Energy concentration and Sommerfeld condition for Helmholtz and Liouville equations

Benoı̂t Perthame; Luis Vega

doi:10.1016/j.crma.2003.09.006

Partial Differential Equations

Energy concentration and Sommerfeld condition for Helmholtz and Liouville equations
[Concentration de l'énergie et condition de Sommerfeld pour les équations de Helmholtz et de Liouville]

Présenté par : Gilles Lebeau

Benoı̂t Perthame ¹ ; Luis Vega 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

Comptes Rendus. Mathématique, Volume 337 (2003) no. 9, pp. 587-592.

Résumé
Abstract

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

\frac{1}{R} \int_{| x | ⩽ R} {|\nabla u - i n_{\infty}^{1 / 2} (\frac{x}{| x |}) u \frac{x}{| x |}|}^{2} d x \to 0, lorsque R \to \infty .

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

\int_{ℝ^{d}} {|\frac{\partial}{\partial ω} n_{\infty} (ω)|}^{2} \frac{{| u |}^{2}}{| x |} d x ⩽ C, ω = \frac{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.

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

\frac{1}{R} \int_{| x | ⩽ R} {|\nabla u - i n_{\infty}^{1 / 2} (\frac{x}{| x |}) u \frac{x}{| x |}|}^{2} d x \to 0, as R \to \infty .

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

\int_{ℝ^{d}} {|\frac{\partial}{\partial ω} n_{\infty} (ω)|}^{2} \frac{{| u |}^{2}}{| x |} d x ⩽ C, ω = \frac{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.

Reçu le : 2003-08-25
Accepté le : 2003-09-15
Publié le : 2003-10-20

DOI : 10.1016/j.crma.2003.09.006

Affiliations des auteurs :

Benoı̂t Perthame ¹ ; Luis Vega 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

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     author = {Beno{\i}̂t Perthame and Luis Vega},
     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},
}

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Benoı̂t Perthame; Luis Vega. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.006/

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