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

 $\frac{1}{R}\underset{|\mathrm{x}|⩽\mathrm{R}}{\int }{\left|\nabla \mathrm{u}-\mathrm{i}{n}_{\infty }^{1/2}\left(\frac{x}{|\mathrm{x}|}\right)u\phantom{\rule{1.69998pt}{0ex}}\frac{x}{|\mathrm{x}|}\right|}^{2}\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}\mathrm{x}\to 0,\phantom{\rule{10.0pt}{0ex}}\mathrm{as}\phantom{\rule{3.30002pt}{0ex}}\mathrm{R}\to \infty .$
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
 $\underset{{ℝ}^{d}}{\int }{\left|\frac{\partial }{\partial \omega }{n}_{\infty }\left(\omega \right)\right|}^{2}\frac{{|\mathrm{u}|}^{2}}{|\mathrm{x}|}\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}\mathrm{x}⩽\mathrm{C},\phantom{\rule{10.0pt}{0ex}}\omega =\frac{x}{|\mathrm{x}|}.$
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

 $\frac{1}{R}\underset{|\mathrm{x}|⩽\mathrm{R}}{\int }{\left|\nabla \mathrm{u}-\mathrm{i}{n}_{\infty }^{1/2}\left(\frac{x}{|\mathrm{x}|}\right)u\phantom{\rule{1.69998pt}{0ex}}\frac{x}{|\mathrm{x}|}\right|}^{2}\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}\mathrm{x}\to 0,\phantom{\rule{10.0pt}{0ex}}\mathrm{lorsque}\phantom{\rule{3.30002pt}{0ex}}\mathrm{R}\to \infty .$
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
 $\underset{{ℝ}^{d}}{\int }{\left|\frac{\partial }{\partial \omega }{n}_{\infty }\left(\omega \right)\right|}^{2}\phantom{\rule{1.69998pt}{0ex}}\frac{{|\mathrm{u}|}^{2}}{|\mathrm{x}|}\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}\mathrm{x}⩽\mathrm{C},\phantom{\rule{10.0pt}{0ex}}\omega =\frac{x}{|\mathrm{x}|}.$
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.

Accepted:
Published online:
DOI: 10.1016/j.crma.2003.09.006

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},
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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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