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 language | English (US) |
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Pages (from-to) | 187-209 |
Number of pages | 23 |
Journal | Revista Matematica Iberoamericana |
Volume | 18 |
Issue number | 1 |
DOIs | |
State | Published - 2002 |
Externally published | Yes |
Keywords
- Geometrical optics
- Helmholtz equations
- High frecuency
- Transport equations
ASJC Scopus subject areas
- General Mathematics