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

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

2012-01-02

arxiv.org/abs/1201.0494v1

Mathematics

Analysis of PDEs

Scientific paper

We study the electromagnetic Helmholtz equation \notag (\nabla + ib(x))^{2}u(x) + n(x)u(x) + Q(x)u(x) = f(x), \quad x\in\Rd with a variable index of refraction $n(x)=\lambda(1+\p(x))$ that satisfies the eikonal equation $|\nabla\varphi|^{2}=\frac{1}{\lambda}n(x)$ and 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 an 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.

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