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

Euan Spence (Bath)

Joint work with Andrea Moiola (Reading)

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. These bounds then imply the existence of a resonance-free strip beneath the real axis. The main novelty of these results is that the only comparable results currently in the literature are for smooth, convex obstacles with strictly positive curvature.