Energy concentration and explicit Sommerfeld radiation condition for the electromagnetic Helmholtz equation

We study the electromagnetic Helmholtz equation ∇ +ib(x) 2 u(x) +n(x)u(x) = f (x), x ∈ 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 dependence like n(x)→n∞( x |x| ) as |x|→∞. We prove an explicit Sommerfeld radiation condition Rd ∇ bu −in 1/2 ∞ x |x|u 2 dx 1 + |x| <+∞ for solutions obtained from the limiting absorption principle and we also give a new energy estimate Rd ∇ωn∞ x |x| 2 |u|2 1 + |x| dx <+∞, 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. © 2012 Elsevier Inc. All rights reserved.