个关于波数之比的自然假设下，我们用数据证明了解的边界，这些边界在所有参数中都是显式的。特别是，解的（加权的）[公式：见文]范数由源项的[公式：见文]范数限定，与波数无关。这些边界意味着在实轴下存在无共振带。主要的新奇之处在于，目前文献中唯一可比较的结果是具有严格正曲率的光滑凸障碍物，而这里我们只假设关于点的Lipschitz规则和星形。此外，我们的边界是使用Morawetz首先引入的恒等式获得的（本质上是分部积分），而现有的边界使用了更复杂的微局部分析和奇点传播技术。我们还利用已有的结果表明，如果取消对波数的假设，则不可能存在与波数多项式相关的界。
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) [Formula: see text] norm of the solution is bounded by the [Formula: see text] 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.