# The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach

François Castella^{[1]}

- [1] IRMAR - Université de Rennes 1 Campus Beaulieu - 35042 Rennes Cedex - France

Journées Équations aux dérivées partielles (2004)

- page 1-18
- ISSN: 0752-0360

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topCastella, 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): 1-18. <http://eudml.org/doc/10596>.

@article{Castella2004,

abstract = {We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter $\alpha >0$. The high-frequency (or: semi-classical) parameter is $\varepsilon >0$. We let $\varepsilon $ and $\alpha $ 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^\varepsilon $ radiates in the outgoing direction, uniformly in $\varepsilon $. In particular, the function $u^\varepsilon $, when conveniently rescaled at the scale $\varepsilon $ 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 $\varepsilon $) 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 $\varepsilon $.},

affiliation = {IRMAR - Université de Rennes 1 Campus Beaulieu - 35042 Rennes Cedex - France},

author = {Castella, François},

journal = {Journées Équations aux dérivées partielles},

language = {eng},

month = {6},

pages = {1-18},

publisher = {Groupement de recherche 2434 du CNRS},

title = {The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach},

url = {http://eudml.org/doc/10596},

year = {2004},

}

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

DA - 2004/6//

PB - Groupement de recherche 2434 du CNRS

SP - 1

EP - 18

AB - We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter $\alpha >0$. The high-frequency (or: semi-classical) parameter is $\varepsilon >0$. We let $\varepsilon $ and $\alpha $ 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^\varepsilon $ radiates in the outgoing direction, uniformly in $\varepsilon $. In particular, the function $u^\varepsilon $, when conveniently rescaled at the scale $\varepsilon $ 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 $\varepsilon $) 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 $\varepsilon $.

LA - eng

UR - http://eudml.org/doc/10596

ER -

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