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October 2014 Radiation condition at infinity for the high-frequency Helmholtz equation: optimality of a non-refocusing criterion
François CASTELLA, Aurélien KLAK
Hokkaido Math. J. 43(3): 275-325 (October 2014). DOI: 10.14492/hokmj/1416837568

Abstract

We consider the high frequency Helmholtz equation with a variable refraction index n2(x) (x ∈ ℝd), supplemented with a given high frequency source term supported near the origin x = 0. A small absorption parameter αε > 0 is added, which prescribes a radiation condition at infinity for the considered Helmholtz equation. The semi-classical parameter is ε > 0. We let ε and αε go to zero simultaneously. We study the question whether the prescribed radiation condition at infinity is satisfied uniformly along the asymptotic process ε Ⅺ 0.

This question has been previously studied by the first autor in [4], where it is proved that the radiation condition is indeed satisfied uniformly in ε, provided the refraction index satisfies a specific non-refocusing condition. The non-refocusing condition requires, in essence, that the rays of geometric optics naturally associated with the high-frequency Helmholtz operator, and that are sent from the origin x = 0 at time t = 0, should not refocus at some later time t > 0 near the origin again.

In the present text we show the optimality of the above mentioned non-refocusing condition. We exhibit a refraction index which does refocus the rays of geometric optics sent from the origin near the origin again, and we show that the limiting solution does not satisfy the natural radiation condition at infinity in that case.

Citation

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François CASTELLA. Aurélien KLAK. "Radiation condition at infinity for the high-frequency Helmholtz equation: optimality of a non-refocusing criterion." Hokkaido Math. J. 43 (3) 275 - 325, October 2014. https://doi.org/10.14492/hokmj/1416837568

Information

Published: October 2014
First available in Project Euclid: 24 November 2014

zbMATH: 1308.35293
MathSciNet: MR3282637
Digital Object Identifier: 10.14492/hokmj/1416837568

Subjects:
Primary: 35Qxx
Secondary: 35J10, 81Q20

Rights: Copyright © 2014 Hokkaido University, Department of Mathematics

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Vol.43 • No. 3 • October 2014
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