We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter . The high-frequency (or: semi-classical) parameter is . We let and 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 radiates in the outgoing direction, uniformly in . In particular, the function , when conveniently rescaled at the scale 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 ) 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 .
@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}, year = {2004}, pages = {1-18}, doi = {10.5802/jedp.4}, zbl = {02161530}, mrnumber = {2135359}, language = {en}, url = {http://dml.mathdoc.fr/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), pp. 1-18. doi : 10.5802/jedp.4. http://gdmltest.u-ga.fr/item/JEDP_2004____A4_0/
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