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

Abstract

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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 .

How to cite

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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): 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 $\alpha &gt;0$. The high-frequency (or: semi-classical) parameter is $\varepsilon &gt;0$. 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 $\alpha &gt;0$. The high-frequency (or: semi-classical) parameter is $\varepsilon &gt;0$. 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 -

References

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