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

Abstract

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_ωn_{\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.