High frequency limit of the Helmholtz equations

Jean David Benamou*, François Castella, Theodoros Katsaounis, Benoit Perthame

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

34 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)187-209
Number of pages23
JournalRevista Matematica Iberoamericana
Volume18
Issue number1
DOIs
StatePublished - Jan 1 2002

Keywords

  • Geometrical optics
  • Helmholtz equations
  • High frecuency
  • Transport equations

ASJC Scopus subject areas

  • Mathematics(all)

Fingerprint

Dive into the research topics of 'High frequency limit of the Helmholtz equations'. Together they form a unique fingerprint.

Cite this