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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 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
@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},
zbl = {02124202},
mrnumber = {1813168},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SEDP_1999-2000____A5_0/}
}
TY - JOUR AU - Benamou, Jean-David AU - Castella, François AU - Katsaounis, Thodoros AU - Perthame, Benoît TI - High Frequency limit of the Helmholtz Equations JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:5 PY - 1999-2000 SP - 1 EP - 25 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://geodesic.mathdoc.fr/item/SEDP_1999-2000____A5_0/ LA - en ID - SEDP_1999-2000____A5_0 ER -
%0 Journal Article %A Benamou, Jean-David %A Castella, François %A Katsaounis, Thodoros %A Perthame, Benoît %T High Frequency limit of the Helmholtz Equations %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:5 %D 1999-2000 %P 1-25 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://geodesic.mathdoc.fr/item/SEDP_1999-2000____A5_0/ %G en %F 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) dit aussi "Séminaire Goulaouic-Schwartz" (1999-2000), Exposé no. 5, 25 p. http://geodesic.mathdoc.fr/item/SEDP_1999-2000____A5_0/
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