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Comptes Rendus Mathématique
Volume 355, n° 7
pages 774-779 (juillet 2017)
Doi : 10.1016/j.crma.2017.05.011
Received : 3 Mars 2017 ;  accepted : 30 May 2017
Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples
Estimations de dispersion pour l'équation des ondes et de Schrödinger à l'extérieur des obstacles strictement convexes et contre-exemples
 

Oana Ivanovici a , Gilles Lebeau b
a CNRS et Université Côte d'Azur, Laboratoire J.-A.-Dieudonné, UMR CNRS 7351, parc Valrose, 06108 Nice cedex 02, France 
b Université Côte d'Azur, Laboratoire J.-A.-Dieudonné, UMR CNRS 7351, parc Valrose, 06108 Nice cedex 02, France 

Abstract

The purpose of this note is to prove dispersive estimates for the wave and the Schrödinger equations outside strictly convex obstacles in  . If  , we show that, for both equations, the linear flow satisfies the (corresponding) dispersive estimates as in  . In higher dimensions   and if the domain is the exterior of a ball in  , we show that losses in dispersion do appear and this happens at the Poisson spot.

The full text of this article is available in PDF format.
Résumé

L'objet de cette note est de démontrer des estimations de dispersion pour l'équation des ondes et de Schrödinger à l'extérieur d'un obstacle strictement convexe de  . Si  , on démontre que, pour chacune des deux équations, le flot linéaire vérifie les estimations de dispersion comme dans  . En dimension plus grande  , on démontre que des pertes dans la dispersion apparaissent à l'extérieur d'une boule de   et que cela arrive au point de Poisson.

The full text of this article is available in PDF format.
Version française abrégée

Pour l'équation des ondes, dans le cas euclidien, la forme explicite du flot permet d'obtenir les estimations de dispersion
‖χ(hDt)e±it−ΔRd‖L1(Rd)→L∞(Rd)≤C(d)h−dmin⁡{1,(h/|t|)d−12},χ∈C0∞(]0,∞[). Pour l'équation de Schrödinger, les estimations de dispersion s'énoncent comme suit :‖e±itΔRd‖L1(Rd)→L∞(Rd)≤C(d)|t|−d/2.

Notre but est d'obtenir des estimations de dispersion à l'extérieur d'un obstacle strictement convexe. Plusieurs résultats positifs sur les effets dispersifs ont été obtenus récemment dans ce contexte : cependant, la question de savoir si les estimations de dispersion étaient vraies ou non est restée ouverte, même à l'extérieur d'une boule. Puisqu'il n'y a pas de concentration apparente d'énergie, comme dans le cas d'un domaine non captant quelconque (pour lequel les portions concaves du bord peuvent agir comme des miroirs et re-focaliser les paquets d'ondes), on pourrait raisonnablement penser que les estimations de dispersion devraient être vérifiées à l'extérieur d'un convexe (voire l'extérieur d'une balle [[2]] dans le cas des fonctions à symétrie spherique). On montre ici que c'est effectivement le cas en dimension 3 ; en revanche, en dimension plus grande, on construit des contre-exemples explicites à l'extérieur d'une boule.

Théorème 1

Soit   un domaine compact avec bord régulier, strictement convexe et soit  . Soit Δ le Laplacien dans Ω avec condition de Dirichlet au bord. Alors

(i)
les estimations de dispersion pour le propagateur des ondes dans Ω sont vérifiées comme dans   :
‖χ(hDt)e±it−Δ‖L1(Ω)→L∞(Ω)≤Ch−3min⁡{1,h|t|}.
(ii)
les estimations de dispersion pour le flot de Schrödinger dans Ω sont vérifiées comme dans  .

On remarque qu'une perte dans la dispersion pourrait être liée (de façon informelle) à la présence d'un point de concentration : ces points apparaissent lorsque des rayons optiques (envoyés d'une même source dans des directions différentes) cessent de diverger. Le principe de Huygens énonce que, lorsque la lumière éclaire un obstacle circulaire, chaque point de l'obstacle se comporte à son tour comme une nouvelle source lumineuse ponctuelle ; tous les rayons lumineux issus des points de la circonférence de l'obstacle se concentrent au centre de l'ombre et décrivent le même chemin optique ; il en résulte une tache lumineuse au centre de l'ombre (le point de Poisson). Par conséquent, l'intuition nous dit que s'il y a une perte dans la dispersion, elle devrait apparaître au point de Poisson.

