# High Frequency limit of the Helmholtz Equations

Jean-David Benamou^{[1]}; François Castella^{[2]}; Thodoros Katsaounis^{[3]}; Benoît Perthame^{[4]}

- [1] INRIA-Rocquencourt, BP 105, 78153 Le Chesnay, France
- [2] CNRS et IRMAR, Campus de Beaulieu, Université de Rennes 1, 35042 Rennes Cédex, France
- [3] IACM, FORTH, P.O. Box 1527, Vassilika Boutwn 71110, Heraklion Crete, Greece
- [4] ENS, DMA, 45, rue d’Ulm, 75230 Paris, France

Séminaire Équations aux dérivées partielles (1999-2000)

- Volume: 1999-2000, page 1-25

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topBenamou, Jean-David, et al. "High Frequency limit of the Helmholtz Equations." Séminaire Équations aux dérivées partielles 1999-2000 (1999-2000): 1-25. <http://eudml.org/doc/11002>.

@article{Benamou1999-2000,

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 $L^\{2\}$ 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.},

affiliation = {INRIA-Rocquencourt, BP 105, 78153 Le Chesnay, France; CNRS et IRMAR, Campus de Beaulieu, Université de Rennes 1, 35042 Rennes Cédex, France; IACM, FORTH, P.O. Box 1527, Vassilika Boutwn 71110, Heraklion Crete, Greece; ENS, DMA, 45, rue d’Ulm, 75230 Paris, France},

author = {Benamou, Jean-David, Castella, François, Katsaounis, Thodoros, Perthame, Benoît},

journal = {Séminaire Équations aux dérivées partielles},

language = {eng},

pages = {1-25},

publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},

title = {High Frequency limit of the Helmholtz Equations},

url = {http://eudml.org/doc/11002},

volume = {1999-2000},

year = {1999-2000},

}

TY - JOUR

AU - Benamou, Jean-David

AU - Castella, François

AU - Katsaounis, Thodoros

AU - Perthame, Benoît

TI - High Frequency limit of the Helmholtz Equations

JO - Séminaire Équations aux dérivées partielles

PY - 1999-2000

PB - Centre de mathématiques Laurent Schwartz, École polytechnique

VL - 1999-2000

SP - 1

EP - 25

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 $L^{2}$ 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.

LA - eng

UR - http://eudml.org/doc/11002

ER -

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