Mesures limites pour l’équation de Helmholtz dans le cas non captif

Bony, Jean-François

doi:10.5802/afst.1210

Cited by 4 publications

(12 citation statements)

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“…This kind of condition already appears in [19]. This is the assumption we are going to use here and, as a consequence, even if we can show that the outgoing solution u h of (1.1) is microlocally zero in the incoming region, the contribution of large times in u h does not vanish when h → 0 as is the case in [4]. In particular the solution can be an infinite sum of lagrangian distributions around some points of the phase space.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 85%

“…15) In [4,Section 4] is given an example of what can happen without a hypothesis of this kind. Note that when Γ = {0}, this assumption is weaker than the assumption ν 0 2] which is used for instance in [29].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…Here we are going to use the point of view of J.-F. Bony [4]. He considers the case of a source which concentrates on one or two points (with V 1 = 0) using a time-dependant method based on a W.K.B.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…We recall that for z ∈ Γ and ζ ∈ T z Γ of norm less than γ 1 , exp z (ζ ) is well defined on Γ 1 . For δ and τ 0 small enough, if x ∈Γ (]δτ 0 , 2δτ 0 ]), t ∈ [0, τ 0 ], and d Γ (z, z x ) δ then |x − z| δ 2 and |x − (z + 2tξ )| δ 4 . As a result we can do partial integrations with L as before and see that modulo O ((h/δ) N ), J 0 δ (x) is given by integration over z in a neighborhood of radius δ around z x :…”

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confidence: 99%

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Semiclassical measure for the solution of the dissipative Helmholtz equation

Royer¹

2010

Journal of Differential Equations

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International audienceWe study the semiclassical measures for the solution of a dissipative Helmholtz equation with a source term concentrated on a bounded submanifold. The potential is not assumed to be non-trapping, but trapped trajectories have to go through the region where the absorption coefficient is positive. In that case, the solution is microlocally written around any point away from the source as a sum (finite or infinite) of lagragian distributions. Moreover we prove and use the fact that the outgoing solution of the dissipative Helmholtz equation is microlocally zero in the incoming region

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 85%

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

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Semiclassical measure for the solution of the dissipative Helmholtz equation

Royer¹

2010

Journal of Differential Equations

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“…The reader may safely skip this fact, since the Helmholtz equation also arises in the spectral analysis of Schrödinger operators, where the refraction index becomes E − V (x) where E is an energy and V (x) is a space-dependent potential, and the term E − V (x) may change sign in that context. 2 The limiting case αε = 0 + can be considered along our analysis, see below.…”

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confidence: 99%

Radiation condition at infinity for the high-frequency Helmholtz equation: optimality of a non-refocusing criterion

Castella¹,

Klak²

2014

Hokkaido Math. J.

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We consider the high frequency Helmholtz equation with a variable refraction index n 2 (x) (x ∈ R d ), supplemented with a given high frequency source term supported near the origin x = 0. A small absorption parameter αε > 0 is added, which somehow 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 simultaneaously. We study the question whether the indirectly prescribed radiation condition at infinity is satisfied uniformly along the asymptotic process ε → 0, or, in other words, whether the conveniently rescaled solution to the considered equation goes to the outgoing solution to the natural limiting Helmholtz equation.This question has been previously studied by the first autor in [4]. In [4], it is proved that the radiation condition is indeed satisfied uniformly in ε, provided the refraction index satisfies a specific non-refocusing condition, a condition that is first pointed out in this reference. 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 mentionned non-refocusing condition, in the following sense. We exhibit a refraction index which does refocus the rays of geometric optics sent from the origin near the origin again, and, on the other hand, we completely compute the asymptotic behaviour of the solution to the associated Helmholtz equation: we show that the limiting solution does not satisfy the natural radiation condition at infinity. More precisely, we show that the limiting solution is a perturbation of the outgoing solution to the natural limiting Helmholtz equation, and that the perturbing term explicitly involves the contribution of the rays radiated from the origin which go back to the origin. This term is also conveniently modulated by a phase factor, which turns out to be the action along the above rays of the hamiltonian associated with the semiclassical Helmholtz equation. (2000): Primary 35QXX, Secondary 35J10, 81Q20 Mathematics subject classification1 Here and below we use the standard notation n 2 (x), a squared term, assuming in doing so that the corresponding term is everywhere non-negative. This is a harmless abuse of notation, since the refraction index n 2 (x) that is eventually chosen in our analysis is negative for certain values of x. The reader may safely skip this fact, since the Helmholtz equation also arises in the spectral analysis of Schrödinger operators, where the refraction index becomes E − V (x) where E is an energy and V (x) is a space-dependent potential, and the term E − V (x) may change sign in that context. 2 The limiting case αε = 0 + can be considered along our analysis, see below.

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Uniform resolvent estimates for a non-dissipative Helmholtz equation

Royer¹

2014

Bul. Soc. Math. France

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We study the high frequency limit for a non-dissipative Helmholtz equation. We first prove the absence of eigenvalue on the upper half-plane and close to an energy which satisfies a weak damping assumption on trapped trajectories. Then we generalize to this setting the resolvent estimates of Robert-Tamura and prove the limiting absorption principle. We finally study the semiclassical measures of the solution when the source term concentrates on a bounded submanifold of R n .

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Mesures limites pour l’équation de Helmholtz dans le cas non captif

Cited by 4 publications

References 20 publications

Semiclassical measure for the solution of the dissipative Helmholtz equation

Semiclassical measure for the solution of the dissipative Helmholtz equation

Radiation condition at infinity for the high-frequency Helmholtz equation: optimality of a non-refocusing criterion

Uniform resolvent estimates for a non-dissipative Helmholtz equation

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