Théorème 2

Pour  , on pose   la boule unité de  . Soit   et soit   le Laplacien dans   avec condition de Dirichlet. Au point de Poisson, les estimations de dispersion précédentes (où   et   sont remplacés par   et  ) ne sont plus vérifiées. Precisément, soient   les points source et d'observation situés à distance r du centre O de la boule  , symétriques par rapport à O ; alors, si  , avec γ à valeurs dans un compact de  ,

pour le propagateur des ondes et pour  
|(χ(hDt)eit|Δ|(δQ+(γh−1/3))|(Q−(γh−1/3))≃h−d(ht)d−12h−d−33,
pour le propagateur de Schrödinger classique et pour  
|(χ(hDt)eitΔ(δQ+(γh−1/6))|(Q−(γh−1/6))≃h−d6−d−36.
Pour  , ces estimations contredisent les estimations du cas plat  .

Introduction

In the Euclidean case, the explicit form of the wave propagator yields the following dispersive estimate
(1)‖χ(hDt)e±it−ΔRd‖L1(Rd)→L∞(Rd)≤C(d)h−dmin⁡{1,(h/|t|)d−12},χ∈C0∞(]0,∞[). Concerning the Schrödinger equation, the dispersive estimates read as follows:(2)‖e±itΔRd‖L1(Rd)→L∞(Rd)≤C(d)|t|−d/2.

Our aim in the present paper is to obtain dispersive estimates outside strictly convex obstacles. While many positive results on dispersive effects had been established lately in this context, the question about whether or not dispersion did hold remained open, even for the exterior of a ball. Since there is no apparent concentration of energy, like in the case of a generic non-trapping obstacle (where concave portions of the boundary can act as mirrors and refocus wave packets), one would expect dispersive estimates to hold outside strictly convex obstacles (see the exterior of a ball [[2]] for spherically symmetric functions). We prove that this is indeed the case in dimension three, while in higher dimensions we provide explicit counterexamples for the exterior of a ball.

Theorem 3

Let   be a compact domain with smooth, strictly convex boundary and let  . Let Δ denote the Dirichlet Laplace operator in Ω. Then

(i)
the dispersive estimates for the wave flow in Ω do hold like in  :
(3)‖χ(hDt)e±it−Δ‖L1(Ω)→L∞(Ω)≤Ch−3min⁡{1,h|t|},
(ii)
the dispersive estimates for the classical Schrödinger flow in Ω hold like in  .

We remark that a loss in dispersion may be informally related to a cluster point: such clusters occur because optical rays (sent from the same source along different directions) are no longer diverging from each other. When light shines on a circular obstacle, Huygens's principle says that every point of the obstacle acts as a new point source of light, so all the light passing close to a perfectly circular object concentrate at the perfect center of the shadow behind it; this results in a bright spot at the shadow's center (the Poisson spot). Therefore, our intuition tells us that, if there is a location where dispersion could fail, this will happen at the Poisson spot.

Theorem 4

Let   and let   be the unit ball in  . Set   and let   denote the Laplace operator in  . Then at the Poisson spot the dispersive estimates ((1)), ((2)) (with   and   replaced by   and  ) fail. Precisely, let   be the source and the observation points at the (same) distance r from the ball  , symmetric with respect to the center of the ball, then, taking  , with γ in a compact subset of   yields

for the wave flow and for  
(4)|(χ(hDt)eit|Δ|(δQ+(γh−1/3))|(Q−(γh−1/3))≃h−d(ht)d−12h−d−33,
for the classical Schrödinger flow and for  
|(χ(hDt)eitΔ(δQ+(γh−1/6))|(Q−(γh−1/6))≃h−d6−d−36.
For  , these estimates contradict the usual ones ((1)), ((2)) in  .

