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.
@article{JEDP_2006____A4_0, author = {Fouassier, Elise}, title = {High frequency limit of Helmholtz equations: the case of a~discontinuous index}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, year = {2006}, pages = {1-19}, doi = {10.5802/jedp.31}, language = {en}, url = {http://dml.mathdoc.fr/item/JEDP_2006____A4_0} }
Fouassier, Elise. High frequency limit of Helmholtz equations: the case of a discontinuous index. Journées équations aux dérivées partielles, (2006), pp. 1-19. doi : 10.5802/jedp.31. http://gdmltest.u-ga.fr/item/JEDP_2006____A4_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.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 -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)