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/

[1] S. Agmon, L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Analyse Math., 30 (1976), 1–38. | MR 466902 | Zbl 0335.35013

[2] J.A. Barcelo, A. Ruiz, L. Vega, Weighted estimates for Helmholtz equation and some applications, J. of Funct. Anal. , 150 (1997), 356–382. | MR 1479544 | Zbl 0890.35028

[3] J.-D. Benamou, Direct computation of multi-valued phase-space solutions of Hamilton-Jacobi equations, to appear in Comm. Pure and Appl. Math. | MR 1702708 | Zbl 0935.35032

[4] F. Castella , On the derivation of a quantum Boltzmann equation from the periodic Von-Neumann equation, Math. Mod. An. Num. 33, N. 2 (1999), 329–350. | Numdam | MR 1700038 | Zbl 0954.82023

[5] F. Castella, P. Degond, From the Von-Neumann equation to the Quantum Boltzmann equation in a deterministic framework, Preprint Université de Rennes 1 and C. R. Acad. Sci., t. 329, sér. I (1999), 231–236. | MR 1711066 | Zbl 0930.35146

[6] F. Castella, B. Perthame, O. Runborg, High frequency limit in the Helmholtz equation: the case of a general source, in preparation.

[7] L. Erdös, H.T. Yau, Linear Boltzmann equation as scaling limit of the quantum Lorentz gas, Preprint (1998). | MR 1605282

[8] I. Gasser, P. Markowich, B. Perthame, Dispersion and moments lemma revisited, to appear in J. Diff. Eq. | Zbl 0931.35135

[9] P. Gérard, Microlocal defect measures, Comm. Partial Diff. Equations 16 (1991), 1761–1794. | MR 1135919 | Zbl 0770.35001

[10] P. Gérard, P.A. Markowich, N.J. Mauser, F. Poupaud, Homogeneisation limits and Wigner transforms, Comm. pure and Appl. Math., 50 (1997), 321–357. | MR 1438151 | Zbl 0881.35099

[11] J.B. Keller, R. Lewis, Asymptotic methods for partial differential equations: the reduced wave equation and Maxwell’s equation, in Surveys in applied mathematics, eds J.B. Keller, D.McLaughlin and G. Papanicolaou, Plenum Press, New York, 1995. | Zbl 0848.35068

[12] C. Kenig, G. Ponce, L. Vega, Small solutions to nonlinear Schroedinger equations, Annales de l’I.H.P., 10, (1993), 255–288. | Numdam | MR 1230709 | Zbl 0786.35121

[13] P.-L. Lions, T. Paul, Sur les mesures de Wigner, Revista Matemática Iberoamericana, 9 (3) (1993), 553–618. | MR 1251718 | Zbl 0801.35117

[14] P.L.Lions, B.Perthame, Lemmes de moments, de moyenne et de dispersion. C. R. Acad. Sc t.314 (série I) (1992), 801–806. | MR 1166050 | Zbl 0761.35085

[15] G. Papanicolaou, L. Ryzhik, Waves and Transport. IAS/ Park City Mathematics series. Volume 5 (1997). | MR 1662832 | Zbl 0930.35172

[16] B. Perthame, L. Vega, Morrey-Campanato estimates for Helmholtz equations. J. Funct. Anal. 164(2) (1999), 340–355. | MR 1695559 | Zbl 0932.35048

[17] B. Perthame, L. Vega, work under progress.

[18] L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Ed., 115 A (1990), 193–230. | MR 1069518 | Zbl 0774.35008

[19] Bo Zhang, Radiation condition and limiting amplitude principle for acoustic propagators with two unbounded media, Proc. Roy. Soc. Ed., 128 A (1998), 173–192. | MR 1606369 | Zbl 0897.35044