Energy concentration and Sommerfeld condition for Helmholtz and Liouville equations

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

$\text{1}\text{R}\text{∫}\text{|x|?R}\text{?u?}\text{i}\text{n}{}_{\mathrm{\infty }}{}^{\mathrm{1/2}}\text{x}\text{|x|}\text{u}\text{x}\text{|x|}{}^2\text{d}\text{x→0,}\text{as}\text{R→∞.}$

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

$\text{∫}\text{R}{}^{\mathrm{d}}\text{?}\text{?ω}\text{n}{}_{\mathrm{\infty }}\text{(ω)}{}^2\text{|u|}{}^2\text{|x|}\text{d}\text{x?C,}\text{ω=}\text{x}\text{|x|}\text{.}$

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).