Abstract

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|⩽R ∇u− in ∞ 1/2 x |x| u x |x| 2 dx→0, 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 ∫ R d ∂ ∂ω n ∞(ω) 2 |u| 2 |x| dx⩽C, 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. To cite this article: B. Perthame, L. Vega, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

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