High Frequency Limit of the Helmholtz Equations - ENSTA Paris - École nationale supérieure de techniques avancées Paris Access content directly
Reports (Research Report) Year : 1999

High Frequency Limit of the Helmholtz Equations

François Castella
  • Function : Author
  • PersonId : 828826
Thodoros Katsaounis
  • Function : Author
Benoît Perthame

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.

Domains

Other [cs.OH]
Fichier principal
Vignette du fichier
RR-3785.pdf (285.89 Ko) Télécharger le fichier

Dates and versions

inria-00072875 , version 1 (24-05-2006)

Identifiers

  • HAL Id : inria-00072875 , version 1

Cite

Jean-David Benamou, François Castella, Thodoros Katsaounis, Benoît Perthame. High Frequency Limit of the Helmholtz Equations. [Research Report] RR-3785, INRIA. 1999. ⟨inria-00072875⟩
240 View
244 Download

Share

Gmail Facebook X LinkedIn More