Energy concentration and Sommerfeld condition for Helmholtz equation with variable index at infinity

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 $n(x)\to n\_\infty(x/|x |)$ as $|x |\to \infty$. Under some appropriate assumptions on this convergence and on $n\_\infty$ we prove that the Sommerfeld condition at infinity still holds true under the explicit form $$ \int\_{\R^d} | \nabla u -i n\_\infty^{1/2} u \xox |^2 \f{dx}{|x |}<+\infty. $$ It is a very striking and unexpected feature that the index $n\_{\infty}$ appears in this formula and not the gradient of the phase as established by Saito in \cite {S} and broadly used numerically. This apparent contradiction is clarified by the existence of some extra estimates on the energy decay. In particular we prove that $$ \int\_{\R^d} | \nabla\_\omega n\_\infty(\xox)|^2 \f{| u |^2}{|x |} dx < +\infty. $$ In fact our main contribution is to show that this can be interpreted as a concentration of the energy along the critical lines of $n\_\infty$. In other words, the Sommerfeld condition hides the main physical effect arising for a variable $n$ at infinity; energy concentration on lines rather than dispersion in all directions.