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

Abstract

We study the electromagnetic Helmholtz equationwith 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 conditionfor solutions obtained from the limiting absorption principle and we also give a new energy estimatewhich 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.

The forward problem for the electromagnetic Helmholtz equation with critical singularities

“…See also Remark 5.4 below in connection with a recent work by Rodnianski and Tao [39] where estimates valid for all λ are obtained. In addition, it is to be expected that the decay of the electric potential V required on assumption (H3) can be relaxed modifying the phase of the Sommerfeld radiation condition using the solution of the eikonal equation |∇K| 2 = λ + V , as done by Saito [40] and Zubeldia [41]. We propose this as an interesting question which will be studied elsewhere.…”

Uniform resolvent estimates and absence of eigenvalues for Lamé operators with complex potentials

“…This tool was introduced in [24] for the Klein-Gordon equation and then it was developed in several other contexts. For example, about the Helmholtz equation's framework, let us mention the seminal works of Perthame and Vega [25], [26] which are concerned with the purely electric case and then [16,14,32,2,5,33], which extend the technique in an electromagnetic setting. We should also quote [9] for an adaptation of multipliers method on exterior domains.…”

A computational method for the Helmholtz equation in unbounded domains based on the minimization of an integral functional

“…There are other conditions of this type available in literature; we shall not discuss about them here and refer to [29], [32], [36] and references therein.…”