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.

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
     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},
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     year = {2003},
     doi = {10.1016/j.crma.2003.09.006},
     language = {en},
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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.

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