- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:43
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
18
2002
1
High Frequency limit of the Helmholtz Equations
Jean-David
Benamou
Domaine de Voluceau, LE CHESNAY CEDEX, FRANCE
François
Castella
Université de Rennes I, RENNES CEDEX, FRANCE
Theodoros
Katsaounis
University of Crete, IRAKLION, GREECE
Benoît
Perthame
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Helmholtz equations, high frecuency, transport equations, geometrical optics
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.
Partial differential equations
Optics, electromagnetic theory
Quantum theory
General
187
209
10.4171/RMI/315
http://www.ems-ph.org/doi/10.4171/RMI/315