High Frequency limit of Helmholtz Equations
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Autores: Jean David Benamou, François Castella, Benoît Perthame
, Theodoros Katsaounis
- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 18, Nº 1, 2002, págs. 187-210
- Idioma: inglés
- DOI: 10.4171/rmi/315
- Títulos paralelos:
- Límite de alta frecuencia de las ecuaciones de Helmholtz.
- Enlaces
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Resumen
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. (A)
