声传输问题：波数显式边界和无共振区
Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions
摘要：
本文研究了具有可穿透星形Lipschitz障碍物的Helmholtz传输问题。在一个自然的假设下的波数的比率，我们证明的解决方案的数据方面的界限，这些界限明确的所有参数。特别地，解的（加权的）$H^1$范数由源项的$L^2$范数限制，与波数无关。这些界限则意味着在实轴下存在一个无共振带。主要的新颖性是，目前在文献中唯一可比的结果是光滑，凸的障碍，严格的正曲率，而在这里，我们假设只有Lipschitz正则性和星形相对于一个点。此外，我们的边界得到使用的身份首先介绍了Morawetz（基本上分部积分法），而现有的边界使用更复杂的技术的微局部分析和奇性传播。我们还回顾了现有的结果表明，如果假设的波数被解除，那么没有约束的多项式依赖的波数是可能的。
英文摘要：
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) $H^1$ norm of the solution is bounded by the $L^2$ 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 recap existing results that show that if the assumption on the wavenumbers is lifted, then no bound with polynomial dependence on the wavenumber is possible.
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