Nous considérons l'équation de Helmholtz avec un indice de réfraction n(x) qui peut varier à l'infini en fonction de l'angle, n(x)→n∞(x/|x|) lorsque |x|→∞. Nous prouvons que la condition de radiation de Sommerfeld reste valable sous la forme faible
We consider the Helmholtz equation with a variable index of refraction n(x), which is not necessarily constant at infinity but can have an angular dependency like n(x)→n∞(x/|x|) as |x|→∞. We prove that the Sommerfeld condition at infinity still holds true under the weaker form
Accepté le :
Publié le :
@article{CRMATH_2003__337_9_587_0, author = {Perthame, Beno{\i}̂t and Vega, Luis}, title = {Energy concentration and {Sommerfeld} condition for {Helmholtz} and {Liouville} equations}, journal = {Comptes Rendus. Math\'ematique}, pages = {587--592}, publisher = {Elsevier}, volume = {337}, number = {9}, year = {2003}, doi = {10.1016/j.crma.2003.09.006}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2003.09.006/} }
TY - JOUR AU - Perthame, Benoı̂t AU - Vega, Luis TI - Energy concentration and Sommerfeld condition for Helmholtz and Liouville equations JO - Comptes Rendus. Mathématique PY - 2003 SP - 587 EP - 592 VL - 337 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2003.09.006/ DO - 10.1016/j.crma.2003.09.006 LA - en ID - CRMATH_2003__337_9_587_0 ER -
%0 Journal Article %A Perthame, Benoı̂t %A Vega, Luis %T Energy concentration and Sommerfeld condition for Helmholtz and Liouville equations %J Comptes Rendus. Mathématique %D 2003 %P 587-592 %V 337 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2003.09.006/ %R 10.1016/j.crma.2003.09.006 %G en %F CRMATH_2003__337_9_587_0
Perthame, Benoı̂t; Vega, Luis. Energy concentration and Sommerfeld condition for Helmholtz and Liouville equations. Comptes Rendus. Mathématique, Tome 337 (2003) no. 9, pp. 587-592. doi : 10.1016/j.crma.2003.09.006. http://www.numdam.org/articles/10.1016/j.crma.2003.09.006/
[1] Generalized Fourier transform for Schrödinger operators with potentials of order zero, J. Funct. Anal., Volume 167 (1999), pp. 345-369
[2] Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math., Volume 30 (1976), pp. 1-37
[3] High frequency limit of the Helmholtz equations, Rev. Mat. Iberoamericana (2002)
[4] Spectral and scattering for Schrödinger operators with potentials independent of |x|, Amer. J. Math., Volume 113 (1991) no. 3, pp. 509-565
[5] The Analysis of Linear Partial Differential Operators, Grundlehren Math. Wiss., 257, Springer, 1990
[6] Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Lineaire, Volume 10 (1993), pp. 255-288
[7] Morrey–Campanato estimates for Helmholtz equation, J. Funct. Anal., Volume 164 (1999) no. 2, pp. 340-355
[8] B. Perthame, L. Vega, Energy decay and Sommerfeld condition for Helmholtz equation with variable index at infinity, Preprint
[9] Sommerfeld condition for a Liouville equation and concentration of trajectories, Bull. Braz. Math. Soc. (N.S.), Volume 34 (2003) no. 1, pp. 1-15
[10] Schrödinger operators with a nonspherical radiation condition, Pacific J. Math., Volume 126 (1987) no. 2, pp. 331-359
[11] Radiation condition and limiting amplitude principle for acoustic propagators with two unbounded media, Proc. Roy. Soc. Edinburgh Sect. A, Volume 128 (1998), pp. 173-192
[12] On transmission problems for wave propagation in two locally perturbed half spaces, Math. Proc. Cambridge Philos. Soc., Volume 115 (1994), pp. 545-558
Cité par Sources :