The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach
| dc.creator | Castella, François | |
| dc.date | 2005-03-16 | |
| dc.date.accessioned | 2026-07-07T05:18:01Z | |
| dc.date.available | 2026-07-07T05:18:01Z | |
| dc.description | We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter $\a>0$. The high-frequency (or: semi-classical) parameter is $\eps>0$. We let $\eps$ and $\a$ go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption. Under these assumptions, we prove that the solution $u^\eps$ radiates in the outgoing direction, {\bf uniformly} in $\eps$. In particular, the function $u^\eps$, when conveniently rescaled at the scale $\eps$ close to the origin, is shown to converge towards the {\bf outgoing} solution of the Helmholtz equation, with coefficients frozen at the origin. This provides a uniform version (in $\eps$) of the limiting absorption principle. Writing the resolvent of the Helmholtz equation as the integral in time of the associated semi-classical Schrödinger propagator, our analysis relies on the following tools: (i) For very large times, we prove and use a uniform version of the Egorov Theorem to estimate the time integral; (ii) for moderate times, we prove a uniform dispersive estimate that relies on a wave-packet approach, together with the above mentioned transversality condition; (iii) for small times, we prove that the semi-classical Schrödinger operator with variable coefficients has the same dispersive properties as in the constant coefficients case, uniformly in $\eps$. | |
| dc.identifier | https://arxiv.org/abs/math/0503331 | |
| dc.identifier | http://arxiv.org/abs/math/0503331 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/74515 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Mathematical Physics | |
| dc.subject | 35Q40 ; 35J10 ; 81Q20 | |
| dc.title | The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach | |
| dc.type | text |