We study the electromagnetic Helmholtz equation \notag (\nabla + ib(x))^{ 2}u(x) + n(x)u(x) = f(x), \quad x\in\Rd with the magnetic vector potential $b(x)$ and $n(x)$ a variable index of refraction that does not necessarily converge to a constant at infinity, but can have an angular dependency like $n(x) \to n_{ \infty}(\frac{ x}{ |x|})$ as $|x|\to\infty$. We prove an explicit Sommerfeld radiation condition \notag \int_{ \Rd} |\D u - in_{ \infty}^{ 1/2}\frac{ x}{ |x|}u|^{ 2} \frac{ dx}{ 1+|x)} < + \infty for solutions obtained from the limiting absorption principle and we also give a new energy estimate \notag \int_{ \Rd}| \nabla_{ \omega}n_{ \infty}(\frac{ x}{ |x|})|^{ 2}\frac{ |u|^{ 2}}{ 1+|x|} dx < +\infty, which explains the main physical effect of the angular dependence of $n$ at infinity and deduces that the energy concentrates in the directions given by the critical points of the potential.
Categories: math.AP