Sommerfeld condition for a liouville equation and concentration of trajectories

Bulletin of the Brazilian Mathematical Society

Volumn 34, Issue 1, 2003, Pages 43-57

  Perthame, Benoît ^a   Vega, Luis ^b  

^a ECOLE NORMALE SUPÉRIEURE   (France)

^b UNIVERSITY OF THE BASQUE COUNTRY UPV EHU   (Spain)

Author keywords

Geometrical optics; Helmholtz equation; Liouville equation; Sommerfeld conditions

Indexed keywords

EID: 0038648304     PISSN: 01003569     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00574-003-0002-7     Document Type: Article

References (20)

1. S. Agmon, J. Cruz-Sampedro and I. Herbst, Generalized Fourier transform for Schrödinger operators with potentials of order zero, J. of Funct. Anal., 167 (1999), 345-369.
2. S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math. 30 (1976), 1-37.
3. J.D. Benamou, F. Castella, T. Katsaounis and B. Perthame, High frequency limit of the Helmholtz equations, Rev. IberoAmer. (2002).
4. F. Castella, B. Perthame and O. Runborg, High frequency limit of the Helmholtz equation. Source on a general manifold, Comm. P.D.E. 27 n. 3-4 (2002), 607-651.
5. R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98(3) (1989), 511-547.
6. D. M. Eidus, The principle of limiting absorption, Math. Sb., 57 (1962), 13-44. Amer. Math. Soc. Transl. (2) 47 (1965), 157-191.
7. D. M. Eidus, The principle of limiting absorption, Math. Sb., 57 (1962), 13-44. Amer. Math. Soc. Transl. (2) 47 (1965), 157-191.
8. D. M. Eidus, The limiting absorption and amplitud principles for the diffraction problem with two unbounded media, Comm. Math. Phys. 107 (1986), 29-38.
9. L. Erdös and H. T. Yau, Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation, Comm. Pure Appl. Math. 53 (2000), 667-735.
10. I. Herbst, Spectral and scattering theory for Schrödinger operators with potentials independent of |x|, Amer. J. Math. 113(3) (1991), 509-565.
11. T. Ikebe and Y. Saito, Limiting absorption method and aboslut continuity for the Schrödinger operator, J. Math. Kyoto Univ. 12-3 (1972), 512-542.
12. C. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincare Anal. Non Lineaire 10 (1993), 255-288.
13. P.L. Lions and B. Perthame, Lemmes de moments, de moyenne et de dispersion, C. R. Acad. Sci. Paris, Série 1314 (1992), 801-806.
14. B. Perthame and L. Vega, Morrey-Campanato estimates for Helmholtz Equation, J. Funct. Anal. 164(2) (1999), 340-355.
15. B. Perthame and L. Vega, Energy decay and Sommerfeld condition for Helmholtz equation with variable index at infinity, preprint (2002).
16. F. Poupaud and A. Vasseur Classical and quantum transport in random media, preprint (2001).
17. M. Reed and B. Simon, Analysis of Operators. Methods of Modern Mathematical Physics IV, Acad. Press, San Diego, 1978.
18. L. Ryzhik, G. Papanicolaou and J. Keller, Energy transport for elastic and other waves in a random medium Wave motion 24 (1996), 327-370.
19. Y. Saito, Schrödinger operators with a nonspherical radiation condition, Pacif. J. of Math. 126(2) (1987), 331-359.
20. Bo Zhang, Radiation condition and limiting amplitude principle for acoustic propagators with two unbounded media, Proc. of the Royal Soc. of Edinburgh, 128 A (1998), 173-192.