High Frequency limit of the Helmholtz Equations
Séminaire Équations aux dérivées partielles (Polytechnique), (1999-2000), Talk no. 5, 25 p.

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.

@article{SEDP_1999-2000____A5_0,
     author = {Benamou, Jean-David and Castella, Fran\c cois and Katsaounis, Thodoros and Perthame, Beno\^\i t},
     title = {High Frequency limit of the Helmholtz Equations},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {1999-2000},
     note = {talk:5},
     mrnumber = {1813168},
     zbl = {02124202},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_1999-2000____A5_0}
}
Benamou, Jean-David; Castella, François; Katsaounis, Thodoros; Perthame, Benoît. High Frequency limit of the Helmholtz Equations. Séminaire Équations aux dérivées partielles (Polytechnique),  (1999-2000), Talk no. 5, 25 p. http://www.numdam.org/item/SEDP_1999-2000____A5_0/

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