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.
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Jean-David Benamou, François Castella, Theodoros Katsaounis, Benoît Perthame, High Frequency limit of the Helmholtz Equations. Rev. Mat. Iberoam. 18 (2002), no. 1, pp. 187–209DOI 10.4171/RMI/315