General setting for the wave flow outside a ball in  

In this note, we give a sketch of the proof of Theorem 3 only in the case of the wave equation outside a ball. The general case will be dealt with in [[1]]. Let  ,  . Let   and   the Dirac distribution at  . Let also   denote the Laplace operator in the whole space   and   be the solution to
(∂t2−ΔRd)U=0 in Ω;U|t=0=δQ0,∂tU|t=0=0;U|∂Ω=0. Then  , where   is the free wave in  ,   is the outward unit normal to ∂Ω pointing towards Ω and N is the Neumann operator. DefineU_(t,Q,Q0):=U(t,Q,Q0), if Q∈Ω;U_(t,Q,Q0):=0, if Q∈Bd(0,1)‾. Then   satisfies the following equation:(∂t2−ΔRd)U_=∂nU|∂Ω⊗δ∂Ω in Rd;U_|t=0=δQ0,∂tU_|t=0=0, which yields  , wheree+−1F(t)=∫−∞tR(t−t′)⁎F(t′)dt′,R(t,ξ)ˆ=sin⁡(t|ξ|)|ξ|.

Sketch of proof of Theorem 3

In dimension  , from the last formula and the form of   in terms of   and N, we find
(5)e+−1((∂nUfree|∂Ω−N(Ufree|∂Ω))⊗δ∂Ω|t>0)=14π∫∂Ω(∂nUfree|∂Ω−N(Ufree|∂Ω))(t−|Q−P|,P)4π|P−Q|dσ(P). In order to prove Theorem 3 (in dimension 3), we are reduced to obtaining bounds for ((5)). For that, we use the Melrose and Taylor parametrix, which provides the form of the solution near the glancing regime in terms of the Airy function. Outside a neighborhood of the glancing region, it is easy to see that the dispersive estimates hold true. The next theorem is due to Melrose and Taylor and holds for ∂Ω strictly concave.

Theorem 5

  phase functions near the glancing region,  symbols (with a elliptic,  ) such that, if V is a solution in Ω to
(τ2+Δ)V∈OC∞(τ−∞), then there exists F such that V(τ,Q)=τ2π∫eiτθ(Q,η)(aA+τ−1/3bA′)(τ2/3ζ(Q,η))Fˆ(τη)dη, where we set  , where Ai is the Airy function, which satisfies  .

For  , we take  . We introduce polar coordinates: since  , a point in Ω can be written as  ,  ,  ,  . We can always assume that the source point   has coordinates  ,  . We define the apparent contour   of a point   as the boundary of the set of points that can be “viewed” from Q . Therefore,  .
Remark 1

When  ,  , the functions θ and ζ can be taken under the following form
θ(φ,η)=φη,ζ(r,η)=η2/3l(rη), where   and  . Then  .

Let   and  , then the trace of the free wave on the boundary   reads as(6)Ufreeˆ(τ,P,Q0)=τ|P−Q0|e−iτ|P−Q0|=τ2π∫eiτφηa0A(τ2/3ζ0(η))Fˆ(τη)dη. Let  ; then   does not vanish for  . We compute∂nUfreeˆ(τ,P,Q0)=τ2π∫eiτφη(a˜A+τ2/3b˜A′)(τ2/3ζ0(η))Fˆ(τη)dη,N(Ufree|S2)ˆ(τ,P,Q0)=τ2π∫eiτφη(a˜A+τ2/3b˜A+′A+A)(τ2/3ζ0(η))Fˆ(τη)dη. We obtain an explicit form for   as followse+−1((∂nUfree|S2−N(Ufree|S2))⊗δS2)=∫eiτtχ(hτ)(IF+ID+IR)(τ,Q,Q0)dτ, where
  is the direct wave: the phase is the phase of the free wave and the amplitude is just the amplitude of the free wave cutoff near the shadow (⇒ OK for dispersion);
  is the diffracted wave: it corresponds to a neighborhood of   on the boundary of size   in φ and of size   in angle around the glancing direction (around  );
  is the reflected wave: the phase has a singular, Airy-type term (easy to deal with).
Since difficulties appear near rays issued from   that hit the boundary without being deviated, only the diffracted wave part (containing  ) will be dealt with here. Notice that this is the regime that provides counter-examples in higher dimensions. We haveID(τ,Q,Q0)=∫P=(1,φ,ω)∈S21|P−Q|e−iτ|P−Q|τ2π∫eiτφητ2/3b˜χ0(τ2/3ζ0(η))(AA+)(τ2/3ζ0(η))Fˆ(τη)dηdφdω, where   is supported near 0,   and   is an elliptic symbol.

