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{c}ois}, 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}, eid = {4}, pages = {1--18}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2004}, doi = {10.5802/jedp.4}, zbl = {02161530}, mrnumber = {2135359}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jedp.4/} }
TY - JOUR AU - Castella, François TI - The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach JO - Journées équations aux dérivées partielles PY - 2004 SP - 1 EP - 18 PB - Groupement de recherche 2434 du CNRS UR - http://www.numdam.org/articles/10.5802/jedp.4/ DO - 10.5802/jedp.4 LA - en ID - JEDP_2004____A4_0 ER -
%0 Journal Article %A Castella, François %T The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach %J Journées équations aux dérivées partielles %D 2004 %P 1-18 %I Groupement de recherche 2434 du CNRS %U http://www.numdam.org/articles/10.5802/jedp.4/ %R 10.5802/jedp.4 %G en %F 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/articles/10.5802/jedp.4/
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