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

2012/01/02 by Miren Zubeldia, Zubeldia, Miren
Mathematics · #35B45 #35J05 #5Q60 #78A40 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1201.0494

openalex publication_date 2012/01/02 · openalex created_date 2022/09/24 · openalex updated_date 2026/07/28

Abstract

We study the electromagnetic Helmholtz equation \(\∇ +\nib(x))²u(x) + n(x)u(x) = f(x), x\∈ Rd with the magnetic vector\npotential b(x) and n(x) a variable index of refraction that does not\nnecessarily converge to a constant at infinity, but can have an angular\ndependency like n(x) \→ n_\∞(\(x)/(|x|)) as |x|\→\∞. We\nprove an explicit Sommerfeld radiation condition ∫ _Rd | D u -\nin_\∞^1/2\(x)/(|x|)u|² \(dx)/(1+|x)) < + \∞ for solutions\nobtained from the limiting absorption principle and we also give a new energy\nestimate ∫ _Rd|\n\∇_\ωn_\∞(\(x)/(|x|))|² frac|u|²1+|x| dx <\n+\∞, which explains the main physical effect of the angular dependence of\nn at infinity and deduces that the energy concentrates in the directions\ngiven by the critical points of the potential.\n

Citations

• Effective limiting absorption principles, and applications

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