High Frequency limit of the Helmholtz Equations
Benamou, Jean-David1 ; Castella, François2 ; Katsaounis, Thodoros3 ; Perthame, Benoît4
1 INRIA-Rocquencourt, BP 105, 78153 Le Chesnay, France
2 CNRS et IRMAR, Campus de Beaulieu, Université de Rennes 1, 35042 Rennes Cédex, France
3 IACM, FORTH, P.O. Box 1527, Vassilika Boutwn 71110, Heraklion Crete, Greece
4 ENS, DMA, 45, rue d’Ulm, 75230 Paris, France
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1999-2000), Exposé no. 5, 25 p.

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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.

Benamou, Jean-David 1 ; Castella, François 2 ; Katsaounis, Thodoros 3 ; Perthame, Benoît 4

1 INRIA-Rocquencourt, BP 105, 78153 Le Chesnay, France
2 CNRS et IRMAR, Campus de Beaulieu, Université de Rennes 1, 35042 Rennes Cédex, France
3 IACM, FORTH, P.O. Box 1527, Vassilika Boutwn 71110, Heraklion Crete, Greece
4 ENS, DMA, 45, rue d’Ulm, 75230 Paris, France 
@article{SEDP_1999-2000____A5_0,
     author = {Benamou, Jean-David and Castella, Fran\c{c}ois 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) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:5},
     pages = {1--25},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {1999-2000},
     mrnumber = {1813168},
     zbl = {02124202},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/SEDP_1999-2000____A5_0/}
}
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Benamou, Jean-David; Castella, François; Katsaounis, Thodoros; Perthame, Benoît. High Frequency limit of the Helmholtz Equations. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1999-2000), Exposé no. 5, 25 p. https://geodesic-test.mathdoc.fr/item/SEDP_1999-2000____A5_0/