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Energy concentration and Sommerfeld condition for Helmholtz and Liouville equations
Authors:Benoı̂t Perthame  Luis Vega
Affiliation: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
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
1R|x|?R?u?in1/2x|x|ux|x|2dx→0,asR→∞.
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
Rd?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).
Keywords:
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