High Frequency limit of the Helmholtz Equations
Jean-David Benamou
Domaine de Voluceau, Le Chesnay, FranceFrançois Castella
Université de Rennes I, Rennes, FranceTheodoros Katsaounis
University of Crete, Iraklion, GreeceBenoît Perthame
Université Pierre et Marie Curie, Paris, France
![High Frequency limit of the Helmholtz Equations cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-rmi-volume-18-issue-1.png&w=3840&q=90)
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 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.
Cite this article
Jean-David Benamou, François Castella, Theodoros Katsaounis, Benoît Perthame, High Frequency limit of the Helmholtz Equations. Rev. Mat. Iberoam. 18 (2002), no. 1, pp. 187–209
DOI 10.4171/RMI/315