# High frequency limit of the Helmholtz equations.

Jean-David Benamou; François Castella; Theodoros Katsaounis; Benoit Perthame

Revista Matemática Iberoamericana (2002)

- Volume: 18, Issue: 1, page 187-209
- ISSN: 0213-2230

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topBenamou, Jean-David, et al. "High frequency limit of the Helmholtz equations.." Revista Matemática Iberoamericana 18.1 (2002): 187-209. <http://eudml.org/doc/39653>.

@article{Benamou2002,

abstract = {We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term ( which does not share the quadratic aspect) in the limit, then, the lack of L2 bounds which can be handled with homogeneous Morrey-Campanato estimates, and finally the problem of uniqueness which, at several stage of the proof, is related to outgoing conditions at infinity. (A)},

author = {Benamou, Jean-David, Castella, François, Katsaounis, Theodoros, Perthame, Benoit},

journal = {Revista Matemática Iberoamericana},

keywords = {Ecuación de Helmholtz; Ecuación de Liouville; Optica geométrica; Transformada de Wigner; Helmholtz equations; high frequuency; transport equations; geometrical optics},

language = {eng},

number = {1},

pages = {187-209},

title = {High frequency limit of the Helmholtz equations.},

url = {http://eudml.org/doc/39653},

volume = {18},

year = {2002},

}

TY - JOUR

AU - Benamou, Jean-David

AU - Castella, François

AU - Katsaounis, Theodoros

AU - Perthame, Benoit

TI - High frequency limit of the Helmholtz equations.

JO - Revista Matemática Iberoamericana

PY - 2002

VL - 18

IS - 1

SP - 187

EP - 209

AB - We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term ( which does not share the quadratic aspect) in the limit, then, the lack of L2 bounds which can be handled with homogeneous Morrey-Campanato estimates, and finally the problem of uniqueness which, at several stage of the proof, is related to outgoing conditions at infinity. (A)

LA - eng

KW - Ecuación de Helmholtz; Ecuación de Liouville; Optica geométrica; Transformada de Wigner; Helmholtz equations; high frequuency; transport equations; geometrical optics

UR - http://eudml.org/doc/39653

ER -

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- Jean-François Bony, Mesures limites pour l’équation de Helmholtz dans le cas non captif
- Elise Fouassier, High frequency limit of Helmholtz equations: the case of a discontinuous index

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