Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions

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.