The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach
Journées équations aux dérivées partielles (2004), article no. 4, 18 p.

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter α>0. The high-frequency (or: semi-classical) parameter is ε>0. We let ε and α 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 ε radiates in the outgoing direction, uniformly in ε. In particular, the function u ε , when conveniently rescaled at the scale ε close to the origin, is shown to converge towards the outgoing solution of the Helmholtz equation, with coefficients frozen at the origin. This provides a uniform (in ε) version 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 ε.

DOI : https://doi.org/10.5802/jedp.4
Classification:  35Q40,  35J10,  81Q20
@article{JEDP_2004____A4_0,
     author = {Castella, Fran\c cois},
     title = {The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2004},
     doi = {10.5802/jedp.4},
     mrnumber = {2135359},
     zbl = {02161530},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_2004____A4_0}
}
Castella, François. The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach. Journées équations aux dérivées partielles (2004), article  no. 4, 18 p. doi : 10.5802/jedp.4. http://www.numdam.org/item/JEDP_2004____A4_0/

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