High frequency limit of the Helmholtz equations.

Benamou, 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 -