Existence of interior trapped-mode resonances

Determine whether the compactly supported interior hydroelastic operator associated with the submerged-plate problem possesses interior resonance frequencies, which are the only frequencies at which loss of uniqueness can occur under the stated radiation and unique-continuation arguments.

Background

The paper reduces possible nonuniqueness of the continuous hydroelastic problem to the existence of compactly supported interior eigenfunctions. Any nontrivial homogeneous solution must have zero reflection and transmission coefficients for all propagating modes and therefore carry no radiated energy.

Unique continuation then implies that the fluid potential vanishes outside the plate region, leaving an interior hydroelastic eigenvalue problem. The authors note that related geometries exhibit trapped modes at isolated frequencies, but do not establish whether such resonances occur for the present submerged-plate configurations. The well-posedness theorem consequently assumes a continuous non-resonance condition.