Revisiting the limiting amplitude principle.
The limiting amplitude principle is a well-known result connecting the solution of the Helmholtz equation with the large-time behavior of time-dependent wave equations with a source term which is periodic in time. After giving a general introduction into the topic, I’ll focus on the version of the principle relevant to the wave equation with non-constant coefficients. Motivated by numerical analysis of time-domain methods for stationary scattering problems in heterogeneous media, we quantify the convergence of the time-dependent wave equation towards the stationary solution under appropriate assumptions. We also generalise the formulation of the limiting amplitude principle to the one-dimensional setting where the classical statement of the principle is known to be violated. Our proof of the limiting amplitude principle and its quantification is due to reduction towards several results concerning temporal decay for wave equations with sufficiently localised initial data or a source term.
The talk is based on recent results of a joint work with A. Arnold, S. Geevers and I. Perugia.