Core Concepts
The authors prove the validity of the limiting amplitude principle (LAP) for the wave equation with nonconstant coefficients in spatial dimensions 1, 2, and 3. They quantify the convergence rates of the time-domain solution to the frequency-domain solution.
Abstract
The paper studies the limiting amplitude principle (LAP) for the wave equation with nonconstant coefficients. The authors consider the following setup:
Frequency-domain problem:
The Helmholtz equation with variable coefficients and a compactly supported source term.
Time-domain problem:
The wave equation with variable coefficients and a time-harmonic source term.
The authors make the following key assumptions:
Smoothness, positivity, and compact support of the variable coefficients.
Non-trapping condition on the coefficients (for d ≥ 2).
Compact support of the source term.
The main results are:
For d = 2, 3, the authors prove the validity of the LAP and establish algebraic convergence rates of the time-domain solution to the frequency-domain solution.
For d = 1, the authors prove the validity of the LAP with an appropriate modification and establish exponential convergence.
The proofs rely on time-decay estimates for solutions of auxiliary problems, which are established using a decomposition approach. The authors avoid a direct study of the resolvent operator and instead leverage recent results on time-decay of solutions.