The phase function of   equals (  phase of  ) and reads as  , where   has coordinates  ,  . The symbol of   is of the form  , where the factor in brackets comes from  , obtained from ((6)) as an oscillatory integral with critical points of order precisely 2 on  . It will be enough to prove that  .

Let   be an observation point in Ω; in  , the only dependence in ω comes from   since
|P−Q|=(1+rQ2−2rQsin⁡(φ)sin⁡(φQ)cos⁡(ω−ωQ)−2rQcos⁡(φ)cos⁡(φQ))1/2. The critical points with respect to ω satisfy  .

If   ⇒ the critical points satisfy   which yields   or  ; this means that the stationary points P on the boundary belong to a circle situated in the plane  ; all the computations are explicit and provide the announced result;
If   the derivative vanishes everywhere. In this case the points Q , O and   are colinear and the integration in ω does not provide negative factors of τ . It is easy to see that the integration with respect to φ provides a power of   (corresponding to the critical points φ such that  , which are degenerate of order 2) and the integration with respect to η provides a factor  , due to the localisation  . It remains to show that the remaining factor   must be bounded by  , for some  ; indeed,   is defined by  , while   is defined by  , and   (since otherwise, by integrations by parts with respect to η , we get a   contribution). Notice that we have also used that  .
This allows us to achieve the proof of Theorem 3.

Sketch of proof of Theorem 4 for the wave flow outside a ball in  ,  

Let S , N denote the south pole and the north pole, respectively. Let   be a point on OS axis, at distance r from O and let   denote its symmetric with respect to O on the ON axis, where   is the centre of the ball. We let   for some γ in a compact set of   and let   and  . The counterexample to dispersion comes from the diffracted part, which, in this case, takes the form
∫eitτχ(hτ)ID(τ,Q−,Q+)dτ, where for  ID(τ,Q,Q0)=∫P=(1,φ,.)∈Sd−1τd−32|P−Q|d−12e−iτ|P−Q|Σd(|P−Q|τ)×τ2π∫Reiτφητ2/3b˜χ0(τ2/3ζ0(η))(AA+)(τ2/3ζ0(η))Fˆ(τη)dηdφ, where   is a symbol of degree 0 that satisfiesUˆfree(τ,P,Q0)=τd−12|P−Q0|d−12e−iτ|P−Q0|Σd(|P−Q0|τ), for |P−Q0|≫τ−1. With  ,  , on the apparent contour  , we haveFˆ(τη)=τ−1/3τd−12|P0−Q0|d−12eiτηφ0e−iτ|P0−Q0|Σ(|P0−Q0|,τ), where Σ is obtained from   after applying the stationary phase with degenerate critical point   on  . Notice that the observation point   is such that  , since  , O and   are on the same line and, due to rotational symmetry, in the integral defining   the phase function does not depend on ω . We obtain by explicit computations|∫eitτχ(hτ)ID(τ,Q,Q0)dτ|≃C1hh−(d−2+13)|P0−Q0|d−1. Since we must have  , it follows that|∫eitτχ(hτ)ID(τ,Q,Q0)dτ|≃Cγd−12h−dhd−12|t|d−12h−d−33. For   this coincide with the usual estimates ((1)) of  . However, for   there is a loss coming from the factor   for γ in a fixed compact of  .


 The authors were partially supported by ERC project SCAPDE (grant 320845). The authors would like to thank Centro di Giorgi, Pisa for the warm welcome during the summer 2015 when this article has started.

References

O. Ivanovici, G. Lebeau, Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples, preprint, 2017.
Li D., Smith H., Zhang X. Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data Math. Res. Lett. 2012 ;  19 (1) : 213-232 [cross-ref]



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