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

Abstract

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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.

How to cite

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Benamou, 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 -

References

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