Open Access
March, 2002 High Frequency limit of the Helmholtz Equations
Jean-David Benamou, François Castella, Theodoros Katsaounis, Benoit Perthame
Rev. Mat. Iberoamericana 18(1): 187-209 (March, 2002).


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.


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Jean-David Benamou. François Castella. Theodoros Katsaounis. Benoit Perthame. "High Frequency limit of the Helmholtz Equations." Rev. Mat. Iberoamericana 18 (1) 187 - 209, March, 2002.


Published: March, 2002
First available in Project Euclid: 18 February 2003

zbMATH: 1090.35165
MathSciNet: MR1924691

Primary: 35J05 , 78A05 , 81S30

Keywords: geometrical optics , Helmholtz equations , high frecuency , transport equations

Rights: Copyright © 2002 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.18 • No. 1 • March, 2002
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