Partial Differential Equations
Energy concentration and Sommerfeld condition for Helmholtz and Liouville equations
Comptes Rendus. Mathématique, Volume 337 (2003) no. 9, pp. 587-592.

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

1 R |x|Ru-in 1/2 x |x|ux |x| 2 dx0, as R.
Our approach consists in proving this estimate in the framework of the limiting absorbtion principle. We use Morrey–Campanato type of estimates and a new inequality on the energy decay, namely
d ωn (ω) 2 |u| 2 |x|dxC,ω=x |x|.
It is a striking feature that the index n appears in this formula and not the phase gradient, in apparent contradiction with existing literature.

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

1 R |x|Ru-in 1/2 x |x|ux |x| 2 dx0, lorsque R.
Nous démontrons cette estimation via le principe d'absorbtion limite. Nous utilisons des inégalités de type Morrey–Campanato et une nouvelle estimation a priori sur le contrôle de l'énergie à l'infini
d ωn (ω) 2 |u| 2 |x|dxC,ω=x |x|.
Le point surprenant de la condition de Sommerfeld ci-dessus est que l'indice n y apparaı̂t et non le gradient de la phase, ce qui contredit apparamment la littérature existante.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.09.006
Perthame, Benoı̂t 1; Vega, Luis 2

1 Département de mathématiques et applications, UMR 8553, École normale supérieure, 45, rue d'Ulm, 75230 Paris cedex 05, France
2 Universidad del Pais Vasco, Apdo. 644, 48080 Bilbao, Spain
@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, Volume 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] Agmon, S.; Cruz-Sampedro, J.; Herbst, I. Generalized Fourier transform for Schrödinger operators with potentials of order zero, J. Funct. Anal., Volume 167 (1999), pp. 345-369

[2] Agmon, S.; Hörmander, L. Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math., Volume 30 (1976), pp. 1-37

[3] Benamou, J.D.; Castella, F.; Katsaounis, T.; Perthame, B. High frequency limit of the Helmholtz equations, Rev. Mat. Iberoamericana (2002)

[4] Herbst, I. Spectral and scattering for Schrödinger operators with potentials independent of |x|, Amer. J. Math., Volume 113 (1991) no. 3, pp. 509-565

[5] Hörmander, L. The Analysis of Linear Partial Differential Operators, Grundlehren Math. Wiss., 257, Springer, 1990

[6] Kenig, C.; Ponce, G.; Vega, L. Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Lineaire, Volume 10 (1993), pp. 255-288

[7] Perthame, B.; Vega, L. 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] Perthame, B.; Vega, L. 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] Saito, Y. Schrödinger operators with a nonspherical radiation condition, Pacific J. Math., Volume 126 (1987) no. 2, pp. 331-359

[11] Zhang, Bo 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] Zhang, Bo On transmission problems for wave propagation in two locally perturbed half spaces, Math. Proc. Cambridge Philos. Soc., Volume 115 (1994), pp. 545-558

Cited by Sources: