High frequency limit of Helmholtz equations: the case of a discontinuous index
Journées équations aux dérivées partielles (2006), Exposé no. 4, 19 p.

In this text, we compute the high frequency limit of the Hemholtz equation with source term, in the case of a refraction index that is discontinuous along a sharp interface between two unbounded media. The asymptotic propagation of energy is studied using Wigner measures.

Publié le :
DOI : https://doi.org/10.5802/jedp.31 
@article{JEDP_2006____A4_0,
     author = {Elise Fouassier},
     title = {High frequency limit of {Helmholtz} equations: the case of a~discontinuous index},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     note = {talk:4},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2006},
     doi = {10.5802/jedp.31},
     language = {en},
     url = {https://jedp.centre-mersenne.org/articles/10.5802/jedp.31/}
}
Elise Fouassier. High frequency limit of Helmholtz equations: the case of a discontinuous index. Journées équations aux dérivées partielles (2006), Exposé no. 4, 19 p. doi : 10.5802/jedp.31. https://jedp.centre-mersenne.org/articles/10.5802/jedp.31/

[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.D. Benamou, F.Castella, T. Katsaounis, B. Perthame, High frequency limit of the Helmholtz equation, Rev. Mat. Iberoamericana 18 (2002), no. 1, 187–209. | MR 1924691 | Zbl 1090.35165

[3] F. Castella, B. Perthame, O. Runborg, High frequency limit of the Helmholtz equation. Source on a general manifold, Comm. P.D.E 3-4 (2002), 607-651. | MR 1900556 | Zbl 01786302

[4] F. Castella, The radiation condition at infinity for the high frequency Helmholtz equation with source term: a wave packet approach, J. Funct. Anal. 223 (2005), no.1, 204-257. | MR 2139886 | Zbl 1072.35159

[5] J. Dereziński, C. Gérard, Scattering theory of classical and quantum N-particle systems, Texts and Monographs in Physics, Springer, Berlin, 1997. | MR 1459161 | Zbl 0899.47007

[6] M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semiclassical limit, London Mathematical Society Lecture Notes Series, vol. 268, Cambridge University Press, Cambridge, 1999. | MR 1735654 | Zbl 0926.35002

[7] E. Fouassier, Morrey-Campanato estimates for Helmholtz equations with two unbounded media, Proc. Roy. Soc. Edinburg Sect. A 135 (2005), no.4, 767-776. | MR 2173338 | Zbl 02242736

[8] E. Fouassier, High frequency analysis of Helmholtz equations: refraction by sharp interfaces, to appear in Journal de Mathématiques Pures et APpliquées.

[9] P. Gérard, Mesures semi-classiques et ondes de Bloch, In Séminaire Equations aux dérivées partielles 1988-1989, exp XVI, Ecole Polytechnique, Palaiseau (1988). | Numdam | MR 1131589 | Zbl 0739.35096

[10] P. Gérard, E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J., 71 (1993), 559-607. | MR 1233448 | Zbl 0788.35103

[11] P. Gérard, P.A. Markowitch, N.J. Mauser, F. Poupaud, Homogeneisation limits and Wigner transforms, Comm. pure and Appl. Math., 50 (1997), 321-357. | Zbl 0881.35099

[12] C. Gérard, A. Martinez, Principe d’absorption limite pour des opérateurs de Schrödingerà longue portée, C. R. Acad. Sci. Paris, Ser. I math, Vol 195, 3, 121-123 (1988). | Zbl 0672.35013

[13] L. Hörmander,The Analysis of Linear Partial Differential Operators I and III, Springer-Verlag. | Zbl 0601.35001

[14] L. Hörmander, Lecture Notes at the Nordic Summer School of mathematics (1968).

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

[16] L. Miller, Propagation d’ondes semi-classiques à travers une interface et mesures 2-microlocales, Doctorat de l’Ecole Polytechnique, Palaiseau (1996).

[17] L. Miller, Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary, J. Math. Pures Appl. (9) 79 (2000), 227–269. | MR 1750924 | Zbl 0963.35022

[18] B. Perthame, L. Vega, Morrey-campanato estimates for the Helmholtz equation, J. Funct. Anal. 164(2) (1999), 340-355. | MR 1695559 | Zbl 0932.35048

[19] X.P. Wang, P. Zhang, High frequency limit of the Helmholtz equation with variable index of refraction, Preprint (2004) | Zbl 05003033

[20] E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932)